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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).

%I A369096 #19 Mar 29 2025 01:37:30
%S A369096 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,
%T A369096 43,45,47,49,51,53,55,57,59,61,63,64,65,67,69,71,73,75,77,79,81,83,85,
%U A369096 87,89,91,93,95,97,99,101,103,107,109,110,111,113,115,117,119,121,123,125,127,128,129
%N 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).
%C A369096 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).
%C A369096 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.
%C A369096 Table with percentage of primes <= 10^k for k = 2..9:
%C A369096   | k | #terms <= 10^k | #primes <= 10^k | %primes <= 10^k |
%C A369096   | 2 |          55    |           25    |      45.45      |
%C A369096   | 3 |         506    |          168    |      33.20      |
%C A369096   | 4 |        4832    |         1229    |      25.43      |
%C A369096   | 5 |       46675    |         9592    |      20.55      |
%C A369096   | 6 |      456155    |        78498    |      17.21      |
%C A369096   | 7 |     4480617    |       664579    |      14.83      |
%C A369096   | 8 |    44081959    |      5761455    |      13.07      |
%C A369096   | 9 |   433916814    |     50847535    |      11.72      |
%C A369096 The percentage of primes decreases asymptotically as 10^k increases.
%C A369096 Conjecture: the asymptotic density of primes in this sequence is 0.
%e A369096 2 is a term since (2+1)^2 = 3^2 = 9 is divisible by omega(2) = 1.
%e A369096 15 is a term since (15+1)^2 = 16^2 = 256 is divisible by omega(15) = 2.
%e A369096 12 is not a term since (12+1)^2 = 13^2 = 169 is not divisible by omega(12) = 2.
%p A369096 isA369096 := proc(k)
%p A369096     if modp((k+1)^2, A001221(k)) = 0 then
%p A369096         true;
%p A369096     else
%p A369096         false;
%p A369096     end if;
%p A369096 end proc:
%p A369096 A369096 := proc(n)
%p A369096     option remember ;
%p A369096     if n = 1 then
%p A369096         2;
%p A369096     else
%p A369096         for a from procname(n-1)+1 do
%p A369096             if isA369096(a) then
%p A369096                 return a;
%p A369096             end if;
%p A369096         end do:
%p A369096     end if;
%p A369096 end proc:
%p A369096 [seq(A369096(n),n=1..100)] ; # _R. J. Mathar_, Jan 18 2024
%o A369096 (PARI) isok(n)=my(x=(n+1)^2,y=omega(n));!(x%y);
%Y A369096 Cf. A000217, A001221.
%Y A369096 Cf. A090777, A356410, A369093, A369097.
%Y A369096 Subsequence: A000040.
%K A369096 nonn,easy
%O A369096 1,1
%A A369096 _Claude H. R. Dequatre_, Jan 13 2024