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 A349699 Triangular numbers with exactly 10 divisors. 1
 496, 3321, 13203, 195625, 780625, 2883601, 11527201, 107186761, 407879641, 3487920481, 39155632561, 250123560121, 47622568443841, 95853663421561, 322876778328721, 403230060146161, 3034217580863041, 6333850463213521, 13292221046055841, 25335401515201441 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified September 17 21:09 EDT 2024. Contains 375990 sequences. (Running on oeis4.)