A369093 -id:A369093 - cp's OEIS frontend

A369096 Numbers k >= 2 such that omega(k) divides the sum of the triangular numbers T(k) and T(k+1), where omega(k) is the number of distinct primes dividing k (A001221).

2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 64, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 107, 109, 110, 111, 113, 115, 117, 119, 121, 123, 125, 127, 128, 129

Offset: 1

Claude H. R. Dequatre, Jan 13 2024

k is a term if (k^2+k)/2 + ((k+1)^2+k+1)/2 = k^2+2*k+1 = (k+1)^2 is divisible by omega(k).
Trivial case: If k is prime, then omega(k) = 1 and (k+1)^2 is always divisible by 1, thus all primes are terms of this sequence.
Table with percentage of primes <= 10^k for k = 2..9:
  | k | #terms <= 10^k | #primes <= 10^k | %primes <= 10^k |
  | 2 |          55    |           25    |      45.45      |
  | 3 |         506    |          168    |      33.20      |
  | 4 |        4832    |         1229    |      25.43      |
  | 5 |       46675    |         9592    |      20.55      |
  | 6 |      456155    |        78498    |      17.21      |
  | 7 |     4480617    |       664579    |      14.83      |
  | 8 |    44081959    |      5761455    |      13.07      |
  | 9 |   433916814    |     50847535    |      11.72      |
The percentage of primes decreases asymptotically as 10^k increases.
Conjecture: the asymptotic density of primes in this sequence is 0.

			2 is a term since (2+1)^2 = 3^2 = 9 is divisible by omega(2) = 1.
15 is a term since (15+1)^2 = 16^2 = 256 is divisible by omega(15) = 2.
12 is not a term since (12+1)^2 = 13^2 = 169 is not divisible by omega(12) = 2.

Cf. A000217, A001221.
Cf. A090777, A356410, A369093, A369097.
Subsequence: A000040.

isA369096 := proc(k)
    if modp((k+1)^2, A001221(k)) = 0 then
        true;
    else
        false;
    end if;
end proc:
A369096 := proc(n)
    option remember ;
    if n = 1 then
        2;
    else
        for a from procname(n-1)+1 do
            if isA369096(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
[seq(A369096(n),n=1..100)] ; # R. J. Mathar, Jan 18 2024

isok(n)=my(x=(n+1)^2,y=omega(n));!(x%y);

cp's OEIS Frontend

A369096 Numbers k >= 2 such that omega(k) divides the sum of the triangular numbers T(k) and T(k+1), where omega(k) is the number of distinct primes dividing k (A001221).

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