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

1,1

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).

Links

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

Example

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

Mathematica

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

Prog

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

Crossrefs

Cf. A000005, A000217, A030628, A178739.

Sequence in context: A302366 A303083 A302888 * A303623 A249200 A249215

Adjacent sequences: A349696 A349697 A349698 * A349700 A349701 A349702

Keyword

nonn

Author

Jon E. Schoenfield, Nov 25 2021

Status

approved