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Numbers k such that (37^k - 5^k)/32 is prime.
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0
COMMENTS
The definition implies that k must be a prime.
a(5) > 10^5.
LINKS
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002. Eric Weisstein's World of Mathematics, Repunit.
MATHEMATICA
Select[Prime[Range[10000]], PrimeQ[(37^# - 5^#)/32] &]
CROSSREFS
Cf. A062587, A062589, A127996, A127997, A128344, A204940, A217320, A225807, A229542, A375161, A375236, A377031.
1, 2, 3, 4, 5, 7, 10, 15, 21, 30, 42, 62, 92, 137, 204, 306, 462, 701, 1066, 1634, 2504, 3842, 5927, 9135, 14132, 21881, 33958, 52779, 82185, 128158, 200065, 312748, 489472, 766792, 1202449, 1887463, 2965181, 4662549, 7337565, 11555256, 18210876, 28719623
EXAMPLE
A394720(10) = 113 is the 30-th prime, so a(10) = 30.
MAPLE
b:= proc(n) option remember;
`if`(n<3, n+1, (s-> ((x, y)-> `if`(y-s<s-x, y, x))(
prevprime(s+1), nextprime(s-1)))(b(n-1)+b(n-2)))
end:
a:= n-> numtheory[pi](b(n)):
1, 1, 128, 279936, 4586471424, 358318080000000, 100306130042880000000, 82606411253903523840000000, 173238200573946282828103680000000, 828592942960967278432052230225920000000, 8285929429609672784320522302259200000000000000, 161469323688736156802100136913438716723200000000000000
CROSSREFS
Cf. A000142, A001015, A001044, A000442, A036740, A010050, A009445, A134366, A134367, A134368, A134369, A134371, A134372, A134373, A134374, A134375, A396224, A396225.
1, 1, 64, 46656, 191102976, 2985984000000, 139314069504000000, 16390160963076096000000, 4296582355504620109824000000, 2283380023591730815784976384000000, 2283380023591730815784976384000000000000, 4045146997974190235742848547815424000000000000
CROSSREFS
Cf. A000142, A001014, A001044, A000442, A036740, A010050, A009445, A134366, A134367, A134368, A134369, A134371, A134372, A134373, A134374, A134375, A396224, A396226.
1, 1, 32, 7776, 7962624, 24883200000, 193491763200000, 3252016064102400000, 106562062388507443200000, 6292383221978976013516800000, 629238322197897601351680000000000, 101339461028293606595289415680000000000, 25216500766592354716319055882485760000000000
CROSSREFS
Cf. A000142, A000584, A001044, A000442, A036740, A010050, A009445, A134366, A134367, A134368, A134369, A134371, A134372, A134373, A134374, A134375, A396225, A396226.
Numbers k such that A081973(k) is prime.
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2, 3, 4, 5, 6, 9, 11, 12, 15, 24, 57, 68, 89, 92, 102, 152, 165, 220, 226, 229
MATHEMATICA
nmax = 165; a = {}; Module[{b = 1}, For[k = 2, k <= nmax, k++, b += DivisorSigma[1, b]; If[PrimeQ[b], AppendTo[a, k]]]]; a
a(1) = 3; a(n) is the smallest odd prime not occurring earlier so that a(n) mod a(n-1) is prime.
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3, 5, 7, 17, 11, 13, 29, 19, 41, 23, 53, 31, 67, 37, 79, 43, 89, 47, 97, 59, 61, 127, 71, 73, 149, 83, 173, 101, 103, 211, 107, 109, 223, 113, 229, 131, 269, 137, 139, 281, 151, 307, 157, 317, 163, 331, 167, 337, 179, 181, 367, 191, 193, 389, 197, 199, 401
COMMENTS
2 cannot be on the list as all numbers are 0 or 1 mod 2, neither of which is prime.
There are three ways that the next term may be generated. If there is at least an odd prime < a(n) not on the list, a(n+1) is the smallest among them. Otherwise, if a(n)+2 is prime and not on the list, a(n+1) = a(n)+2. Otherwise, a(n+1) takes the form 2*k*a(n)+p with prime p. It is conjectured that k is always 1, which has been confirmed until n = 10000.
