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A369093 Numbers k >= 1 such that sigma(k) divides the sum of the triangular numbers T(k) and T(k+1), where sigma(k) = A000203(k) is the sum of the divisors of k.

%I A369093 #14 Jan 18 2024 09:29:01
%S A369093 1,2,3,5,7,11,13,17,19,23,29,31,35,37,41,43,47,53,59,61,67,71,73,79,
%T A369093 83,89,97,101,103,107,109,113,119,127,131,137,139,149,151,157,163,167,
%U A369093 173,179,181,191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,293
%N A369093 Numbers k >= 1 such that sigma(k) divides the sum of the triangular numbers T(k) and T(k+1), where sigma(k) = A000203(k) is the sum of the divisors of k.
%C A369093 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 sigma(k).
%C A369093 Trivial case: If k is prime, then sigma(k) = k+1 and (k+1)^2 is divisible by k+1, thus all primes are terms of this sequence.
%C A369093 Table with the percentage of primes <= 10^k compared with the number of terms and the number of primes <= 10^k, for k = 2..8:
%C A369093 .
%C A369093  | k | #terms <= 10^k | #primes <= 10^k | %primes <= 10^k |
%C A369093  | 2 |          27    |            25   |        92.59    |
%C A369093  | 3 |         175    |           168   |        96.00    |
%C A369093  | 4 |        1248    |          1229   |        98.48    |
%C A369093  | 5 |        9627    |          9592   |        99.64    |
%C A369093  | 6 |       78565    |         78498   |        99.91    |
%C A369093  | 7 |      664707    |        664579   |        99.98    |
%C A369093  | 8 |     5761724    |       5761455   |        99.99    |
%C A369093 .
%C A369093 The percentage of primes increases asymptotically as 10^k increases.
%C A369093 Conjecture: The asymptotic density of primes in this sequence is 1.
%C A369093 Contains terms like 2, 399, 935, 1539,.. which are not in A210494. Does not contain terms like 775, 819, 3335, 6815,.. which are in A210494. - _R. J. Mathar_, Jan 18 2024
%e A369093 3 is a term since (3+1)^2 = 4^2 = 16 is divisible by sigma(3) = 4.
%e A369093 35 is a term since (35+1)^2 = 36^2 = 1296 is divisible by sigma(35) = 48.
%e A369093 42 is not a term since (42+1)^2 = 43^2 = 1849 is not divisible by sigma(42) = 96.
%p A369093 isA369093 := proc(k)
%p A369093     if modp((k+1)^2, numtheory[sigma](k)) = 0 then
%p A369093         true;
%p A369093     else
%p A369093         false;
%p A369093     end if;
%p A369093 end proc:
%p A369093 A369093 := proc(n)
%p A369093     option remember ;
%p A369093     if n = 1 then
%p A369093         1;
%p A369093     else
%p A369093         for a from procname(n-1)+1 do
%p A369093             if isA369093(a) then
%p A369093                 return a;
%p A369093             end if;
%p A369093         end do:
%p A369093     end if;
%p A369093 end proc:
%p A369093 [seq(A369093(n),n=1..100)] ; # _R. J. Mathar_, Jan 18 2024
%o A369093 (PARI) isok(n) = my(x=(n+1)^2,y=sigma(n));!(x%y);
%Y A369093 Cf. A000203, A000217.
%Y A369093 Cf. A090777, A356410, A369096, A369097.
%Y A369093 Subsequence: A000040.
%K A369093 nonn,easy
%O A369093 1,2
%A A369093 _Claude H. R. Dequatre_, Jan 13 2024