A318661 Numerators of the sequence whose Dirichlet convolution with itself yields A055653, sum of phi(d) over all unitary divisors d of n.

1, 1, 3, 1, 5, 3, 7, 3, 19, 5, 11, 3, 13, 7, 15, 5, 17, 19, 19, 5, 21, 11, 23, 9, 59, 13, 95, 7, 29, 15, 31, 9, 33, 17, 35, 19, 37, 19, 39, 15, 41, 21, 43, 11, 95, 23, 47, 15, 123, 59, 51, 13, 53, 95, 55, 21, 57, 29, 59, 15, 61, 31, 133, 67, 65, 33, 67, 17, 69, 35, 71, 57, 73, 37, 177, 19, 77, 39, 79, 25, 2019, 41, 83, 21, 85, 43, 87, 33, 89

Offset: 1

Author

Antti Karttunen, Sep 03 2018

Keywords

• nonn
• frac

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Crossrefs

Cf. A055653, A318662 (denominators).

Programs

PARI

up_to = 1+(2^16);
A055653(n) = sumdiv(n, d, if(gcd(n/d, d)==1, eulerphi(d))); \\ From A055653
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA055653(n)));
A318661(n) = numerator(v318661_62[n]);
A318662(n) = denominator(v318661_62[n]);
A318663(n) = valuation(A318662(n),2);

PARI

for(n=1, 100, print1(numerator(direuler(p=2, n, ((1 + X^2 - p*X^2 - X)/((1-X)*(1-p*X)))^(1/2))[n]), ", ")) \\ Vaclav Kotesovec, May 10 2025

Formula

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A055653(n) - Sum_{d|n, d>1, d 1.

From Vaclav Kotesovec, May 10 2025: (Start)

Let f(s) = Product_{primes p} (1 + 1/p^(2*s) - 1/p^(2*s-1) - 1/p^s).

Sum_{k=1..n} A318661(k) / A318662(k) ~ n^2 * sqrt(Pi*f(2)/(24*log(n))) * (1 - ((gamma - 1)/2 + f'[2]/(2*f(2)) + 3*zeta'(2)/Pi^2) / (2*log(n))), where

f(2) = Product_{primes p} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A330523 = 0.5358961538283379998085026313185459506482223745141452711510108346133288119...

f'(2)/f(2) = Sum_{primes p} (p^2 + 2*p - 2) * log(p) / (p^4 - p^2 - p + 1) = 0.8249574883141571786856463180997569604486048593127391054584235479395133668...

and gamma is the Euler-Mascheroni constant A001620. (End)