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A370685 Semiprimes that are also the sums of two, three and four successive semiprimes.

%I A370685 #10 Feb 27 2024 09:43:41
%S A370685 2045,2705,2855,14614,18838,28437,31299,43603,68807,76841,77386,88041,
%T A370685 108415,116822,194605,213679,218729,252094,255202,269653,290449,
%U A370685 294683,302761,305362,310799,339382,348242,361055,398111,445066,445174,459761,464567,489809,496081,501386,515981,534777,544405
%N A370685 Semiprimes that are also the sums of two, three and four successive semiprimes.
%H A370685 Robert Israel, <a href="/A370685/b370685.txt">Table of n, a(n) for n = 1..2734</a>
%e A370685 a(3) = 2855 is a term because 2855 = 5 * 571 is a semiprime, A001358(423) = 1418 = 2*709 and A001358(424) = 1437 = 3 * 479 are two successive semiprimes whose sum is 2855, A001358(285) = 949 = 13 * 73, A001358(286) = 951 = 3 * 317 and A001358(287) = 955 = 5 * 191 are three successive semiprimes whose sum is 2855, and A001358(216) = 707 = 7 * 101, A001358(217) = 713 = 23 * 31, A001358(218) = 717 = 3 * 239, A001358(219) = 718 = 2 * 359 are four successive semiprimes whose sum is 2855.
%p A370685 N:= 10^6: # for terms <= N
%p A370685 P:= select(isprime, [2, seq(i, i=3..N/2, 2)]):
%p A370685 nP:= nops(P):
%p A370685 SP:= 0:
%p A370685 for i from 1 while P[i]^2 <= N do
%p A370685   m:= ListTools:-BinaryPlace(P, N/P[i]);
%p A370685   SP:= SP, op(P[i]*P[i..m]);
%p A370685 od:
%p A370685 SP:= sort([SP]):
%p A370685 SS:= ListTools:-PartialSums(SP):
%p A370685 SS2:= {seq(SS[i]-SS[i-2], i=3..nops(SS))}:
%p A370685 SS3:= {seq(SS[i]-SS[i-3], i=4..nops(SS))}:
%p A370685 SS4:= {seq(SS[i]-SS[i-4], i=5..nops(SS))}:
%p A370685 A:=SS2 intersect SS3 intersect SS4 intersect convert(SP, set):
%p A370685 A:= sort(convert(A, list)):
%Y A370685 Cf. A001358, A370162. Intersection of A092192, A131610 and A158339.
%K A370685 nonn
%O A370685 1,1
%A A370685 _Robert Israel_, Feb 26 2024