A287093 a(0) = 0, a(1) = 2; a(2*n) = sopf(a(n)), a(2*n+1) = a(n) + a(n+1), where sopf() is the sum of the distinct prime factors (A008472).
0, 2, 2, 4, 2, 6, 2, 6, 2, 8, 5, 8, 2, 8, 5, 8, 2, 10, 2, 13, 5, 13, 2, 10, 2, 10, 2, 13, 5, 13, 2, 10, 2, 12, 7, 12, 2, 15, 13, 18, 5, 18, 13, 15, 2, 12, 7, 12, 2, 12, 7, 12, 2, 15, 13, 18, 5, 18, 13, 15, 2, 12, 7, 12, 2, 14, 5, 19, 7, 19, 5, 14, 2, 17, 8, 28, 13, 31, 5, 23, 5, 23, 5, 31, 13, 28, 8, 17, 2, 14, 5
Offset: 0
Examples
a(0) = 0; a(1) = 2; a(2) = a(2*1) = sopf(a(1)) = 2; a(3) = a(2*1+1) = a(1) + a(2) = 4; a(4) = a(2*2) = sopf(a(2)) = 2; a(5) = a(2*2+1) = a(2) + a(3) = 6; a(6) = a(2*3) = sopf(a(3)) = 2, etc.
Links
- G. C. Greubel, Table of n, a(n) for n = 0..5000
- Michael Gilleland, Some Self-Similar Integer Sequences
- Ilya Gutkovskiy, Extended graphical example
- Eric Weisstein's World of Mathematics, Stern's Diatomic Series
- Index entries for sequences related to Stern's sequences
Programs
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Mathematica
a[0] = 0; a[1] = 2; a[n_] := If[EvenQ[n], DivisorSum[a[n/2], # &, PrimeQ[#] &], a[(n - 1)/2] + a[(n + 1)/2]]; Table[a[n], {n, 0, 90}] -
PARI
a(n) = if (n==0, 0, if (n ==1, 2, if (n%2, a((n-1)/2) + a((n+1)/2), vecsum(factor(a(n/2))[,1])))); \\ Michel Marcus, Dec 17 2017
Comments