cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Original entry on oeis.org

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

Views

Author

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

Keywords

Comments

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

Examples

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

References

  • 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).

Links

Crossrefs

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)).

Programs

  • Mathematica
    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 *)

Extensions

More terms from David W. Wilson

A263025 n is the a(n)-th positive integer having its sum of divisors.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 3, 1, 1, 1, 1, 2, 3, 1, 1, 1, 3, 1, 2, 1, 4, 2, 1, 2, 3, 1, 1, 2, 1, 1, 2, 1, 1, 1, 3, 2, 5, 1, 1, 1, 2, 1, 4, 2, 2, 1, 1, 2, 3, 1, 1, 1, 3
Offset: 1

Views

Author

Paul Tek, Oct 09 2015

Keywords

Comments

Sum of divisors is given by A000203.
This can also be described as the ordinal transform of A000203. - Franklin T. Adams-Watters, Oct 09 2015
a(n) > 1 iff n is in A069822.

Examples

			The numbers with sum of divisors 72 are: 30, 46, 51, 55, 71.
Hence: a(30)=1, a(46)=2, a(51)=3, a(55)=4, a(71)=5.
More generally: the terms of each row of A085790 (say of length i) map to 1, 2, ..., i.
Also: for any n>0, the n terms of the n-th row of A201915 map to 1, 2, ..., n.

Links

Crossrefs

Cf. A000203, A034885, A069822, A078898, A085790, A201915.

Programs

  • Maple
    N:= 1000: # to get a(1) to a(N)
Sigmas:= [seq(numtheory:-sigma(i),i=1..N)]:
seq(numboccur(Sigmas[n], Sigmas[1..n]),n=1..N); # Robert Israel, Oct 09 2015
  • Mathematica
    t = DivisorSigma[1, #] & /@ Range@ 10000; s = Position[t, #] & /@ Range@ Max@ t; Flatten[Position[s, #, {3}]][[2]] & /@ Range@ 87 (* Michael De Vlieger, Oct 09 2015 *)
  • PARI
    cnt = vector(224); for (n=1, 87, s=sigma(n); cnt[s] = cnt[s]+1; print1(cnt[s] ", "))

Formula

a(A034885(k))=1 for k>0.

A184393 The smallest 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, 6, 14, 42, 30, 60, 114, 132, 120, 204, 210, 480, 408, 390, 264, 930, 1080, 1860, 864, 870, 552, 3120, 4080, 1140, 1380, 1020, 2460, 2184, 840, 2040, 3480, 4140, 1560, 2208, 1320, 3780, 1848, 5544, 7590, 6468, 8544, 13500, 8280, 8190, 4872, 4620, 8856
Offset: 1

Views

Author

Jaroslav Krizek, Jan 12 2011

Keywords

Comments

A184394 is the sequence of the largest such numbers. Row n of A201915 gives all n values satisfying sigma(x) = A007368(n).

Examples

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

Links

Extensions

Extended by T. D. Noe, Jan 24 2012

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

Views

Author

Jaroslav Krizek, Jan 12 2011

Keywords

Comments

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

Examples

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

Links

Extensions

Extended by T. D. Noe, Jan 24 2012
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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