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 1-8 of 8 results.

A131902 Smallest positive integer k with the same sum of divisors as the n-th integer for which such a k exists.

Original entry on oeis.org

6, 14, 10, 14, 16, 20, 21, 33, 24, 28, 20, 30, 33, 30, 34, 30, 54, 40, 24, 42, 44, 42, 66, 30, 48, 42, 60, 57, 68, 44, 54, 40, 60, 66, 54, 52, 63, 85, 102, 74, 66, 104, 88, 66, 80, 60, 84, 99, 93, 96, 86, 114, 76, 132, 105, 102, 60, 88, 111, 90, 138, 105, 114, 102, 105, 138, 96
Offset: 1

Views

Author

Peter Pein (petsie(AT)dordos.net), Jul 26 2007

Keywords

Examples

			a(3)=10 because 17 is the third integer for which a smaller integer with same sum of divisors exists and sigma(17)=1+17=18 and sigma(10)=1+2+5+10=18 and there is no k>0 less than 10 with sigma(k)=18.

Links

Crossrefs

Cf. A069822, A131903, A131904, A131905, A131906, A131907, A131908.

Programs

  • Maple
    N:= 1000: # to use values of sigma <= N
V:= Vector(N): A:= Vector(N):
for n from 1 to N do
  v:= numtheory:-sigma(n);
  if v <= N then
    if V[v] = 0 then V[v]:= n
    else A[n]:= V[v]
    fi
  fi
od:
subs(0=NULL, convert(A,list)); # Robert Israel, Mar 30 2018
  • Mathematica
    Clear[tmp]; Function[n,If[Head[ #1]===tmp,#1=n;Unevaluated[Sequence[]],#1]& [tmp[DivisorSigma[1,n]]]]/@Range[200]
  • PARI
    list(lim) = my(m); for(k = 1, lim, m = invsigmaMin(sigma(k)); if(m < k, print1(m, ", "))); \\ Amiram Eldar, Dec 20 2024, using Max Alekseyev's invphi.gp

Formula

Let S = {n>0 : there exists a k>0 and k0: sigma(k) = sigma(n-th element of S)).

A110928 Pairs of distinct numbers m and n, m

Original entry on oeis.org

6, 7, 24, 26, 30, 35, 40, 47, 66, 77, 78, 91, 102, 119, 114, 133, 120, 130, 120, 141, 130, 141, 136, 157, 138, 161, 150, 175, 168, 182, 174, 203, 186, 215, 186, 217, 215, 217, 222, 259, 230, 249, 246, 287, 258, 301, 264, 286, 280, 282, 280, 329, 282, 329, 318
Offset: 1

Views

Author

Walter Kehowski, Sep 23 2005

Keywords

Comments

There do not appear to be any pairs (m,n) such that sigma_k(m)=sigma_k(n) for k>2.
For sigma_3, the first pair is (184926, 194315). Other terms may be found in A131907 and A131908. See A158915.

Examples

			sigma_2(30)=1^1+2^2+3^2+5^2+6^2+10^2+15^2+30^2=1300 and sigma_2(35)=1^2+5^2+7^2+35^2=1300.

Links

Crossrefs

Cf. A001157, A002025, A002046, A063990.
Cf. A110926, A110927, A110929.

Programs

  • Maple
    with(numtheory); sigmap := proc(p,n) convert(map(proc(z) z^p end, divisors(n)),`+`) end; SA2:=[]: for z from 1 to 1 do for m to 1500 do M:=sigmap(2,m); for n from m+1 to 1500 do N:=sigmap(2,n); if N=M then SA2:=[op(SA2),[m,n,N]] fi od od od; SA2; select(proc(z) z[1]<=1000 end, SA2); #just to shorten it a bit
  • Mathematica
    a[n_] := Module[{s = DivisorSigma[2, n], ans = {}}, kmax = Ceiling[Sqrt[s]]; Do[If[DivisorSigma[2, k] == s, AppendTo[ans, k]], {k, n + 1, kmax}]; ans];  s = {}; Do[v = a[n]; Do[s = Join[s, {n, v[[k]]}], {k, 1, Length[v]}], {n, 1, 400}]; s (* Amiram Eldar, Sep 08 2019 *)

Formula

sigma_2(m)=sigma_2(n), m

A131907 Integers for which a smaller positive integer exists which has the same sum of cubes of its divisors.

