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 1-10 of 12 results. Next

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

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)).

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.

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

A275987 Least k such that sigma(k) = sigma(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 6, 12, 13, 14, 14, 16, 10, 18, 19, 20, 21, 22, 14, 24, 16, 20, 27, 28, 29, 30, 21, 32, 33, 34, 33, 36, 37, 24, 28, 40, 20, 42, 43, 44, 45, 30, 33, 48, 49, 50, 30, 52, 34, 54, 30, 54, 57, 40, 24, 60, 61, 42, 63, 64, 44, 66, 67, 68, 42, 66, 30, 72, 73, 74, 48
Offset: 1

Views

Author

Altug Alkan, Aug 15 2016

Keywords

Examples

			a(11) = 6 because sigma(6) = sigma(11) = 12.

Links

Crossrefs

Cf. A000203, A069822, A286603.

Programs

  • Mathematica
    With[{nn = 76}, With[{s = Values@ PositionIndex@ Array[DivisorSigma[1, #] &, nn]}, Array[s[[FirstPosition[s, #][[1]], 1 ]] &, nn]]] (* Michael De Vlieger, Nov 16 2017 *)
  • PARI
    a(n) = {my(k = 1); while(sigma(k) != sigma(n), k++); k; }
  • PARI
    a(n) = invsigmaMin(sigma(n)); \\ Amiram Eldar, Dec 20 2024, using Max Alekseyev's invphi.gp

A296214 Numbers k for which there is at least one x < k such that phi(x) = phi(k).

Original entry on oeis.org

2, 4, 6, 8, 9, 10, 12, 14, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 86, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115
Offset: 1

Views

Author

Antti Karttunen, Dec 08 2017

Keywords

Comments

Numbers k for which A081373(k) > 1.
Apart from the initial term 2, this is the complement of union of A000040 (primes) and A069823.

Links

Crossrefs

Cf. A000010, A069822, A069823, A081373.
Cf. A296087 (a subsequence).

Programs

  • PARI
    for(n=1,200,y=0;s=eulerphi(n);for(k=1,(n-1),if(eulerphi(k)==s,y=1;break)); if(y,print1(n,",")));
  • PARI
    is(k) = invphiMin(eulerphi(k)) < k; \\ Amiram Eldar, Nov 15 2024, using Max Alekseyev's invphi.gp

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

A296087 Numbers n such that there is k < n for which A003557(k) = A003557(n), A048250(k) = A048250(n) and A173557(k) = A173557(n).

Original entry on oeis.org

15265, 27962, 30217, 30530, 45795, 50541, 54379, 54905, 57598, 60434, 61060, 64255, 66526, 72357, 72713, 89585, 90651, 91590, 101082, 101949, 108758, 109810, 120868, 122120, 128510, 136555, 137385, 137883, 138761, 144714, 145426, 149739, 151085, 152633, 161386, 163137, 164715, 166315, 179170, 181302, 181543, 182942
Offset: 1

Views

Author

Antti Karttunen, Dec 08 2017

Keywords

Comments

Because Euler phi(n) = A000010(n) = A003557(n) * A173557(n), Dedekind psi(n) = A001615(n) = A003557(n) * A048250(n), and because also sigma(n) (A000203) can be computed from those three elements (see A291750), these numbers form also a subset of the positions of such duplicated occurrences of values computed for those functions. See for example A069822 and A296214.
a(11) = 61060 is the first term that is not squarefree.

Examples

			15265 is a term because A003557(15265) = 1 = A003557(15169), A048250(15265) = 19008 = A048250(15169), A173557(15265) = 11760 = A173557(15169).
27962 is a term because A003557(27962) = 1 = A003557(26355), A048250(27962) = 48384 = A048250(26355), A173557(27962) = 12000 = A173557(26355).

Links

Crossrefs

Cf. A000010, A001615, A003557, A048250, A173557.
Subsequence of A069822 and of A296214.

Programs

  • PARI
    search_up_to = (2^23);
A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = max(0,f[i, 2]-1)); factorback(f); };
A048250(n) = if(n<1, 0, sumdiv(n, d, if(core(d)==d, d)));
A173557(n) = my(f=factor(n)[, 1]); prod(k=1, #f, f[k]-1); \\ This function from Michel Marcus, Oct 31 2017
Anotsubmitted1(n) = (1/2)*(2 + ((A003557(n)+A173557(n))^2) - A003557(n) - 3*A173557(n));
Akaikki3(n) = (1/2)*(2 + ((A048250(n)+Anotsubmitted1(n))^2) - A048250(n) - 3*Anotsubmitted1(n));
om = Map(); m = 0; i=0; for(n = 1, search_up_to, k = Akaikki3(n); if(!mapisdefined(om,k), mapput(om,k,n), i++; write("b296087.txt", i, " ", n)));

A124141 Numbers k such that there is a number m < k satisfying A000203(k) = A000203(m) = m + k - gcd(m,k).

Original entry on oeis.org

38, 92, 153, 284, 332, 459, 494, 885, 956, 1035, 1358, 1784, 2295, 2528, 2678, 5434, 5607, 6027, 6255, 7564, 7928, 8235, 8648, 9729, 10325, 10413, 12008, 14104, 15813, 16198, 17794, 22712, 22936, 23247, 27082, 27626, 28917, 30938, 33082, 34688, 37790, 37816
Offset: 1

Views

Author

Yasutoshi Kohmoto, Dec 01 2006

Keywords

Examples

			38 is in the sequence because A000203(24) = A000203(38) = 60 = 24 + 38 - gcd(24,38).

Links

Crossrefs

Cf. A000203, A124140.
Subsequence of A069822.

Programs

  • PARI
    isok(k) = {my(sk = sigma(k)); for (m=1, k, if ((sk == sigma(m)) && (sk == m + k - gcd(m,k)), return (1));); return (0);} \\ Michel Marcus, Oct 27 2019
  • PARI
    is(k) = {my(s = sigma(k), v = invsigma(s)); for(i = 1, #v, if(v[i] < k && s == v[i] + k - gcd(v[i], k), return(1))); 0;} \\ Amiram Eldar, Dec 20 2024, using Max Alekseyev's invphi.gp

Extensions

Edited by Stefan Steinerberger, Aug 14 2007
More terms from Jinyuan Wang, Feb 07 2022

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
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