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A349699 Triangular numbers with exactly 10 divisors.

%I A349699 #12 Nov 27 2021 06:49:19
%S A349699 496,3321,13203,195625,780625,2883601,11527201,107186761,407879641,
%T A349699 3487920481,39155632561,250123560121,47622568443841,95853663421561,
%U A349699 322876778328721,403230060146161,3034217580863041,6333850463213521,13292221046055841,25335401515201441
%N A349699 Triangular numbers with exactly 10 divisors.
%C A349699 All terms are of the form p^4 * q, with primes p < q.
%C A349699 a(3) = 13203 = 3^4 * 163 is the only term for which q = 2*p^4 + 1; for all other terms, q is either 2*p^4 - 1 (e.g., a(1) = 496 = 2^4 * 31) or (p^4 + 1)/2 (e.g., a(2) = 3321 = 3^4 * 41).
%H A349699 Jon E. Schoenfield, <a href="/A349699/b349699.txt">Table of n, a(n) for n = 1..10000</a>
%e A349699 Table showing the first 20 terms and their prime factorizations. Because of the different relationships between the prime factors p and q for different terms (see Comments), neither the values of p nor those of q are nondecreasing.
%e A349699 .
%e A349699    n               a(n) =   p^4 *         q
%e A349699   --  -------------------------------------
%e A349699    1                496 =   2^4 *        31
%e A349699    2               3321 =   3^4 *        41
%e A349699    3              13203 =   3^4 *       163
%e A349699    4             195625 =   5^4 *       313
%e A349699    5             780625 =   5^4 *      1249
%e A349699    6            2883601 =   7^4 *      1201
%e A349699    7           11527201 =   7^4 *      4801
%e A349699    8          107186761 =  11^4 *      7321
%e A349699    9          407879641 =  13^4 *     14281
%e A349699   10         3487920481 =  17^4 *     41761
%e A349699   11        39155632561 =  23^4 *    139921
%e A349699   12       250123560121 =  29^4 *    353641
%e A349699   13     47622568443841 =  47^4 *   9759361
%e A349699   14     95853663421561 =  61^4 *   6922921
%e A349699   15    322876778328721 =  71^4 *  12705841
%e A349699   16    403230060146161 =  73^4 *  14199121
%e A349699   17   3034217580863041 =  79^4 *  77900161
%e A349699   18   6333850463213521 = 103^4 *  56275441
%e A349699   19  13292221046055841 = 113^4 *  81523681
%e A349699   20  25335401515201441 = 103^4 * 225101761
%t A349699 t[n_] := n*(n + 1)/2; Select[t /@ Range[10^5], DivisorSigma[0, #] == 10 &] (* _Amiram Eldar_, Nov 26 2021 *)
%o A349699 (PARI) select(x->(numdiv(x)==10), vector(10^5, k, k*(k+1)/2)) \\ _Michel Marcus_, Nov 26 2021
%Y A349699 Cf. A000005, A000217, A030628, A178739.
%K A349699 nonn
%O A349699 1,1
%A A349699 _Jon E. Schoenfield_, Nov 25 2021