A318321 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A003961.

1, 3, 5, 27, 7, 15, 11, 135, 75, 21, 13, 135, 17, 33, 35, 2835, 19, 225, 23, 189, 55, 39, 29, 675, 147, 51, 625, 297, 31, 105, 37, 15309, 65, 57, 77, 2025, 41, 69, 85, 945, 43, 165, 47, 351, 525, 87, 53, 14175, 363, 441, 95, 459, 59, 1875, 91, 1485, 115, 93, 61, 945, 67, 111, 825, 168399, 119, 195, 71, 513, 145, 231, 73

Offset

1,2

Comments

Multiplicative because A003961 is.

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Index entries for sequences computed from indices in prime factorization

Formula

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A003961(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.

Prog

(PARI)
up_to = 16384;
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
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&&d<n, u[d]*u[n/d], 0)))/2); u}; \\ From A317937.
v318321aux = DirSqrt(vector(up_to, n, A003961(n)));
A318321(n) = numerator(v318321aux[n]);

Crossrefs

Cf. A003961, A046644 (denominators).

Cf. also A318319.

Sequence in context: A182030 A348509 A282141 * A279510 A301315 A272148

Adjacent sequences: A318318 A318319 A318320 * A318322 A318323 A318324

Keyword

nonn,frac,mult

Author

Antti Karttunen, Aug 24 2018

Status

approved