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 A343106 Smallest number having exactly n divisors of the form 8*k + 5. 5
 1, 5, 45, 315, 585, 2205, 2925, 14175, 9945, 17325, 28665, 178605, 45045, 190575, 240975, 143325, 135135, 3189375, 225225, 93002175, 405405, 1403325, 1715175, 2401245, 675675, 3583125, 3239775, 1576575, 3468465, 94918019805, 2027025, 15436575, 2297295, 11609325, 16769025, 27286875, 3828825, 42879375, 117661005 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Smallest index of n in A188171. a(n) exists for all n, since 5*3^(2n-2) has exactly n divisors of the form 8*k + 1, namely 5*3^0, 5*3^2, ..., 5*3^(2n-2). This actually gives an upper bound (which is too far from reality when n is large) for a(n). LINKS Bert Dobbelaere, Table of n, a(n) for n = 0..200 FORMULA a(2n-1) <= 3^(2n-2) * 35, since 3^(2n-2) * 35 has exactly 2n-1 divisors congruent to 5 modulo 8: 3^0 * 5, 3^2 * 5, ..., 3^(2n-2) * 5, 3^1 * 7, 3^3 * 7, ..., 3^(2n-3) * 7. a(2n) <= 3^(n-1) * 455, since 3^(n-1) * 455 has exactly 2n divisors congruent to 5 modulo 8: 3^0 * 5, 3^2 * 5, ..., 3^b * 5, 3^1 * 7, 3^3 * 7, ..., 3^a * 7, 3^0 * 13, 3^2 * 13, ..., 3^b * 13, 3^1 * 455, 3^3 * 455, ... 3^a * 455, where a is the largest odd number <= n-1 and b is the largest even number <= n-1. EXAMPLE a(4) = 585 since it is the smallest number with exactly 4 divisors congruent to 5 modulo 8, namely 5, 13, 45 and 585. PROG (PARI) res(n, a, b) = sumdiv(n, d, (d%a) == b) a(n) = for(k=1, oo, if(res(k, 8, 5)==n, return(k))) CROSSREFS Smallest number having exactly n divisors of the form 8*k + i: A343104 (i=1), A343105 (i=3), this sequence (i=5), A188226 (i=7). Cf. A188171. Sequence in context: A303408 A241275 A343136 * A081070 A247494 A043025 Adjacent sequences:  A343103 A343104 A343105 * A343107 A343108 A343109 KEYWORD nonn AUTHOR Jianing Song, Apr 05 2021 EXTENSIONS More terms from Bert Dobbelaere, Apr 09 2021 STATUS approved

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Last modified August 3 04:47 EDT 2021. Contains 346435 sequences. (Running on oeis4.)