The On-Line Encyclopedia of Integer Sequences (OEIS)

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Sequence	Status	First Edited	Last Active	Draft by	Reviewed or edited by
Showing entries 1-100 \| older changes

A397049	editing +13 −2	Jul 16 23:40	Jul 22 19:29	John Tyler Rascoe (4/10)
	Number of sets S of nonempty words with n total letters such that no words in S share letters and the concatenation of all words in S covers an initial interval.

A197834	editing +4 −3	Jul 22 16:09	Jul 22 19:27	Charles R Greathouse IV (1/∞)	Robert C. Lyons
	Decimal expansion of the maximum of (cos(x))^2+(sin(3Pix))^2.

A398098	editing +7 −2	Jul 22 18:42	Jul 22 18:43	Andrew Howroyd (8/∞)
	Number of nonisomorphic n X n symmetric nonnegative integer matrices with each row sum nonzero and at most 4, under row and column permutations.

A398097	editing +7 −2	Jul 22 18:34	Jul 22 18:36	Andrew Howroyd (8/∞)
	Number of nonisomorphic n X n symmetric nonnegative integer matrices with each row sum nonzero and at most 3, under row and column permutations.

A398096	editing +14 −2	Jul 22 18:22	Jul 22 18:33	Andrew Howroyd (8/∞)
	Number of nonisomorphic n X n symmetric nonnegative integer matrices with each row sum nonzero and at most 2, under row and column permutations.

A394615	editing +26 −2	Jul 17 18:05	Jul 22 18:26	John Tyler Rascoe (4/10)
	Triangle read by rows: T(n,k) is the number of sets S of nonempty words with n total letters and k words such that no two words in S share letters and the concatenation of all words in S covers an initial interval.

A397098	editing +27 −2	Jul 17 20:11	Jul 22 18:26	John Tyler Rascoe (4/10)
	Triangle read by rows: T(n,k) is the number of sets S of nonempty words with n total letters such that no two words in S share letters and the concatenation of all words in S covers [k].

A056886	editing +2 −1	Jul 22 17:56	Jul 22 17:56	Michael De Vlieger (15/∞)
	Determinant of the n X n Hankel matrix whose entries are s_2 (i+j), 0 <= i, j < n, where s_2 is the sum of the base-2 bits.

A398157	editing +11 −2	Jul 22 13:22	Jul 22 17:47	Mark Anthony Larman (1/3)	Jianing Song
	a(n) = n^4 - 2n^3 + n^2 - 2n + 2, a fixed point of the Kaprekar (digit-sorting subtraction) map on 4-digit numerals in bijective base n, for n >= 5.

A226525	editing +3 −2	Jul 22 17:42	Jul 22 17:42	Michael M. Ross (2/3)
	Fortunate semiprimes: least m > 1 such that m + sp(n)# is semiprime, where sp# denotes the product of the semiprimes <= sp.

A398022	editing +34 −2	Jul 17 03:22	Jul 22 17:14	Thomas Scheuerle (5/7)	Sean A. Irvine
	Irregular triangular array T(n, k) read by rows: Row n gives coefficients for calculating the coefficient of x^n in the power series S(x) = c0+c1x+c2x^2+..., that satisfies Mish(S(x)) = x. Mish is the machine learning activation function explained in A396907. See comments for further details.

A397675	editing +55 −2	Jul 04 19:07	Jul 22 17:04	Alex Weslowski (2/3)	Sean A. Irvine
	Numbers whose multiplicative partition covers exactly 1/2 of the integers.

A147680	editing +2 −1	Jul 20 12:42	Jul 22 16:54	M. F. Hasler (9/19)	Bradley Klee
	Number of disk polyominoes of order n (see Comments for definition).

A295236	editing +3 −3	Jul 22 16:33	Jul 22 16:33	Max Alekseyev (11/∞)
	Hemi-imperfect numbers: numbers such that the denominator of k/A206369(k) is equal to 2.

