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A318319
Numerators of rational valued sequence whose Dirichlet convolution with itself yields A064989.
4
1, 1, 1, 3, 3, 1, 5, 5, 3, 3, 7, 3, 11, 5, 3, 35, 13, 3, 17, 9, 5, 7, 19, 5, 27, 11, 5, 15, 23, 3, 29, 63, 7, 13, 15, 9, 31, 17, 11, 15, 37, 5, 41, 21, 9, 19, 43, 35, 75, 27, 13, 33, 47, 5, 21, 25, 17, 23, 53, 9, 59, 29, 15, 231, 33, 7, 61, 39, 19, 15, 67, 15, 71, 31, 27, 51, 35, 11, 73, 105, 35, 37, 79, 15, 39, 41, 23, 35, 83, 9, 55, 57
OFFSET
1,4
COMMENTS
Multiplicative because A064989 is.
No negative terms among the first 2^20 terms.
FORMULA
a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A064989(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
PROG
(PARI)
up_to = 16384;
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1, 1]==2), f[1, 2] = 0); for (i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f)};
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.
v318319aux = DirSqrt(vector(up_to, n, A064989(n)));
A318319(n) = numerator(v318319aux[n]);
CROSSREFS
Cf. A064989, A317932 (seems to give denominators, see A261179).
Cf. also A318321.
Sequence in context: A039739 A160496 A105663 * A321785 A021755 A272295
KEYWORD
nonn,frac,mult
AUTHOR
Antti Karttunen, Aug 24 2018
STATUS
approved