A144486 Triangular numbers n*(n+1)/2 with n and n+1 composite, where number of prime factors in n = number of prime factors in n+1. (Prime factors are counted with multiplicity.)

45, 105, 231, 325, 378, 561, 595, 741, 990, 1653, 2850, 3655, 3741, 4371, 4465, 4851, 6786, 7021, 7381, 7503, 7750, 8911, 9180, 10011, 10153, 10585, 10878, 11781, 12561, 13530, 14535, 14706, 15225, 15753, 20301, 20503, 21115, 22791, 23005, 23653

Offset: 1

Author

Juri-Stepan Gerasimov, Dec 09 2008

Keywords

• nonn
• easy

Examples

			The first 11 values of n that satisfy the definition are 9, 14, 21, 25, 27, 33, 34, 38, 44, 57 and 75, so
a(1) = 9*10/2 = 45; 9 = 3*3, 10 = 2*5.
a(2) = 14*15/2 = 105; 14 = 2*7, 14 = 3*5.
a(3) = 21*22/2 = 231; 21 = 3*7, 22 = 2*11.
a(4) = 25*26/2 = 325; 25 = 5*5, 26 = 2*13.
a(5) = 27*28/2 = 378; 27 = 3*3*3, 28 = 2*2*7.
a(6) = 33*34/2 = 561; 33 = 3*11, 34 = 2*17.
a(7) = 34*35/2 = 595; 34 = 2*17, 35 = 5*7.
a(8) = 38*39/2 = 741; 38=2*19, 39=3*13.
a(9) = 44*45/2 = 990; 44=2*2*11, 45=3*3*5.
a(10) = 57*58/2 = 1653; 57=3*19, 58=2*29.
a(11) = 75*76/2 = 2850; 75=3*5*5, 76=2*2*19.
		

Crossrefs

Cf. A000217, A045920, A144291.

Programs

Maple

isA045920 := proc(n) if numtheory[bigomega](n) = numtheory[bigomega](n+1) then true; else false; fi; end: A045920 := proc(n) option remember ; local a; if n =1 then 2; else for a from procname(n-1)+1 do if isA045920(a) then RETURN(a) ; fi; od: fi; end: A000217 := proc(n) n*(n+1)/2 ; end: A144486 := proc(n) A000217(A045920(n+1)) ; end: for n from 1 to 100 do printf("%d,",A144486(n)) ; od: # R. J. Mathar, Dec 10 2008

Mathematica

Times@@#/2&/@Select[Partition[Range[500],2,1],!PrimeQ[#[[1]]] && !PrimeQ[#[[2]]] && PrimeOmega[#[[1]]]==PrimeOmega[#[[2]]]&] (* Harvey P. Dale, May 23 2013 *)

PARI

for(n=2, 1e3, if(bigomega(n+1) == bigomega(n+2) && k = (n+1)*(n+2)/2, print1(k", "))) \\ Altug Alkan, Oct 18 2015

Formula

a(n) = A000217(A045920(n+1)).

Extensions

Corrected and extended by R. J. Mathar and Ray Chandler, Dec 10 2008