Because of the first generation step, this is a permutation of A065091.
EXAMPLE
a(2) = 5 as 5 is prime and 5 mod 3 = 2 is prime.
a(4) = 17 as 17 is prime and 17 mod 7 = 3 is prime.
a(5) = 11 as 11 is prime and 11 mod 17 = 11 is prime.
MATHEMATICA
a[n_] := a[n] = Module[{ps = Array[a, n - 1], p = 5}, While[MemberQ[ps, p] || !PrimeQ[Mod[p, a[n - 1]]], p = NextPrime[p]]; p]; a[1] = 3; Array[a, 57] (* Amiram Eldar, May 21 2026 *)
PROG
(Julia)
using Primes
function a(n)
prime = [3]
print("3, ")
for _ in 1:n-1
num = 3
while true
num += 2
if isprime(num) && isprime(num % prime[end]) && !(num in prime)
push!(prime, num)
print("$num, ")
break
end
end
end
end
(Python)
from itertools import count, islice
from sympy import isprime, nextprime, sieve
def agen(): # generator of terms
an, nextp, seen = 3, 5, set()
while True:
yield an
seen.add(an)
while nextp in seen: nextp = nextprime(nextp)
if nextp < an: an = nextp
elif an+2 not in seen and isprime(an+2): an += 2
else: an = next(m for k in count(1) for p in sieve.primerange(3, an) if (m:=2*k*an+p) not in seen and isprime(m))
Smallest number k such that reverse(k)^i <= reverse(k^i), for i=1..n, but not for i=n+1, or -1 if no such number exists.
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8, 4, 3, 2, 66, 474, 4053, 33
COMMENTS
If any of the values a(9), a(13), a(14), a(17) and a(18) exist, they exceed 10^10.
31 is not a term since reverse(31)^i = 13^i <= reverse(31^i) for all i >= 1.
Proof: inspection shows it holds for i <= 2. For i > 2, 13^i has less digits than 31^i. As 31 is no multiple of 10, 31^i and reverse(31^i) have the same number of digits. So reverse(31)^i = 13^i <= reverse(31^i) and 31 is no term. - David A. Corneth, May 21 2026 No power of 10 is a term since reverse(10^j)^i = reverse((10^j)^i) = 1 for all i, j >= 0.
No multiple of 10 can be a term since reverse(m*10) = reverse(m) so reverse(m*10)^i = reverse(m)^i and reverse((m*10)^i) = reverse(m^i * 10^i) = reverse(m^i). Thus, a(n) = m*10 is not possible since 0 < m < 10*m would have satisfied the same condition. (End)
Thus, the sequence contains neither the number 1 nor any number whose last digit is 0.
Some other terms: a(10) = 51894, a(11) = 360133, a(12) = 68186, a(15) = 63136, a(16) = 31813, a(19) = 33913, a(20) = 31713, a(25) = 640136, a(39) = 334713, a(85) = 63269036. - Michael S. Branicky, May 21 2026
EXAMPLE
Let R(k) = reverse(k) = A004086(k). a(1) = 8 since R(8)^1 = 8 <= 8 = R(8^1), but R(8)^2 = 64 > 46 = R(8^2);
a(2) = 4 since R(4)^1 = 4 <= 4 = R(4^1), R(4)^2 = 16 <= 61 = R(4^2), but R(4)^2 = 64 > 46 = R(4^3);
a(3) = 3 since R(3)^1 = 3 <= 3 = R(3^1), R(3)^2 = 9 <= 9 = R(3^2), R(3)^3 = 27 <= 72 = R(3^3), but R(4)^4 = 81 > 18 = R(4^4);
and no lesser numbers have this property.
PROG
(PARI) a(n) = {for(i = 1, oo, if(iscan(i, n), return(i)))}
iscan(c, n) = {my(rc = rev(c)); for(i = 1, n, if(rc^i > rev(c^i), return(0))); rc^(n+1) > rev(c^(n+1))}
Numbers that are neither the sum nor difference of two 5-smooth numbers.