A393267	editing +25 −11	Jul 12 02:17	Jul 22 16:29	Diego Lago Gómez (2/3)	Andrew Howroyd
	Array read by descending antidiagonals T(n,k) is the minimum total surface area of k-element sets of distinct integer-sided cuboids that fill an n X n X n cube.

A397850	editing +90 −2	Jul 12 11:27	Jul 22 14:07	Javier Alejandro Cabrera Otten (1/3)
	Number of tilings of a 4 X n rectangle using straight trominoes and Z-tetrominoes.

A398154	editing +31 −2	Jul 22 12:09	Jul 22 12:10	Rémy Sigrist (2/∞)
	The canonical snowball representation of A398153(n), when read backwards, corresponds to that of A398153(a(n)).

A398153	editing +35 −2	Jul 22 12:00	Jul 22 12:01	Rémy Sigrist (2/∞)
	Nonnegative numbers whose canonical snowball representation, when read backwards, corresponds to a legit canonical snowball representation.

A258336	editing +11 −10	Jul 21 21:45	Jul 22 11:18	Max Alekseyev (11/∞)
	a(n) = smallest solution of the equation x' = x + p, where x' is the arithmetic derivative of x and p is the n-th prime. a(n)= -1 if no such solution exists.

A398149	editing +5 −2	Jul 22 08:36	Jul 22 08:36	Scott R. Shannon (2/19)
	wip

A398132	editing +9 −2	Jul 21 10:55	Jul 22 01:58	Mauro Torelli (3/3)	Michel Marcus
	Primes which are internal nodes in the tree of primes.

A090181	editing +14 −11	Jul 21 13:34	Jul 22 01:57	Oliver Seipel (3/3)	Michel Marcus
	Triangle of Narayana (A001263) with 0 <= k <= n, read by rows.

A008459	editing +3 −1	Jul 21 13:51	Jul 22 01:57	Oliver Seipel (3/3)	Michel Marcus
	Square the entries of Pascal's triangle.

A068994	editing +2 −1	Jul 21 23:48	Jul 21 23:48	Kyle Marek-Spartz (1/3)
	Powers of 2 with all even digits.

A398144	editing +6 −2	Jul 21 23:03	Jul 21 23:03	Daniel E Flath (1/3)
	#{(i,j): mu(i) = mu(j), 1 <= i <= n, 1 <= j <= n}, where mu=A008683 (Moebius function).

A397810	editing +56 −2	Jul 10 20:32	Jul 21 23:03	Russ Curry (1/3)
	Numbers that are both the sum of two positive cubes and the difference of two non-zero fourth powers.

A185222	editing +3 −2	Jul 21 21:51	Jul 21 22:28	Max Alekseyev (11/∞)
	Composite numbers m such that (m'+1)' = m', where m' = A003415(m) is the arithmetic derivative of m.

A200729	editing +3 −2	Jul 21 21:58	Jul 21 21:58	Max Alekseyev (11/∞)
	Smallest nontrivial positive power x such that the number of even powers (i.e., even base) not exceeding x exceeds by n the number of odd powers not exceeding x.

A397137	editing +91 −2	Jul 20 18:05	Jul 21 19:30	Marko Radulovic (2/3)	Alois P. Heinz
	Sum of defects of all permutations of [n] under reflected interval shading.

A397813	editing +26 −2	Jul 10 22:47	Jul 21 19:01	Jean Lauro Muller (1/3)	Sean A. Irvine
	a(0) = 1; for n > 0, with p = the n-th prime, a(n) = a(n-1) - p if that value is positive, previously unused, and not prime, otherwise a(n) is the least unused number >= a(n-1) + p,.

A397896	editing +15 −2	Jul 14 12:47	Jul 21 17:22	Ali Sada (1/3)	Sean A. Irvine
	For each positive integer n, let a(n) be the minimum number of nonnegative integers (suffixes) needed so that, for every positive integer m, at least one of those suffixes can be appended to the decimal representation of m to produce a multiple of n. We must prioritize the shortest possible suffix lengths.

A398075	editing +6 −2	Jul 21 16:22	Jul 21 16:22	Andrew Howroyd (8/∞)
	Number of nonisomorphic n X n symmetric nonnegative integer matrices with each row sum nonzero and at most n, under row and column permutations.

