A317931 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A002487, Stern's Diatomic sequence.

1, 1, 1, 3, 3, 1, 3, 5, 3, 3, 5, 3, 5, 3, 1, 35, 5, 3, 7, 9, 5, 5, 7, 5, 19, 5, 5, 9, 7, 1, 5, 63, 1, 5, 9, 9, 11, 7, 5, 15, 11, 5, 13, 15, 13, 7, 9, 35, 27, 19, 7, 15, 13, 5, 7, 15, 3, 7, 11, 3, 9, 5, -7, 231, -1, 1, 11, 15, 7, 9, 13, 15, 15, 11, 47, 21, 19, 5, 13, 105, 27, 11, 19, 15, 27, 13, 11, 25, 17, 13, 23, 21, 11, 9, 1, 63

Offset

1,4

Links

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

Index entries for sequences related to Stern's sequences

Formula

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A002487(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.

Prog

(PARI)
A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A317931perA317932(n) = if(1==n, n, (A002487(n)-sumdiv(n, d, if((d>1)&&(d<n), A317931perA317932(d)*A317931perA317932(n/d), 0)))/2);
A317931(n) = numerator(A317931perA317932(n));
(PARI)
\\ Memoized implementation:
memo = Map();
A317931perA317932(n) = if(1==n, n, if(mapisdefined(memo, n), mapget(memo, n), my(v = (A002487(n)-sumdiv(n, d, if((d>1)&&(d<n), A317931perA317932(d)*A317931perA317932(n/d), 0)))/2); mapput(memo, n, v); (v)));

Crossrefs

Cf. A002487, A317932 (denominators, conjectured).

Cf. also A299151, A317927, A317839, A317843.

Sequence in context: A202511 A080094 A201873 * A002332 A332547 A302694

Adjacent sequences: A317928 A317929 A317930 * A317932 A317933 A317934

Keyword

sign,frac,look

Author

Antti Karttunen, Aug 11 2018

Status

approved