|
%I #32 Jul 17 2017 03:53:39
%S 10,15,49,121,143,159,187,235,287,301,319,329,371,395,407,471,519,533,
%T 551,565,581,589,633,679,689,713,731,749,771,789,803,817,841,961,985,
%U 1079,1099,1119,1135,1169,1207,1271,1285,1315,1349,1391,1457,1477,1585
%N Products of two primes (i.e., "semiprimes") that are the sum of three consecutive primes.
%C Or, semiprimes in A034961 (Sums of three consecutive primes).
%C Subsequence of square semiprimes: {49, 121, 841, 961, 1849, 22801, 24649, 36481, 69169, ...} = {7, 11, 29, 31, 43, 151, 157, 191, 263, ...}^2 that is also a subsequence of A080665 (Squares in A034961). Cf. also A034962 (Primes A034961).
%C Somewhat surprisingly, the sum of two consecutive primes is never a semiprime. This follows from that fact that if p+q = 2r for primes p,q,r, then r must between p and q. So if p and q are consecutive, then r does not exist.
%H Zak Seidov, <a href="/A188651/b188651.txt">Table of n, a(n) for n = 1..1000</a>
%e a(1) = 10 = 2*5 = A034961(1) = prime(1) + prime(2) + prime(3) = 2 + 3 + 5,
%e a(2) = 15 = 3*5 = A034961(2) = prime(2) + prime(3) + prime(4) = 3 + 5 + 7,
%e a(3) = 49 = 7*7 = A080665(1) = A034961(6) = prime(6) + prime(7) + prime(8) = 13 + 17 + 19.
%t semiPrimeQ[n_Integer] := Total[FactorInteger[n]][[2]] == 2; Select[Total /@ Partition[Prime[Range[100]], 3, 1], semiPrimeQ] (* _T. D. Noe_, Apr 20 2011 *)
%K nonn
%O 1,1
%A _Zak Seidov_, Apr 16 2011
|
