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A358261
a(n) is the number of noninfinitary square divisors of n.
4
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0
OFFSET
1
COMMENTS
First differs from A295884 at n = 64.
The first occurrence of 1 is at n = 16, of 2 is at n = 144, of 3 is at n = 256, ... (see A358262).
LINKS
FORMULA
a(n) = A046951(n) - A358260(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi^2/6 - Product_{p prime} ((1-1/p) * Sum_{k>=1} a(p^k)/p^k) = 0.09038522017381649992... .
MATHEMATICA
f1[p_, e_] := 1 + Floor[e/2]; f2[p_, e_] := 2^DigitCount[If[OddQ[e], e - 1, e], 2, 1]; a[1] = 0; a[n_] := Times @@ f1 @@@ (fct = FactorInteger[n]) - Times @@ f2 @@@ fct; Array[a, 100]
PROG
(PARI) a(n) = {my(f = factor(n)); prod(i=1, #f~, 1+f[i, 2]\2) - prod(i=1, #f~, 2^hammingweight(if(f[i, 2]%2, f[i, 2]-1, f[i, 2])))};
CROSSREFS
Similar sequences: A046951, A056624, A056626, A358260.
Sequence in context: A023974 A277164 A011730 * A295884 A101637 A337380
KEYWORD
nonn
AUTHOR
Amiram Eldar, Nov 06 2022
STATUS
approved