A056458 Number of primitive (aperiodic) palindromes using a maximum of two different symbols.

2, 0, 2, 2, 6, 4, 14, 12, 28, 24, 62, 54, 126, 112, 246, 240, 510, 476, 1022, 990, 2030, 1984, 4094, 4020, 8184, 8064, 16352, 16254, 32766, 32484, 65534, 65280, 131006, 130560, 262122, 261576, 524286, 523264, 1048446, 1047540, 2097150, 2094988, 4194302, 4192254

Offset: 1

Author

Marks R. Nester

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 1..6610
• M. R. Nester, Mathematical investigations of some plant interaction designs, PhD Thesis, (1999), University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Crossrefs

Cf. A016116, A056476.

Column 2 of A284823.

Programs

Maple

f:= proc(n) local d; add(numtheory:-mobius(d)*2^floor((1+n/d)/2), d = numtheory:-divisors(n)) end proc:
map(f, [$1..50]); # Robert Israel, Feb 18 2025

PARI

a(n) = sumdiv(n, d, moebius(d)*2^((1 + n/d)\2)); \\ Michel Marcus, Apr 24 2020

Python

from sympy import mobius, divisors
def A056458(n): return sum(mobius(n//d)<<(1+d>>1) for d in divisors(n, generator=True)) # Chai Wah Wu, Feb 18 2024

Formula

a(n) = Sum_{d|n} mu(d)*A016116(1 + n/d).

a(n) = 2 * A056476(n). - Alois P. Heinz, Feb 18 2025

Extensions

More terms from Michel Marcus, Apr 24 2020