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%I #26 Aug 20 2021 04:22:21
%S 1,2,9,19,29,44,632,11829,19262,25286,26606,29824,247273,310556,
%T 491240,1419166,1601984,9509238,113333959,220531559,1034662494,
%U 8323088842,13102043650,14053673678,23505911647
%N Numbers k that divide the k-th partial sum of all semiprimes.
%C a(26) > 3*10^10. - _Donovan Johnson_, Nov 26 2010
%F {k: k | Sum_{i=1..k} A001358(i)}.
%e a(1) = 1 because 1 divides the first semiprime 4, trivially also the first partial sum of all semiprimes.
%e a(2) = 2 because A062198(2) = A001358(1) + A001358(2) = 4 + 6 = 10 is divisible by 2.
%e a(3) = 9 because A062198(9) = 126 = 2 * 3^2 * 7 is divisible by 9.
%e a(4) = 19 because A062198(19) = 532 = 2^2 * 7 * 19 is divisible by 19.
%e a(5) = 29 because A062198(29) = 1247 = 29 * 43 is divisible by 29.
%e a(6) = 44 because A062198(44) = 2904 = 44 * 66.
%t 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]]
%o (PARI) s=0; p=0; for(n=1, 1e9, until(bigomega(p++)==2,); (s+=p)%n | print1(n", ")) \\ _M. F. Hasler_, Nov 24 2010
%Y Cf. A001358, A007506.
%K nonn,more
%O 1,2
%A _Jonathan Vos Post_, Nov 24 2010
%E Extended by _T. D. Noe_, Nov 24 2010
%E a(1)-a(17) double-checked and a(18) from _M. F. Hasler_, Nov 25 2010
%E a(19) from _Ray Chandler_, Nov 25 2010
%E a(20)-a(25) from _Donovan Johnson_, Nov 26 2010
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