A144581 Triangular numbers k*(k+1)/2 such that (number of prime factors of k) + (number of prime factors of k+1) is composite.

28, 45, 66, 78, 105, 120, 153, 171, 190, 231, 300, 325, 378, 406, 435, 465, 496, 561, 595, 630, 741, 780, 861, 903, 946, 990, 1128, 1378, 1485, 1540, 1596, 1653, 2016, 2080, 2211, 2278, 2485, 2556, 2628, 2850, 3081, 3160, 3240, 3321, 3570, 3655, 3741

Offset

1,1

Comments

Prime factors here are counted with multiplicity. - Robert Israel, Jul 31 2025

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

7 has one prime factor and 8 = 2*2*2 has three prime factors; 1+3 = 4 is composite, hence 7*8/2 = 28 is in the sequence.
15 = 3*5 has two prime factors and 16=2*2*2*2 has four prime factors; 2+4 = 6 is composite, hence 15*16/2 = 120 is in the sequence.
18 = 2*3*3 has three prime factors and 19 has one prime factors; 3+1 = 4 is composite, hence 18*19/2 = 171 is in the sequence.

Maple

N:= 200: W:= map(numtheory:-bigomega, [$1..N]):
V:= W[1..-2] + W[2..-1]:
K:=remove(i -> isprime(V[i]), [$1..N-1]):
map(t -> t*(t+1)/2, K[2..-1]); # Robert Israel, Jul 31 2025

Mathematica

(#(#+1))/2&/@Rest[Select[Range[110], !PrimeQ[PrimeOmega[#] + PrimeOmega[ #+1]]&]] (* Harvey P. Dale, Mar 18 2012 *)

Prog

(Magma) [ k*(k+1)/2: k in [2..86] | not IsPrime(s) where s is &+[ f[2]: f in Factorization(k) ] + &+[ f[2]: f in Factorization(k+1) ] ];

Crossrefs

Cf. A000217 (triangular numbers), A001222, A002808 (composite numbers).

See A144552 for another version.

Sequence in context: A169962 A219685 A180045 * A075875 A332764 A116541

Adjacent sequences: A144578 A144579 A144580 * A144582 A144583 A144584

Keyword

nonn

Author

Juri-Stepan Gerasimov, Dec 31 2008

Extensions

Edited and corrected by Klaus Brockhaus, Jan 03 2009

Edited by N. J. A. Sloane, Jan 08 2009

Status

approved