A387217 The smallest prime factor of (10*n + 5)^4 + 4^(10*n + 5).

17, 29153, 29, 34350564553, 53, 36028782255016913, 17, 17, 13, 17, 13, 3613, 29, 13, 204613, 41, 17, 13, 13, 29, 29, 17, 293, 9949, 13, 13, 157, 13, 13, 13, 73, 853, 257, 13, 17, 13, 59077, 137, 409, 97, 17, 17, 17317, 17, 17, 109, 97, 13, 281, 13, 17, 9171715198301

Offset

0,1

Comments

All integers of the form 4^n + n^4 are composite for n > 1. If n is even, the smallest prime factor is 2. If n is odd and not a multiple of 5, the smallest prime factor is 5. The integers of the form 4^n + n^4 are never divisible by 3, 7, or 11.

a(n) divides at least one of (10*n+5)^2-(10*n+5)*2^(5*n+3)+2^(10*n+5) and (10*n+5)^2+(10*n+5)*2^(5*n+3)+2^(10*n+5).

Links

Andrew Eberlein, Table of n, a(n) for n = 0..659

Andrew Eberlein, 999 of the first 1001 values of a(n)

Michael Penn, a prime problem

Formula

a(n) = A020639(A017332(n) + 4^A017329(n)).

Example

For n=2, 25^4 + 4^25 = 1125899907233249 = 29*373*3121*33350257, so a(2) = 29.

Mathematica

FactorInteger[(10*#+5)^4+4^(10*#+5)][[1, 1]]&/@Range[0, 10]

Crossrefs

Cf. A001589, A017329, A017332, A020639.

Sequence in context: A171687 A178050 A110915 * A068733 A066161 A125040

Adjacent sequences: A387214 A387215 A387216 * A387218 A387219 A387220

Keyword

nonn

Author

Andrew Eberlein, Sep 24 2025

Status

approved