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A318669 Numerators of the sequence whose Dirichlet convolution with itself yields A065769 ("Prime cascade"). 2
1, 1, 1, 7, 3, 1, 5, 25, 5, 3, 7, 7, 11, 5, 3, 363, 13, 5, 17, 21, 5, 7, 19, 25, 51, 11, 13, 35, 23, 3, 29, 1335, 7, 13, 15, 35, 31, 17, 11, 75, 37, 5, 41, 49, 15, 19, 43, 363, 115, 51, 13, 77, 47, 13, 21, 125, 17, 23, 53, 21, 59, 29, 25, 9923, 33, 7, 61, 91, 19, 15, 67, 125, 71, 31, 51, 119, 35, 11, 73, 1089, 139, 37, 79, 35, 39, 41, 23 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Multiplicative because A065769 and A317932 are.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

FORMULA

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A065769(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.

PROG

(PARI)

up_to = 1+(2^16);

A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = max(0, f[i, 2]-1)); factorback(f); };

A065769(n) = { my(f=factor(n>>valuation(n, 2))[, 1]~); (A003557(n) * factorback(vector(#f, i, precprime(f[i]-1)))); }; \\ Antti Karttunen, Sep 03 2018

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};

v318669_aux = DirSqrt(vector(up_to, n, A065769(n)));

A318669(n) = numerator(v318669_aux[n]);

CROSSREFS

Cf. A065769, A317932 (denominators).

Sequence in context: A124603 A199722 A178149 * A110636 A125681 A021899

Adjacent sequences:  A318666 A318667 A318668 * A318670 A318671 A318672

KEYWORD

nonn,frac,mult

AUTHOR

Antti Karttunen, Sep 03 2018

STATUS

approved

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Last modified June 16 13:31 EDT 2019. Contains 324152 sequences. (Running on oeis4.)