A398074	editing +25 −2	Jul 21 16:06	Jul 21 16:18	Andrew Howroyd (8/∞)
	Array read by antidiagonals: T(n,k) is the number of nonisomorphic n X n symmetric nonnegative integer matrices with each row sum nonzero and at most k, under row and column permutations, n >= 0, k >= 0.

A047085	editing +2 −1	Jul 21 15:51	Jul 21 15:51	Steven E Landsburg (3/3)
	a(n) = T(2*n, n), array T as in A047080.

A398116	editing +1 −2	Jul 20 15:45	Jul 21 13:53	Marcel Augusto Calassa Alcântara (1/3)	Alois P. Heinz


A001263	editing +2	Jul 21 13:38	Jul 21 13:43	Oliver Seipel (3/3)	Michel Marcus
	Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle.

A397656	editing +9 −4	Jul 07 15:50	Jul 21 13:38	David A. Corneth (1/30)	Chai Wah Wu
	a(n) is the smallest k such that A397224(k) = n or -1 if no such k exists.

A397972	editing +15 −3	Jul 18 10:44	Jul 21 11:42	Clark Kimberling (7/7)
	Lower (1,1/3) midsequence of triangular numbers (A000217) and squares (A000290); see Comments.

A020714	editing +2 −1	Jul 21 11:05	Jul 21 11:06	Ranjan Kumar Dhani (3/3)
	a(n) = 5 * 2^n.

A397086	editing +17 −2	Jun 29 08:44	Jul 21 10:03	Tony Hernandez (3/3)
	Partial sums of A397085.

A398134	editing +17 −2	Jul 21 09:30	Jul 21 09:30	Peter Luschny (4/∞)
	Denominators of the binomial matrix of the Bernoulli numbers read by ascending antidiagonals.

A398133	editing +26 −2	Jul 21 09:20	Jul 21 09:21	Peter Luschny (4/∞)
	Numerators of the binomial matrix of the Bernoulli numbers read by ascending antidiagonals.

A229878	editing +3 −2	Jul 21 08:07	Jul 21 08:07	Max Alekseyev (11/∞)
	Number of undirected circular permutations tau(1), ..., tau((p_n-1)/2) of 1, ..., (p_n-1)/2 such that the (p_n-1)/2 numbers tau(1)^2 + tau(2)^2, tau(2)^2 + tau(3)^2, ..., tau((p_n-3)/2)^2 + tau((p_n-1)/2)^2, tau((p_n-1)/2)^2 + tau(1)^2 give all the (p_n-1)/2 quadratic residues modulo p_n, where p_n is the n-th prime.

A397239	editing +6 −2	Jul 21 07:38	Jul 21 07:38	Andrey Samosyuk (1/3)
	Primes p such that p^3-2*p^2+2 is prime.

A308619	editing +2 −2	Jul 21 04:50	Jul 21 04:50	João D. R. Camacho (1/3)
	Negative van Eck's sequence: For n >= 1, if there exists an m < n such that a(m) = a(n), take the largest such m and set a(n+1) = n-m, otherwise a(n+1) = -a(n). Start with a(1)=0.

A019442	editing +1 −1	Jul 20 08:50	Jul 21 02:50	Bassey Godwin Bassey (2/3)	Michel Marcus
	Numbers m such that a Hadamard matrix of order m exists.

A171155	editing +4 −2	Jul 20 22:51	Jul 21 02:47	Steven E Landsburg (3/3)	Michel Marcus
	For two strings of length n, this is the number of pairwise alignments that do not have an insertion adjacent to a deletion. (Duplicate of A047085.)

A397553	editing +6 −2	Jun 30 12:09	Jul 21 00:21	Davis McDowell (1/3)
	Maximum number of isolated nodes removed from a square grid graph with a hamiltonian path

A006003	editing +3 −3	Jun 30 01:42	Jul 20 23:22	Gus Michel (1/3)	Sean A. Irvine
	a(n) = n*(n^2 + 1)/2.

