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.

A088847 a(n) = sigma(A087979(n)) / phi(A087979(n)).

Original entry on oeis.org

1, 1, 3, 4, 4, 6, 6, 8, 9, 8, 10, 12, 12, 14, 15, 16, 16, 18, 18, 20, 21, 20, 22, 24, 25, 24, 27, 28, 28, 30, 30, 32, 33, 32, 35, 36, 36, 36, 39, 40, 40, 42, 42, 44, 45, 46, 46, 48, 49, 50, 51, 52, 52, 54, 55, 56, 57, 56, 58, 60, 60, 62, 63, 64, 65, 66, 66, 68, 69, 70, 70, 72, 72, 74, 75, 76, 77, 78, 79, 80, 81, 82, 82, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100
Offset: 1

Views

Author

Labos Elemer, Nov 17 2003

Keywords

Comments

Note that A087979(n) is a balanced number (A020492), hence this sequence is well-defined. - Max Alekseyev, Oct 10 2024
For all n, a(n) <= n and a(n) divides A299822(n). - Max Alekseyev, Oct 11 2024

Examples

			While A088830 includes special balanced numbers, A087979 does not include per definition. Nevertheless, it seems that A087979 consists only of balanced numbers. This is provable at least for special cases.

Crossrefs

Cf. A000010, A000203, A087979, A020492, A088830, A299822.

Extensions

a(24)-a(35) from Amiram Eldar, Dec 05 2019 (calculated from the data at A087979)
Terms a(36) onward from Max Alekseyev, Oct 10 2024

A088830 a(n) = Min{x : sigma(x) = n*phi(x), x is not a prime}, the least nonprime solutions to sigma(x) = n*phi(x); special balanced numbers.

Original entry on oeis.org

1, 35, 15, 14, 56, 6, 12, 42, 30, 168, 2580, 210, 630, 420, 840, 20790, 416640, 9240, 291060, 83160, 120120, 5165160, 1719277560, 43825320, 26860680, 277560360, 1304863560, 569729160, 587133466920, 16522145640, 33044291280, 563462139240, 1140028049160
Offset: 1

Views

Author

Labos Elemer, Nov 03 2003

Keywords

Comments

a(33) > 10^12. - Donovan Johnson, Sep 03 2013
a(34) <= 9015394227840, a(35) <= 1255683068640. - Giovanni Resta, May 08 2017

Links

Crossrefs

Compare A087979, which has a slightly different definition.
Cf. A074891, A000203, A000010, A087979, A020492.
Cf. A055234.

Programs

  • Mathematica
    ds[x_, de_] := DivisorSigma[1, x]-de*EulerPhi[x] a[n_] := Block[{m=1, s=ds[m, n]}, While[(s !=0||PrimeQ[m])&&!Greater[m, 100000], m++ ]; m]; Table[a[n], {n, 22}]

Formula

For n > 3, a(n) = A055234(n). - David Wasserman, Aug 18 2005

Extensions

More terms from David Wasserman, Aug 18 2005
a(32) from Donovan Johnson, Sep 03 2013
a(33) from Giovanni Resta, May 08 2017

A256527 a(n) is the least number k > 0 such that sigma(k) = phi(n*k).

Original entry on oeis.org

1, 1, 15, 3, 14, 6, 6, 42, 30, 42, 168, 210, 210, 420, 840, 20790, 20790, 9240, 9240, 83160, 120120, 3984120, 5165160, 43825320, 26860680, 43825320, 1304863560, 569729160, 569729160, 16522145640, 18176198040, 563462139240, 1140028049160, 3844800479520, 1255683068640, 65361608151840, 65361608151840, 65361608151840, 413956851628320, 1241870554884960, 1241870554884960
Offset: 1

Views

Author

Paolo P. Lava, Apr 01 2015

Keywords

Examples

			sigma(1)  = phi(1*1)  = 1;
sigma(1)  = phi(2*1)  = 1;
sigma(15) = phi(3*15) = 24;
sigma(3)  = phi(4*3)  = 4;
sigma(14) = phi(5*14) = 24;
sigma(6)  = phi(6*6)  = 12;
sigma(6)  = phi(7*6)  = 12;
sigma(42) = phi(8*42) = 96;
sigma(30) = phi(9*30) = 72; etc.

Crossrefs

Except for a(4), same as A087979.
Cf. A000010, A000203, A241762.

Programs

  • Maple
    with(numtheory): P:=proc(q) local k, n;
for n from 1 to q do for k from 1 to q do
if sigma(k)=phi(k*n) then lprint(n,k); break; fi;
od; od; end: P(10^5);
  • Mathematica
    Table[k = 1; While[DivisorSigma[1, k] != EulerPhi[n k], k++]; k, {n, 20}] (* Michael De Vlieger, May 28 2015 *)
  • PARI
    a(n) = {k=1; while(sigma(k) != eulerphi(n*k), k++); k;} \\ Michel Marcus, Apr 01 2015

Formula

For n >= 5, a(n) = A087979(n). - Conjectured by Manfred Scheucher, May 28 2015; proved by Max Alekseyev, Sep 29 2023

Extensions

a(21)-a(23) from Michel Marcus, Apr 01 2015
a(24)-a(26) from Jon E. Schoenfield, Jun 28 2015
a(27)-a(35) from Giovanni Resta, May 24 2016
a(36)-a(41) from Max Alekseyev, Oct 10 2024

A241762 a(n) is the least number k > 0 such that sigma(k/n) = phi(k).

Original entry on oeis.org

1, 2, 45, 12, 70, 36, 42, 336, 270, 420, 1848, 2520, 2730, 5880, 12600, 332640, 353430, 166320, 175560, 1663200, 2522520, 87650640, 118798680, 1051807680, 671517000, 1139458320, 35231316120, 15952416480, 16522145640, 495664369200, 563462139240, 18030788455680, 37620925622280, 130723216303680, 43948907402400
Offset: 1

Views

Author

Paolo P. Lava, Apr 28 2014

Keywords

Examples

			For n=11, the least number is 1848. In fact, sigma(1848/11) = phi(1848) = 480.

Crossrefs

Cf. A000010, A000203, A087979.

Programs

  • Maple
    with(numtheory): P:=proc(q) local k,n;
for k from 1 to q do for n from k by k to q do
if sigma(n/k)=phi(n) then print(n); break; fi;
od; od; end: P(10^5);
  • PARI
    for(k=1,29,n=0;for(i=1,2^64,if(sigma(i)==eulerphi(i*k),n=i*k;break)); print(k,"  ",n)) \\ Dana Jacobsen, May 02 2014
  • Perl
    use Math::Prime::Util qw/:all/; for $k (1..29) { $i=1; $i++ while divisor_sum($i) != euler_phi($i*$k); say "$k  ",$i*$k; } # Dana Jacobsen, May 02 2014

Formula

a(n) = n * A256527(n). - Max Alekseyev, Sep 29 2023

Extensions

a(22)-a(26) from Giovanni Resta, Apr 29 2014
a(27)-a(29) from Dana Jacobsen, May 02 2014
a(30)-a(35) from Max Alekseyev, Sep 29 2023
Showing 1-4 of 4 results.

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