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 1-3 of 3 results.

A145899 Numbers n such that sigma(x) = n has more solutions x than any smaller n.

Original entry on oeis.org

1, 12, 24, 72, 168, 240, 336, 360, 504, 576, 720, 1440, 2880, 4320, 5760, 8640, 10080, 15120, 17280, 20160, 30240, 40320, 60480, 120960, 181440, 241920, 362880, 483840, 604800, 725760, 1088640, 1209600, 1451520, 2177280, 2419200, 2903040, 3628800
Offset: 1

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Author

Douglas E. Iannucci, Oct 22 2008

Keywords

Comments

Sequence A206027 has the number of solutions.

Examples

			sigma(m)=1 has only one solution: m=1.
sigma(m)=12 has two solutions, m=6 and m=11; 12 is the smallest number with more than one such solutions.
sigma(m)=24 has three solutions, m=14,m=15 and m=23; 24 is the smallest number with more than two such solutions.
sigma(m)=72 has five solutions, m=30, m=46, m=51, m=55 and m=71; 72 is the smallest number with more than three such solutions.

Links

Crossrefs

Cf. A000203 (sum of divisors of n), A054973 (number of numbers whose divisors sum to n), A007368 (smallest k such that sigma(x) = k has exactly n solutions).
Cf. A206027.
Cf. Untouchable numbers (A005114), sigma-untouchable numbers (A007369) and highly touchable numbers (A238895).

Programs

  • Mathematica
    t = DivisorSigma[1, Range[10^6]]; t2 = Sort[Tally[t]]; mn = 0; t3 = {}; Do[If[t2[[n]][[2]] > mn, mn = t2[[n]][[2]]; AppendTo[t3, t2[[n]][[1]]]], {n, Length[t2]}]; t3 (* T. D. Noe, Feb 03 2012 *)
  • PARI
    {m=3650000; v=vectorsmall(m); for(n=1, m, s=sigma(n); if(s<=m, v[s]++)); g=0; j=1; while(j<=m, if(v[j]<=g, j++, g=v[j]; print1(j, ",")))} \\ Klaus Brockhaus, Oct 27 2008

Extensions

Extended beyond a(15) by Klaus Brockhaus, Oct 27 2008

A206026 a(n) = smallest number m such that sigma(k) = m has at least n positive solutions k.

Original entry on oeis.org

1, 12, 24, 72, 72, 168, 240, 336, 360, 504, 576, 720, 720, 720, 720, 1440, 1440, 1440, 1440, 1440, 1440, 2880, 2880, 2880, 2880, 2880, 2880, 2880, 2880, 4320, 4320, 4320, 4320, 4320, 4320, 5760, 5760, 8640, 8640, 8640, 8640, 8640, 8640, 8640, 8640, 8640, 8640
Offset: 1

Views

Author

Jaroslav Krizek, Feb 03 2012

Keywords

Comments

Sequence of numbers from A145899.

Examples

			a(6) = 168 because 168 is the smallest value of sigma(k) for n = 6 positive integers k such that sigma(k) = 168 has solution; k = 60, 78, 92, 123, 143, 167.

Links

Crossrefs

Cf. A000203, A054973, A085790, A145899, A206027.

Programs

  • PARI
    list(len) = {my(v = vector(len), k = 1, c = 0, i); while(c < len, i = invsigmaNum(k); for(j = 1, i, if(j <= len && v[j] == 0, v[j] = k; c++)); k++); v;} \\ Amiram Eldar, Dec 15 2024, using Max Alekseyev's invphi.gp

A362403 Number of times that the number A362402(n) occurs as a sum of divisors that have a square factor (A162296).

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 9, 10, 13, 15, 16, 20, 22, 23, 28, 34, 46, 53, 60, 62, 78, 81, 113, 115, 122, 132, 154, 184, 185, 222, 248, 254, 343, 346, 350, 354, 497, 569, 701, 711, 860, 941, 1088, 1221, 1222, 1235, 1263, 1306, 1572, 1721, 1737, 1948, 2191, 2315, 2418, 2877
Offset: 1

Views

Author

Amiram Eldar, Apr 18 2023

Keywords

Links

Crossrefs

Cf. A162296, A362401, A362402.
Similar sequences: A131934, A101373, A206027, A238896.

Programs

  • Mathematica
    s[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1)]; s[1] = 0; seq[max_] := Module[{v = Select[Array[s, max], 0 < # <= max &], sq = {0}, t, tmax = 0}, t = Sort[Tally[v]]; Do[If[t[[k]][[2]] > tmax, tmax = t[[k]][[2]]; AppendTo[sq, t[[k]][[2]]]], {k, 1, Length[t]}]; sq]; seq[10^5]
  • PARI
    s(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; ((p^(e + 1) - 1)/(p - 1))) -  prod(i = 1, #f~, f[i, 1] + 1);}
lista(kmax) = {my(v = vector(kmax), vmax = 0, i); for(k=1, kmax, i = s(k); if(i > 0 && i <= kmax, v[i]++)); print1(0, ", "); for(k=1, kmax, if(v[k] > vmax, vmax = v[k]; print1(v[k], ", "))); }
Showing 1-3 of 3 results.

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