A397880	editing +13 −2	Jul 13 15:45	Jul 20 23:15	Michael M. Ross (2/3)	Sean A. Irvine
	Numerators of c(n), the expected value of an entry at depth n in the Gilbreath (iterated absolute difference) array generated by independent standard exponential random variables.

A396945	editing +18 −2	Jul 15 02:39	Jul 20 23:12	Rafael Vidal Lykova (2/3)	Sean A. Irvine
	Number of divisors of n^3 + 1.

A144836	editing +4 −2	Jul 11 18:12	Jul 20 19:05	Peter Bala (18/19)
	a(n) = round(phi^(4^n)) where phi is the golden ratio (A001622).

A135927	editing +8 −2	Jul 17 09:28	Jul 20 19:04	Peter Bala (18/19)
	a(n) = a(n-1)^2 - 2 with a(1) = 10.

A102847	editing +7 −2	Jul 11 15:42	Jul 20 19:00	Peter Bala (18/19)
	a(0) = 1, a(n) = a(n-1)*a(n-1) + 2.

A374015	editing +4 −3	Jul 14 06:25	Jul 20 18:38	Cezary Glowacz (1/3)	Sean A. Irvine
	Residue modulo 5 of n! divided by the highest power of 10 which divides n!.

A074006	editing +4 −2	Jul 20 16:43	Jul 20 16:44	Andrei Zabolotskii (3/∞)
	Number of elements of GF(5^n) with trace 0 and subtrace 0.

A397781	editing +23 −2	Jul 10 03:22	Jul 20 09:46	Lee M. J. Rich (3/3)
	Numerator of the product of the n largest values of prime(k+1)/prime(k), k >= 1.

A397710	editing +35 −2	Jul 07 22:54	Jul 20 06:44	Michael De Vlieger (15/∞)	Peter Munn
	Composite numbers k such that k = rad(k) * lpf(k), where rad = A007947 and lpf is the least prime factor of k.

A397273	editing +128 −2	Jul 18 16:32	Jul 19 19:55	Rhys Feltman (1/3)
	Maximum number of squares on an n X n board without forming any lines of n-1 consecutive squares horizontally, vertically, or diagonally.

A085096	editing +2 −2	Jul 19 17:24	Jul 19 17:24	Caleb Westfall (1/3)
	Index of the first occurrence of n in A082744, or 0 if n does not occur in the sequence A082744.

A001415	editing +9 −4	Jul 15 01:56	Jul 19 17:05	Hunter Hogan (1/3)
	Number of ways of folding a 2 X n strip of stamps.

A329471	editing +6 −3	Jul 10 06:03	Jul 19 16:45	Peter Bala (18/19)
	a(n) = a(n-1)^2 + 3 for n >= 2 , where a(0) = 1, a(1) = 3.

A135361	editing +13 −1	Jul 13 08:56	Jul 19 16:40	Peter Bala (18/19)
	a(n) = a(n-1)^3 + 1 with a(0) = 0.

A072191	editing +10 −1	Jul 10 05:41	Jul 19 16:37	Peter Bala (18/19)
	a(n) = a(n-1)^2 + 2.

A278837	editing +5 −4	Jul 17 16:53	Jul 19 15:49	Alonso del Arte (2/7)
	Primes p such that the ring of algebraic integers of Q(sqrt(p)) does not have unique factorization.

A033999	editing +3 −2	Jul 19 10:13	Jul 19 10:13	Andrew Lenker (1/3)
	a(n) = (-1)^n.

A398004	editing +50 −2	Jul 16 18:44	Jul 19 09:12	Eric Trébuchon (1/3)
	Iteratively defined symmetric multiplication f(a,b) read by antidiagonals

A073112	editing +3 −2	Jul 18 11:33	Jul 18 11:33	Max Alekseyev (11/∞)
	Number of permutations p from (1,2,3,...,n) to (1,2,3,...,n) such that 1/(1+p(1)) + 1/(2+p(2)) + ... + 1/(n+p(n)) is an integer.

