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A240914
Semiprimes of the form S(n) + T(n) where S(n) and T(n) are the n-th square and the n-th triangular numbers.
1
15, 26, 57, 77, 155, 187, 301, 551, 737, 1027, 1457, 1751, 3197, 3337, 5251, 6767, 7597, 8251, 13301, 22387, 24257, 25807, 32047, 34277, 41417, 41917, 48151, 61307, 63757, 66887, 68801, 85801, 103097, 112751, 136957, 141527, 145237, 179747, 180787, 196747
OFFSET
1,1
COMMENTS
The n-th triangular number T(n) = n/2*(n+1).
Semiprimes (biprimes) in the sequence are product of two primes, simultaneously sum of n-th square & triangular numbers.
All the terms in the sequence, except a(2), are odd numbers.
LINKS
EXAMPLE
a(1) = 15: 3^2 + 3/2*(3+1) = 15 = 3*5, which is product of two primes. Hence it is semiprime.
a(3) = 57: 6^2 + 6/2*(6+1) = 57 = 3*19, which is product of two primes. Hence it is semiprime.
MAPLE
with(numtheory):KD:= proc() local a, b; a:=(n)^2 + n/2*(n+1); b:=bigomega(a); if b=2 then RETURN (a); fi; end: seq(KD(), n=1..500);
MATHEMATICA
KD = {}; Do[t = n^2 + n/2*(n + 1); If[PrimeOmega[t] == 2, AppendTo[KD, t]], {n, 500}]; KD
c=0; Do[t=n^2 + n/2*(n+1); If[PrimeOmega[t]==2, c=c+1; Print[c, " ", t]], {n, 1, 500000}];
Module[{nn=500, s, t}, s=Range[nn]^2; t=Accumulate[Range[nn]]; Select[ Total/@ Thread[{s, t}], PrimeOmega[#]==2&]] (* or *) Select[ Table[ (n(1+3n))/2, {n, 500}], PrimeOmega[#]==2&](* Harvey P. Dale, Feb 07 2018 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
K. D. Bajpai, Apr 14 2014
STATUS
approved