A184393 -id:A184393 - cp's OEIS frontend

A007368 Smallest k such that sigma(x) = k has exactly n solutions.

2, 1, 12, 24, 96, 72, 168, 240, 336, 360, 504, 576, 1512, 1080, 1008, 720, 2304, 3600, 5376, 2520, 2160, 1440, 10416, 13392, 3360, 4032, 3024, 7056, 6720, 2880, 6480, 10800, 13104, 5040, 6048, 4320, 13440, 5760, 18720, 20736, 19152, 22680, 43680

Offset: 0

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

nonn

It's not obvious that a(n) exists for all n; I'd like to see a proof. - David Wasserman, Jun 07 2002
Note that k-1 is frequently prime. See A115374 for the least prime. For each n, it appears that there are an infinite number of k such that sigma(x)=k has exactly n solutions. - T. D. Noe, Jan 21 2006
According to Sierpiński, H. J. Kanold proved that there is a k such that sigma(x)=k has n or more solutions. Sierpiński states that Erdős proved that if, for some k, sigma(x)=k has exactly n solutions, then there are an infinite number of such k. - T. D. Noe, Oct 18 2006
Index of the first occurrence of n in A054973. - Jaroslav Krizek, Apr 25 2009

			a(10) = 504; {204, 220, 224, 246, 284, 286, 334, 415, 451, 503} is the set of x such that sigma(x) = 504.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe and Donovan Johnson, Table of n, a(n) for n = 0..5000 (terms up to a(429) from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Kevin Ford, Sergei Konyagin, On two conjectures of Sierpiński concerning the arithmetic functions σ and ϕ, arXiv:1910.08452 [math.NT], 2019.
W. Sierpiński, Elementary Theory of Numbers, Warszawa 1964, page 166.
R. G. Wilson, V, Letter to N. J. A. Sloane, Jul. 1992

Cf. A000203, A054973, A002191, A007609.
Cf. A115374 (least prime p such that sigma(x)=sigma(p) has exactly n solutions).
Cf. A007369, A007370, A007371, A007372 (n such that sigma(x)=k has 0, 1, 2 and 3 solutions).
Cf. A184393, A184394, A201915 (smallest solution, largest solution, triangle of solutions for sigma(x)=a(n)).

Needs["Statistics`DataManipulation`"]; s=DivisorSigma[1, Range[10^5]]; f=Frequencies[s]; fs=Sort[f]; tfs=Transpose[fs][[1]]; utfs=Union[tfs]; firstMissing=First[Complement[Range[Last[utfs]], utfs]]; pos=1; Table[While[tfs[[pos]]T. D. Noe *)
terms = 100; cnt = DivisorSigma[1, Range[terms^3]] // Tally // Sort; a[0] = 2; a[n_] := SelectFirst[cnt, #[[2]] == n&][[1]]; Table[a[n], {n, 0, terms - 1}] (* Jean-François Alcover, Jul 18 2017 *)

More terms from David W. Wilson

A201915 Each row of triangle T(n,k) has a sorted list of n values such that sigma(T(n,k)) = A007368(n).

Original entry on oeis.org

1, 6, 11, 14, 15, 23, 42, 62, 69, 77, 30, 46, 51, 55, 71, 60, 78, 92, 123, 143, 167, 114, 135, 158, 177, 203, 209, 239, 132, 140, 182, 188, 195, 249, 287, 299, 120, 174, 184, 190, 267, 295, 319, 323, 359, 204, 220, 224, 246, 284, 286, 334, 415, 451, 503

Offset: 1

T. D. Noe, Jan 24 2012

The first and last terms of each row are given in A184393 and A184394.

Note that the integers in the 10th row have sigma(n)=504 (A180164(1)) and thus include A002025(1) and A002046(1). - Michel Marcus, Oct 22 2013

			Triangle:
1
6,    11
14,   15,  23
42,   62,  69,  77
30,   46,  51,  55,  71
60,   78,  92, 123, 143, 167
114, 135, 158, 177, 203, 209, 239
132, 140, 182, 188, 195, 249, 287, 299
120, 174, 184, 190, 267, 295, 319, 323, 359
204, 220, 224, 246, 284, 286, 334, 415, 451, 503

T. D. Noe, Rows n = 1..100 of triangle, flattened

A184394 The largest number m such that sigma(m) = A007368(n), where A007368(n) = the smallest k such that sigma(x) = k has exactly n solutions.

Original entry on oeis.org

1, 11, 23, 77, 71, 167, 239, 299, 359, 503, 527, 1511, 1007, 943, 719, 2201, 3427, 5207, 2419, 2059, 1439, 10187, 12811, 3359, 3901, 3023, 6887, 6719, 2879, 6319, 10799, 13103, 5039, 6047, 4189, 13193, 5609, 18719, 20437, 18871, 22679, 43259, 27707, 25853

Offset: 1

Jaroslav Krizek, Jan 12 2011

nonn

A184393(n) is the sequence of the smallest such numbers. Row n of A201915 gives all n values satisfying sigma(x) = A007368(n).

			For n = 5, sequence of defined numbers m_5: 30, 46, 51, 55, 71; a(5) = 71.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Extended by T. D. Noe, Jan 24 2012

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A007368 Smallest k such that sigma(x) = k has exactly n solutions.

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A201915 Each row of triangle T(n,k) has a sorted list of n values such that sigma(T(n,k)) = A007368(n).

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A184394 The largest number m such that sigma(m) = A007368(n), where A007368(n) = the smallest k such that sigma(x) = k has exactly n solutions.

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