A144524 Triangular numbers n*(n+1)/2 with n composite, where number of prime factors of n, counted with multiplicity, is less than the number of prime factors in n+1.
120, 351, 630, 780, 1225, 1326, 1540, 1953, 2016, 2145, 2415, 2775, 3003, 3828, 4186, 4560, 4950, 6216, 6670, 7140, 7626, 7875, 8385, 9045, 10296, 10731, 12090, 12720, 13041, 14365, 15400, 16836, 17205, 17578, 17766, 18915, 19110, 20706, 21321, 21528
Offset: 1
Keywords
Examples
If n = 15 = 2*3 (number of prime factors = 2) and n+1 = 16 = 2*2*2*2 (number of prime factors = 4), then 15*16/2 = 120 = a(1). If n = 26 = 2*13 (number of prime factors = 2) and n+1 = 27 = 3*3*3 (number of prime factors = 3), then 26*27/2 = 351 = a(2). If n = 35 = 5*7 (number of prime factors = 2) and n+1 = 36 = 2*2*3*3 (number of prime factors = 4), then 35*36/2 = 630 = a(3), etc.
Programs
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Mathematica
fQ[n_] := !PrimeQ@ n && Plus @@ Last /@ FactorInteger@n < Plus @@ Last /@ FactorInteger[n + 1]; # (# + 1)/2 & /@ Select[ Range@ 209, fQ@# &] (* Robert G. Wilson v, Dec 21 2008 *)
Extensions
Edited by Robert G. Wilson v, Dec 21 2008