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

%I A343511 #35 Apr 28 2021 20:41:04
%S A343511 1,2,2,6,2,10,2,42,6,10,2,146,2,10,10,1806,2,146,2,146,10,10,2,23226,
%T A343511 6,10,42,146,2,314,2,3263442,10,10,10,42814,2,10,10,23226,2,314,2,146,
%U A343511 146,10,2,542731938,6,146,10,146,2,23226,10,23226,10,10,2,141578,2,10,146,10650056950806,10
%N A343511 a(n) = 1 + Sum_{d|n, d < n} a(d)^2.
%C A343511 a(n) depends only on the prime signature of n (see formulas). - _Bernard Schott_, Apr 24 2021
%H A343511 Alois P. Heinz, <a href="/A343511/b343511.txt">Table of n, a(n) for n = 1..3000</a>
%H A343511 <a href="/index/Pri#prime_signature">Index entries for sequences related to prime signature</a>
%F A343511 G.f.: x / (1 - x) + Sum_{n>=1} a(n)^2 * x^(2*n) / (1 - x^n).
%F A343511 a(p^k) = A007018(k) for p prime.
%F A343511 From _Bernard Schott_, Apr 24 2021: (Start)
%F A343511 a(A006881(n)) = 10 for signature [1, 1].
%F A343511 a(A054753(n)) = 146 for signature [2, 1].
%F A343511 a(A007304(n)) = 314 for signature [1, 1, 1].
%F A343511 a(A065036(n)) = 23226 for signature [3, 1].
%F A343511 a(A085986(n)) = 42814 for signature [2, 2].
%F A343511 a(A085987(n)) = 141578 for signature [2, 1, 1]. (End)
%p A343511 a:= proc(n) option remember;
%p A343511       1+add(a(d)^2, d=numtheory[divisors](n) minus {n})
%p A343511     end:
%p A343511 seq(a(n), n=1..65);  # _Alois P. Heinz_, Apr 17 2021
%t A343511 a[n_] := a[n] = 1 + Sum[If[d < n, a[d]^2, 0], {d, Divisors[n]}]; Table[a[n], {n, 65}]
%o A343511 (Python)
%o A343511 from functools import lru_cache
%o A343511 from sympy import divisors
%o A343511 @lru_cache(maxsize=None)
%o A343511 def A343511(n): return 1+sum(A343511(d)**2 for d in divisors(n) if d < n) # _Chai Wah Wu_, Apr 17 2021
%o A343511 (PARI) lista(nn) = {my(va = vector(nn)); for (n=1, nn, va[n] = 1 + sumdiv(n, d, if (d<n, va[d]^2));); va;} \\ _Michel Marcus_, Apr 18 2021
%Y A343511 Cf. A025487, A006881, A007018, A007304, A067824, A082588, A333120.
%K A343511 nonn
%O A343511 1,2
%A A343511 _Ilya Gutkovskiy_, Apr 17 2021