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583, 823, 853, 877, 941, 953, 983, 1031, 1063, 1067, 1073, 1091, 1139, 1166, 1193, 1237, 1261, 1303, 1309, 1313, 1337, 1339, 1367, 1379, 1391, 1403, 1409, 1417, 1427, 1447, 1451, 1469, 1477, 1481, 1487, 1507, 1517, 1543, 1549, 1553, 1559, 1567, 1571, 1577, 1579
COMMENTS
Numbers k such that A391077(k) = 0.
PROG
(PARI)
M(v, u, lim)={vecsort(concat(vector(#v, i, my(m=lim\v[i]); v[i]*select(t->t<=m, u))))}
Gen(lim, k)={my(v=[1]); forprime(p=2, k, v=M(v, vector(logint(lim, p)+1, e, p^(e-1)), lim)); v}
lista(n, lim=10^30)={my(v=vector(n), G=Gen(lim, 5)); for(i=1, #G, my(t=G[i], k=i+1); while(k<=#G && G[k]-t<=n, v[G[k]-t]=G[k]; k++)); v}
A006530(n)=if(n>1, vecmax(factor(n)[, 1]), 1)
b(n)=for(t=1, n, A006530(t*(n-t))<=5&return(t)) A391077(n)=if(lista(n)[n]!=0, return(lista(n)[n]); lista(n)[n]==0&b(n)<n, return(n-b(n)); lista(n)[n]==0&b(n)==n, return(0))
for(n=1, 2000, A391077(n)==0&print1(n", "))
Odd integers k such that k+2 is prime and the sequence of primes congruent to 2 modulo k is strictly stable under iterated absolute differences (all leading entries equal k).
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3, 5, 9, 57, 65, 147, 155, 315, 387, 395, 807, 917, 1305, 1695, 1757, 1995, 3615, 4137, 4335, 4655, 4875, 4917, 4965, 5439, 5985, 6315, 6827, 6897, 6905, 7125, 7127, 7875, 8177, 9135, 9275, 9435
COMMENTS
All terms are currently conjectural.
A number k is included if: (i) k is odd, (ii) k+2 is prime, (iii) letting P be the increasing sequence of primes p == 2 (mod k), the triangle formed by iterated absolute differences of P has all leading entries equal to k.
This defines a "strictly stable" condition reminiscent of Gilbreath-type phenomena.
FORMULA
Let P = (p_1, p_2, ...) be primes with p_i == 2 (mod k).
Define rows: R^{(0)} = P, R^{(m+1)}_i = |R^{(m)}_{i+1} - R^{(m)}_i|.
Then k is included if for all m >= 1, R^{(m)}_1 = k.
EXAMPLE
For k = 3: primes == 2 mod 3 are: 2, 5, 11, 17, 23, 29, 41, 47, 53, 59, ...
First absolute differences are: 3, 6, 6, 6, 6, 12, 6, 6, 6, ...
Then they are: 3, 0, 0, 0, 6, 6, 0, 0, ...
Then: 3, 0, 0, 6, 0, 6, 0, 6, 6, ...
Then: 3, 0, 6, 6, 6, 6, 6, 0, ...
...
The leading entries remain 3, so k=3 is a term.
MATHEMATICA
CheckStrictStability[k_, limit_] := Module[{primes, diffs, rowFirsts, \
currentSeq, isStable}, primes = Select[Prime[Range[PrimePi[limit]]],
Mod[#, k] == 2 &];
If[Length[primes] < 5 || primes[[2]] ≠ k + 2, Return[False]];
currentSeq = primes;
rowFirsts = {};
While[Length[currentSeq] > 1, currentSeq = Abs[
Drop[currentSeq, 1] - Drop[currentSeq, -1]];
AppendTo[rowFirsts, First[currentSeq]];
If[First[currentSeq] ≠ k, Return[False]]; ];
Return[And @@ Thread[rowFirsts == k]]; ]
startK = 3;
endK = 500000;
searchLimit = 1000000;
results = {};
Do[If[OddQ[k] && PrimeQ[k + 2],
If[CheckStrictStability[
k, searchLimit], AppendTo[results, k]; ]], {k, startK, endK, 2}];
Print[results];
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