A257391 Numbers of the form 4^p*(4^p+1)*(2^p-1) with p an odd prime.

29120, 32537600, 34093383680, 36011213418659840, 36888985097480437760, 38685331082014736871587840, 39614005699412557795646504960, 41538369916519054182462860998737920, 44601490313984496701256699111250939955118080, 45671926145323068271210017365594287580527984640

Offset: 1

Author

Danny Rorabaugh, Apr 21 2015

Keywords

• nonn
• easy

Comments

5 divides (4^m+1) for odd m, so every term in this sequence is a multiple of 5 (A008587).

A064487(k) = 4^(2k+1)*(4^(2k+1)+1)*(2^(2k+1)-1), so this sequence is a subsequence of A064487.

Every non-solvable number (A056866) is divisible by 12 or 20. All non-solvable numbers not divisible by 12 (A008594) are divisible by a member of this sequence. In particular, every primitive non-solvable number (A257146) not divisible by 12 is in this sequence.

All terms are divisible by 320 and have at least 4 distinct prime factors. - Jianing Song, Apr 04 2022

References

• See A056866 and A064487 for references.

Links

• Danny Rorabaugh, Table of n, a(n) for n = 1..100

Crossrefs

Subsequence of A008587, A008602, A056866, and A064487.

Programs

Mathematica

Table[4^p (4^p+1)(2^p-1),{p,Prime[Range[2,20]]}] (* Harvey P. Dale, Jul 17 2024 *)

PARI

a(n)=my(p=prime(n+1)); 4^p*(4^p+1)*(2^p-1) \\ Charles R Greathouse IV, Apr 21 2015

Sage

[4^nth_prime(n)*(4^nth_prime(n)+1)*(2^nth_prime(n)-1) for n in range(2,12)]

Formula

a(n) = 4^p*(4^p+1)*(2^p-1) where p = A065091(n) = A000040(n+1).