A343511 a(n) = 1 + Sum_{d|n, d < n} a(d)^2.

1, 2, 2, 6, 2, 10, 2, 42, 6, 10, 2, 146, 2, 10, 10, 1806, 2, 146, 2, 146, 10, 10, 2, 23226, 6, 10, 42, 146, 2, 314, 2, 3263442, 10, 10, 10, 42814, 2, 10, 10, 23226, 2, 314, 2, 146, 146, 10, 2, 542731938, 6, 146, 10, 146, 2, 23226, 10, 23226, 10, 10, 2, 141578, 2, 10, 146, 10650056950806, 10

Offset: 1

Author

Ilya Gutkovskiy, Apr 17 2021

Keywords

• nonn

Comments

a(n) depends only on the prime signature of n (see formulas). - Bernard Schott, Apr 24 2021

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..3000
• Index entries for sequences related to prime signature

Crossrefs

Cf. A025487, A006881, A007018, A007304, A067824, A082588, A333120.

Programs

Maple

a:= proc(n) option remember;
      1+add(a(d)^2, d=numtheory[divisors](n) minus {n})
    end:
seq(a(n), n=1..65);  # Alois P. Heinz, Apr 17 2021

Mathematica

a[n_] := a[n] = 1 + Sum[If[d < n, a[d]^2, 0], {d, Divisors[n]}]; Table[a[n], {n, 65}]

PARI

lista(nn) = {my(va = vector(nn)); for (n=1, nn, va[n] = 1 + sumdiv(n, d, if (dMichel Marcus, Apr 18 2021

Python

from functools import lru_cache
from sympy import divisors
@lru_cache(maxsize=None)
def A343511(n): return 1+sum(A343511(d)**2 for d in divisors(n) if d < n) # Chai Wah Wu, Apr 17 2021

Formula

G.f.: x / (1 - x) + Sum_{n>=1} a(n)^2 * x^(2*n) / (1 - x^n).

a(p^k) = A007018(k) for p prime.

From Bernard Schott, Apr 24 2021: (Start)

a(A006881(n)) = 10 for signature [1, 1].

a(A054753(n)) = 146 for signature [2, 1].

a(A007304(n)) = 314 for signature [1, 1, 1].

a(A065036(n)) = 23226 for signature [3, 1].

a(A085986(n)) = 42814 for signature [2, 2].

a(A085987(n)) = 141578 for signature [2, 1, 1]. (End)