A173663 Numbers k that divide the k-th partial sum of all semiprimes.

1, 2, 9, 19, 29, 44, 632, 11829, 19262, 25286, 26606, 29824, 247273, 310556, 491240, 1419166, 1601984, 9509238, 113333959, 220531559, 1034662494, 8323088842, 13102043650, 14053673678, 23505911647

Offset: 1

Author

Jonathan Vos Post, Nov 24 2010

Keywords

• nonn
• more

Comments

a(26) > 3*10^10. - Donovan Johnson, Nov 26 2010

Examples

			a(1) = 1 because 1 divides the first semiprime 4, trivially also the first partial sum of all semiprimes.
a(2) = 2 because A062198(2) = A001358(1) + A001358(2) = 4 + 6 = 10 is divisible by 2.
a(3) = 9 because A062198(9) = 126 = 2 * 3^2 * 7 is divisible by 9.
a(4) = 19 because A062198(19) = 532 = 2^2 * 7 * 19 is divisible by 19.
a(5) = 29 because A062198(29) = 1247 = 29 * 43 is divisible by 29.
a(6) = 44 because A062198(44) = 2904 = 44 * 66.
		

Crossrefs

Cf. A001358, A007506.

Programs

Mathematica

SemiprimeQ[n_Integer] := If[Abs[n]<2, False, (2==Plus@@Transpose[FactorInteger[Abs[n]]][[2]])]; nn=10^6; sm=0; cnt=0; Reap[Do[If[SemiprimeQ[n], cnt++; sm=sm+n; If[Divisible[sm, cnt], Sow[cnt]]], {n, nn}]][[2, 1]]

PARI

s=0; p=0; for(n=1, 1e9, until(bigomega(p++)==2,); (s+=p)%n || print1(n", ")) \\ M. F. Hasler, Nov 24 2010

Formula

{k: k | Sum_{i=1..k} A001358(i)}.

Extensions

Extended by T. D. Noe, Nov 24 2010

a(1)-a(17) double-checked and a(18) from M. F. Hasler, Nov 25 2010

a(19) from Ray Chandler, Nov 25 2010

a(20)-a(25) from Donovan Johnson, Nov 26 2010