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.

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A325654 Numbers m with a divisor d satisfying sigma(d) = 2*m.

Original entry on oeis.org

6, 28, 496, 8128, 60480, 65520, 4357080, 33550336, 47139840, 91065600, 285981696, 2758909440, 8589869056, 87722956800, 132867440640, 137438691328, 306007080960, 806062473216, 1409150457792, 363485766938112, 12177456042320640, 29884246553283840
Offset: 1

Views

Author

Jaroslav Krizek, May 12 2019

Keywords

Comments

Even perfect numbers from A000396 are terms.
Numbers of the form A007691(k)*A054030(k)/2 when A054030(k) is even.
Subsequence of A323652.
Numbers of the form sigma(A325637(k))/2. - Jinyuan Wang, Jun 09 2019

Examples

			60480 is a term because 30240 divides 60480 and sigma(30240) = 120960 = 2*60480.

Links

Crossrefs

Cf. A000203, A000396, A007691, A054030, A323652, A325637.

Programs

  • Magma
    [n: n in [1..100000] | #[d: d in Divisors(n) | SumOfDivisors(d) eq 2*n] gt 0];
  • PARI
    isok(n) = fordiv(n, d, if (sigma(d) == 2*n, return(1))); 0; \\ Michel Marcus, May 12 2019

Extensions

More terms from Jinyuan Wang, Jun 09 2019

A132629 Sigma(n)/Sum_digits(n) for n such that sigma(n) is divisible by Sum_digits(n).

Original entry on oeis.org

1, 2, 18, 6, 4, 2, 21, 9, 10, 24, 8, 8, 6, 16, 12, 14, 28, 12, 12, 9, 9, 5, 8, 26, 217, 51, 72, 26, 42, 32, 11, 108, 62, 40, 18, 120, 28, 32, 63, 56, 27, 24, 32, 21, 18, 19, 62, 26, 54, 24, 24, 12, 32, 30, 16, 36, 21, 26
Offset: 0

Views

Author

Paolo P. Lava and Giorgio Balzarotti, Aug 24 2007

Keywords

Examples

			n=147 -> sigma(n)=1+3+7+21+49+147=228 Sum_digits(n)=1+4+7=12 -> 228/12 = 19
n=177 -> sigma(n)=1+3+59+177=240 Sum_digits(n)=1+7+7=15 -> 240/15 = 16

Crossrefs

Cf. A007691, A054030, A132628.

Programs

  • Maple
    with(numtheory); P:=proc(n) local a,i,j,k,w; for i from 1 by 1 to n do w:=0; k:=i; while k>0 do w:=w+k-(trunc(k/10)*10); k:=trunc(k/10); od; j:=sigma(i)/w; if trunc(j)=j then print(j); fi; od; end: P(200);
  • Mathematica
    Select[Table[DivisorSigma[1,n]/Total[IntegerDigits[n]],{n,300}], IntegerQ] (* Harvey P. Dale, Oct 01 2015 *)

A232702 Sigma(2*m)/m for m such that sigma(2*m) is divisible by m (these m are in A227302).

Original entry on oeis.org

3, 4, 5, 4, 6, 4, 6, 7, 7, 4, 7, 8, 8, 6, 8, 9, 9, 7, 8, 4, 8, 5, 8, 7, 6, 8, 6, 9, 4, 5, 10, 7, 10, 8, 6, 7, 9, 8, 4, 8, 9, 8, 10, 8, 10, 11, 10, 8, 10, 10, 9, 8, 8, 9
Offset: 1

Views

Author

Alex Ratushnyak, Nov 28 2013

Keywords

Crossrefs

Cf. A227302, A054030.

Programs

  • PARI
    for(n=1,10^8,s=sigma(2*n);if(s%n==0,print1(s/n,","))) \\ Ralf Stephan, Nov 30 2013

Extensions

a(30)-a(54) from Jinyuan Wang, Mar 03 2020

A318781 A188999(m)/m for the integers m such that A188999(m) is divisible by m, where A188999 is the bi-unitary sigma function.

Original entry on oeis.org

1, 2, 2, 2, 3, 3, 3, 3, 3, 4, 3, 3, 4, 3, 4, 3, 4, 3, 3, 4, 4, 3, 3, 3, 4, 4, 4, 3, 3, 4, 3, 4, 4, 3, 4, 3, 3, 4, 4, 4, 4, 3
Offset: 1

Views

Author

Michel Marcus, Sep 03 2018, following a suggestion from Felix Fröhlich

Keywords

Comments

10496266260480 is a term of A189000 and it is the smallest known value x such that A188999(x)/x is 5.

Crossrefs

Cf. A188999 (bi-unitary sigma), A189000 (multiply perfect for bi-unitary sigma).
Cf. A054030 (analog for sigma), A007691 (multiply perfect for sigma).

Programs

  • PARI
    a188999(n) = my(f = factor(n)); for (i=1, #f~, p = f[i, 1]; e = f[i, 2]; f[i, 1] = if (e % 2, (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1) -p^(e/2)); f[i, 2] = 1; ); factorback(f) \\ after Michel Marcus in A189000
is_a189000(n) = ! frac(a188999(n)/n) \\ after Michel Marcus in A189000
for(n=1, oo, if(is_a189000(n), print1(a188999(n)/n, ", "))) \\ Felix Fröhlich, Sep 03 2018

Formula

a(n) = A188999(A189000(n))/A189000(n).

Extensions

a(33)-a(42) from Giovanni Resta, Sep 03 2018
Previous Showing 11-14 of 14 results.

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