Original entry on oeis.org

194315, 295301, 2953010, 1181204, 1476505, 1886920, 2067107, 2362408, 2526095, 2953010, 3248311, 3691985, 3838913, 4134214, 4469245, 4724816, 5020117, 5610719, 5635135, 5906020, 6023765, 6496622, 6791923, 7382525, 7677826
Offset: 1

Views

Author

Peter Pein (petsie(AT)dordos.net), Jul 26 2007, Jul 28 2007

Keywords

Comments

If i < j and A001158(i) = A001158(j) then j is in this sequence. - Jason Yuen, Sep 22 2024

Examples

			194315 is in this sequence because A001158(184926) = A001158(194315) = 7401260364550416.

Links

Crossrefs

Cf. A001158, A069822, A131902-A131908.

Programs

  • Mathematica
    First@Transpose[Reap[For[n = 1, n <= 5*10^6, n++, (If[Head[ #1] === tmp, #1 = n, Sow[{n, #1}]] & )[ tmp[DivisorSigma[3, n]]]]][[2, 1]]]

Formula

n-th element of {x>0: there exists a k with 0

Extensions

More terms from Max Alekseyev and Daniel Lichtblau (danl(AT)wolfram.com), Jul 28 2007

A131905 Integers x such that sigma_2(k)=sigma_2(x) for some 0A001157=sigma_2 is the sum of squares of divisors.

Original entry on oeis.org

7, 26, 35, 47, 77, 91, 119, 130, 133, 141, 157, 161, 175, 182, 203, 215, 217, 249, 259, 282, 286, 287, 301, 329, 371, 385, 413, 423, 427, 434, 442, 455, 469, 471, 494, 497, 511, 517, 553, 581, 595, 598, 611, 623, 650, 651, 665, 679, 707, 721, 749, 754, 763, 785
Offset: 1

Views

Author

Peter Pein (petsie(AT)dordos.net), Jul 26 2007

Keywords

Examples

			This sequence contains 35, because sigma_2(35) = 1^2+5^2+7^2+35^2 = 1+25+49+1225 = 1300, and the sum of the squares of the divisors of 30<35 is sigma_2(30) = 1+4+9+25+36+100+225+900 = 1300.

Links

Crossrefs

Cf. A069822, A131902-A131908.

Programs

  • Maple
    N:= 100: # to get a(1)..a(N)
count:= 0: Res:= NULL:
for n from 1 while count < N do
  v:= numtheory:-sigma[2](n);
  if assigned(V[v]) then count:= count+1; Res:= Res, n;
  else V[v]:= n
  fi
od:
Res; # Robert Israel, Mar 30 2018
  • Mathematica
    Clear[tmp]; First@Transpose[ Function[n, (If[Head[ #1] === tmp, #1 = n; Unevaluated[Sequence[]], {n, #1}] & )[tmp[DivisorSigma[2, n]]]] /@ Range[500]]
Module[{nn=800,ds2,c},ds2=DivisorSigma[2,Range[nn]];Table[c=TakeDrop[Take[ds2,n],-1];If[ MemberQ[c[[2]],c[[1,1]]],n,Nothing],{n,nn}]] (* Harvey P. Dale, May 22 2024 *)
  • PARI
    isok(n) = {sn = sigma(n,2); for (k=1, n-1, if (sigma(k,2) == sn, return (1)););} \\ Michel Marcus, Apr 03 2015

Formula

a(n) = n-th element of {x: there exists some k with 0A001157=sigma_2 is the sum of squares of divisors.

Extensions

a(37)-a(54) from Michel Marcus, Apr 03 2015
Edited by Danny Rorabaugh, Apr 03 2015

A131903 Integers x such that d(k)=d(x) for some 0A000005 is the number of divisors.

Original entry on oeis.org

3, 5, 7, 8, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78
Offset: 1

Views

Author

Peter Pein (petsie(AT)dordos.net), Jul 26 2007

Keywords

Comments

Complement of A007416. - T. D. Noe, Jul 26 2007

Examples

			This sequence contains 8 because 8 has |{1,2,4,8}|=4 divisors and 6<8 has |{1,2,3,6}|=4 divisors.

Crossrefs

Cf. A069822, A131902-A131908.

