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 1-6 of 6 results.

A131908 Smallest positive integer k with the same sum of cubes of divisors as the n-th integer for which such a k exists.

Original entry on oeis.org

184926, 291741, 583482, 1166964, 1458705, 1880574, 2042187, 2333928, 2404038, 2917410, 3209151, 3513594, 3792633, 4084374, 4253298, 4667856, 4959597, 5543079, 5362854, 5834820, 5732706, 6418302, 6710043, 7293525, 7585266
Offset: 1

Views

Author

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

Keywords

Examples

			a(1)=184926 because A131907(1)=194315 with the sum of cubes of its divisors being 7401260364550416. This is the same as the sum of cubes of divisors of 184926 and there is no positive integer less than 184926 with this cubesum of its divisors

Links

Crossrefs

Cf. A069822, A131902-A131907.

Programs

  • Mathematica
    Last@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

a(n)=min(k>0, k has the same sum of cubes of divisors as A131907(n))

Extensions

More terms from Max Alekseyev, Jul 28 2007

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
Showing 1-6 of 6 results.

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