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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.)

%I A144486 #16 Jun 09 2025 04:42:39
%S A144486 45,105,231,325,378,561,595,741,990,1653,2850,3655,3741,4371,4465,
%T A144486 4851,6786,7021,7381,7503,7750,8911,9180,10011,10153,10585,10878,
%U A144486 11781,12561,13530,14535,14706,15225,15753,20301,20503,21115,22791,23005,23653
%N 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.)
%F A144486 a(n) = A000217(A045920(n+1)).
%e A144486 The first 11 values of n that satisfy the definition are 9, 14, 21, 25, 27, 33, 34, 38, 44, 57 and 75, so
%e A144486 a(1) = 9*10/2 = 45; 9 = 3*3, 10 = 2*5.
%e A144486 a(2) = 14*15/2 = 105; 14 = 2*7, 14 = 3*5.
%e A144486 a(3) = 21*22/2 = 231; 21 = 3*7, 22 = 2*11.
%e A144486 a(4) = 25*26/2 = 325; 25 = 5*5, 26 = 2*13.
%e A144486 a(5) = 27*28/2 = 378; 27 = 3*3*3, 28 = 2*2*7.
%e A144486 a(6) = 33*34/2 = 561; 33 = 3*11, 34 = 2*17.
%e A144486 a(7) = 34*35/2 = 595; 34 = 2*17, 35 = 5*7.
%e A144486 a(8) = 38*39/2 = 741; 38=2*19, 39=3*13.
%e A144486 a(9) = 44*45/2 = 990; 44=2*2*11, 45=3*3*5.
%e A144486 a(10) = 57*58/2 = 1653; 57=3*19, 58=2*29.
%e A144486 a(11) = 75*76/2 = 2850; 75=3*5*5, 76=2*2*19.
%p A144486 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
%t A144486 Times@@#/2&/@Select[Partition[Range[500],2,1],!PrimeQ[#[[1]]] && !PrimeQ[#[[2]]] && PrimeOmega[#[[1]]]==PrimeOmega[#[[2]]]&] (* _Harvey P. Dale_, May 23 2013 *)
%o A144486 (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
%Y A144486 Cf. A000217, A045920, A144291.
%K A144486 nonn,easy
%O A144486 1,1
%A A144486 _Juri-Stepan Gerasimov_, Dec 09 2008
%E A144486 Corrected and extended by _R. J. Mathar_ and _Ray Chandler_, Dec 10 2008