A349699 Triangular numbers with exactly 10 divisors.

496, 3321, 13203, 195625, 780625, 2883601, 11527201, 107186761, 407879641, 3487920481, 39155632561, 250123560121, 47622568443841, 95853663421561, 322876778328721, 403230060146161, 3034217580863041, 6333850463213521, 13292221046055841, 25335401515201441

Offset: 1

Author

Jon E. Schoenfield, Nov 25 2021

Keywords

• nonn

Comments

All terms are of the form p^4 * q, with primes p < q.

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).

Examples

			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.
.
   n               a(n) =   p^4 *         q
  --  -------------------------------------
   1                496 =   2^4 *        31
   2               3321 =   3^4 *        41
   3              13203 =   3^4 *       163
   4             195625 =   5^4 *       313
   5             780625 =   5^4 *      1249
   6            2883601 =   7^4 *      1201
   7           11527201 =   7^4 *      4801
   8          107186761 =  11^4 *      7321
   9          407879641 =  13^4 *     14281
  10         3487920481 =  17^4 *     41761
  11        39155632561 =  23^4 *    139921
  12       250123560121 =  29^4 *    353641
  13     47622568443841 =  47^4 *   9759361
  14     95853663421561 =  61^4 *   6922921
  15    322876778328721 =  71^4 *  12705841
  16    403230060146161 =  73^4 *  14199121
  17   3034217580863041 =  79^4 *  77900161
  18   6333850463213521 = 103^4 *  56275441
  19  13292221046055841 = 113^4 *  81523681
  20  25335401515201441 = 103^4 * 225101761
		

Links

• Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A000217, A030628, A178739.

Programs

Mathematica

t[n_] := n*(n + 1)/2; Select[t /@ Range[10^5], DivisorSigma[0, #] == 10 &] (* Amiram Eldar, Nov 26 2021 *)

PARI

select(x->(numdiv(x)==10), vector(10^5, k, k*(k+1)/2)) \\ Michel Marcus, Nov 26 2021