A077043	editing +2 −1	Jul 17 12:32	Jul 18 08:14	Charles Obler (1/3)
	"Three-quarter squares": a(n) = n^2 - A002620(n).

A397676	editing +54 −2	Jul 04 19:16	Jul 18 04:02	Alex Weslowski (2/3)	Michel Marcus
	Numbers whose multiplicative partition covers exactly 1/3 of the integers.

A005700	editing +7 −5	Jul 16 14:18	Jul 18 03:37	David Speyer (1/3)	Michel Marcus
	a(n) = C(n)*C(n+2) - C(n+1)^2 where C() are the Catalan numbers A000108.

A397153	editing +16 −2	Jul 17 22:21	Jul 17 22:21	Alain Goupil (2/3)
	Maximum length of snake polyominoes in 12xn rectangles.

A396016	editing +15 −2	Jul 17 19:28	Jul 17 20:19	Lal Su Narinc (1/3)	Andrew Howroyd
	Number of domino tilings of the star strip S(n): n star-shaped cells, each a 4 X 4 square with two-cell tabs on top and bottom, consecutive cells joined by two-cell necks.

A393920	editing +4 −3	Jul 14 18:06	Jul 17 15:47	Ludovic Schwob (1/5)	Stefano Spezia
	Number of full subcategories of the category of finite dimensional linear representations of a totally ordered set with n elements (over any field) that are closed under extensions and direct summands.

A398034	editing +11 −2	Jul 17 15:09	Jul 17 15:17	Frédéric G. Speyser (3/3)	Andrew Howroyd
	Number of unlabeled rooted strict 6-gonal cactus graphs (every block a 6-cycle, no combinatorial embedding fixed — free/non-plane case), with k blocks, k = 0, 1, 2, ...

A398035	editing +10 −2	Jul 17 15:12	Jul 17 15:13	Frédéric G. Speyser (3/3)	Stefano Spezia
	Number of unlabeled (unrooted) strict 6-gonal cactus graphs (every block a 6-cycle, free/non-plane case), with k blocks, up to isomorphism, k = 1, 2, ...

A397249	editing +32 −2	Jul 17 14:01	Jul 17 14:32	Ibne Raihan (2/3)	Andrew Howroyd
	Self-describing digit expansion sequence

A363056	editing +2 −2	Jul 14 20:36	Jul 17 07:16	Eric W. Weisstein (1/19)
	Graph bandwidth of the n X n queen graph.

A119240	editing +3 −1	Jul 13 05:53	Jul 17 01:17	Hiroyuki Ogawa (1/3)	Sean A. Irvine
	Least odd number k such that sigma(k)/k >= n.

A204420	editing +11 −2	Jul 11 11:16	Jul 16 19:32	Natalia L. Skirrow (3/3)	Sean A. Irvine
	Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.

A260335	editing +5 −2	Jul 16 18:24	Jul 16 18:24	Alonso del Arte (2/7)
	Prime determinants of forms with class number greater than 2.

A397145	editing +33 −2	Jul 15 08:16	Jul 16 16:14	Alois P. Heinz (1/∞)
	Number T(n,k) of permutations p of [n] where k is the product of the number of odd weak excedances and the number of even weak excedances of p; triangle T(n,k), n>=0, 0 <= k <= floor(n^2/4), read by rows.

A397899	editing +24 −2	Jul 14 15:08	Jul 16 15:55	Christian Felix Bürckert (3/3)	Alois P. Heinz
	a(n) = max over d of (d + c_d), where Sum_d c_dX_(d) is the square of M_n(X) = Sum_{d=0}^{n-1} (n-d)X_(d), expanded in the falling factorial basis X_(d) = X(X-1)...*(X-d+1).

A397900	editing +30 −2	Jul 14 15:09	Jul 16 15:45	Christian Felix Bürckert (3/3)	Alois P. Heinz
	a(n) is the number of digit sequences (a_1,...,a_n) with 0 <= a_i <= i-1 for all i and Sum_{i=1..n} a_i*(n+1)!/(i+1)! >= n!.