Programs

  • Mathematica
    Clear[tmp]; Function[n, If[Head[ #1] === tmp, #1 = n; Unevaluated[Sequence[]], n] & [tmp[DivisorSigma[0, n]]]] /@ Range[64]
  • PARI
    isok(n) = {my(nd = numdiv(n)); for (k=1, n-1, if (numdiv(k) == nd, return (1)););}

Formula

a(n) = n-th element of the set {x>0 : there exists a k with 0A000005 is the number of divisors.

Extensions

a(54)-a(67) from Michel Marcus, Apr 03 2015
Edited by Danny Rorabaugh, Apr 03 2015

A131904 Smallest positive integer k with the same number of divisors as the n-th integer for which such a k exists.

Original entry on oeis.org

2, 2, 2, 6, 4, 6, 2, 2, 6, 6, 2, 12, 2, 12, 6, 6, 2, 4, 6, 6, 12, 2, 24, 2, 12, 6, 6, 6, 2, 6, 6, 24, 2, 24, 2, 12, 12, 6, 2, 4, 12, 6, 12, 2, 24, 6, 24, 6, 6, 2, 2, 6, 12, 6, 24, 2, 12, 6, 24, 2, 60, 2, 6, 12, 12, 6, 24, 2, 48, 16, 6, 2, 60, 6, 6, 6, 24, 2
Offset: 1

Views

Author

Peter Pein (petsie(AT)dordos.net), Jul 26 2007

Keywords

Examples

			a(4)=6 because A131903(4)=8, which has four divisors, and 6 is the least positive integer with four divisors

Crossrefs

Cf. A069822, A131902-A131908.

Programs

  • Mathematica
    Clear[tmp]; Function[n, If[Head[ #1] === tmp, #1 = n; Unevaluated[Sequence[]], # ] & [tmp[DivisorSigma[0, n]]]] /@ Range[64]
  • PARI
    lista(nn) = {for (n=1, nn, my(nd = numdiv(n)); for (k=1, n-1, if (numdiv(k) == nd, print1(k, ", "); break);););} \\ Michel Marcus, Apr 03 2015

Formula

a(n)=min(k>0, k has the same number of divisors as A131903(n))

Extensions

More terms from Michel Marcus, Apr 03 2015

A131906 Smallest positive integer k with the same sum of squares of divisors as the n-th integer for which such a k exists.

Original entry on oeis.org

6, 24, 30, 40, 66, 78, 102, 120, 114, 120, 136, 138, 150, 168, 174, 186, 186, 230, 222, 280, 264, 246, 258, 280, 318, 330, 354, 360, 366, 430, 408, 390, 402, 408, 456, 426, 438, 440, 474, 498, 510, 552, 520, 534, 600, 645, 570, 582, 606, 618, 642, 696, 654, 680
Offset: 1

Views

Author

Peter Pein (petsie(AT)dordos.net), Jul 26 2007

Keywords

Examples

			a(5)=66 because A131905(5)=77 and the sum of the squares of the divisors of 77 is 1+49+121+5929=6100 and the sum of the squares of the divisors of 66 is 1+4+9+36+121+484+1089+4356=6100 and there is no smaller positive integer with this squaresum of its divisors

Crossrefs

Cf. A069822, A131902-A131908.

Programs

  • Mathematica
    Clear[tmp]; Last@Transpose[ Function[n, (If[Head[ #1] === tmp, #1 = n; Unevaluated[Sequence[]], {n, #1}] & )[tmp[DivisorSigma[2, n]]]] /@ Range[500]]
  • PARI
    lista(nn) = {for (n=1, nn, my(sn = sigma(n,2)); for (k=1, n-1, if (sigma(k, 2) == sn, print1(k, ", "); break);););} \\ Michel Marcus, Apr 03 2015

Formula

a(n)=min(k>0, k has the same sum of squares of divisors as A131905(n))

Extensions

More terms from Michel Marcus, Apr 03 2015

A158915 Least number k such that there is an m > k with sigma_n(k) = sigma_n(m).

Original entry on oeis.org

2, 6, 6, 184926
Offset: 0

Views

Author

T. D. Noe, Mar 30 2009

Keywords

Comments

It is easy to verify the first three terms using sequences A000005, A000203, and A001157. The first term of A131908 gives a(3). The a(4) term is greater than 10^8. Does a(n) exist for all n?

Examples

			sigma_0(2)=sigma_0(3); sigma_1(6)=sigma_1(12); sigma_2(6)=sigma_2(7); sigma_3(184926)=sigma_3(194315).
Showing 1-8 of 8 results.

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