A397809	editing +62 −2	Jul 10 16:53	Jul 16 15:36	Ian R.C. Buckley (1/3)	Sean A. Irvine
	Z_TF plethystic exponential coefficients Coefficients of the plethystic exponential PE[Q^3(1+Q^4)/(1-Q^4)^2], which is the q-expansion of the Thomas-Fermi Selberg-type zeta function Z_TF(s) = Product_{k>=0} (1 - Q^(4k+3))^(-(2k+1)) at Q = exp(-pis/2).

A397898	editing +35 −2	Jul 14 15:07	Jul 16 15:31	Christian Felix Bürckert (3/3)	Alois P. Heinz
	a(g) is the least m such that Sum_{d=0}^{m-g-1} (m-g-d)*(m+1)!/(m+1-d)! >= m!.

A397928	editing +7 −2	Jul 16 10:03	Jul 16 10:09	Andrew Howroyd (8/∞)
	Number of nonnegative integer matrices with a total of n rows and columns with each row sum and column sum nonzero and at most 2.

A279667	editing +3 −2	Jul 16 08:51	Jul 16 08:51	Max Alekseyev (11/∞)
	Number of subparts (also number of odd divisors) of the smallest number k such that the symmetric representation of sigma(k) has n layers.

A000219	editing +6 −2	Jul 06 04:26	Jul 15 23:15	Nicholas Boichuk (2/3)	Sean A. Irvine
	Number of plane partitions (or planar partitions) of n.

A143405	editing +6 −3	Jul 06 04:39	Jul 15 23:15	Nicholas Boichuk (2/3)	Sean A. Irvine
	Number of forests of labeled rooted trees of height at most 1, with n labels, where any root may contain >= 1 labels, also row sums of A143395, A143396 and A143397.

A394672	editing +37 −2	Jul 10 07:42	Jul 15 22:23	Roger Salmerón Vitó (1/3)	Sean A. Irvine
	Numbers m that belong to A003622, and such that A022342(m) and m are coprime.

A114714	editing +1 −1	Jul 15 22:15	Jul 15 22:15	Woosuk Kwak (1/3)
	Number of linear extensions of a 2 X 2 X n lattice.

A397624	editing +45 −2	Jul 02 17:46	Jul 15 19:41	Alan Orwick (1/3)	Sean A. Irvine
	Number of reachable states from the directed star with n labeled in-neighbors under the retain-largest complete-triangle rewrite rule.

A027696	editing +2 −1	Jul 15 15:27	Jul 15 15:27	Ian Olivant (1/3)
	Numbers k >= 2 such that for some m >= 2, the sum of the first m k-gonal numbers is again a k-gonal number, excluding the parametric solution m = (k^2-4*k-2)/3 when k==2 (mod 3).

A397807	editing +14 −2	Jul 10 13:19	Jul 15 13:24	Romain Malécot (2/3)	Hugo Pfoertner
	The Gaston's second-class numbers.

A397806	editing +14 −2	Jul 10 13:15	Jul 15 12:53	Romain Malécot (2/3)	Hugo Pfoertner
	The Gaston's first class numbers.

A054859	editing +5 −2	Jul 15 11:26	Jul 15 12:40	Eleanor Waiss (1/3)	Michel Marcus
	Smallest positive integer that can be expressed as the sum of consecutive primes in exactly n ways.

A394032	editing +11 −2	Jul 15 12:19	Jul 15 12:35	Pierre Colmant (1/3)
	Integers k such that 30k-19 and 30k+19 are both prime numbers.

A397866	editing +28 −2	Jul 13 11:08	Jul 15 11:51	Albert ten Oever (1/3)	Michel Marcus
	a(n) is the maximum number of distinct values in any recurring cycle of the n-digit Kaprekar iteration.

A234517	editing +15 −6	May 25 09:44	Jul 15 09:46	Peter Munn (3/19)
	Numbers k > 1 such that for all m > k, e(k) >= e(m), where e(k) = log_k(sigma(k)), the exponent such that k^e(k) = sigma(k), and sigma is the sum of divisors function, A000203.