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 1-3 of 3 results.

A066218 Numbers k such that sigma(k) = Sum_{j|k, j

Original entry on oeis.org

198, 608, 11322, 20826, 56608, 3055150, 565344850, 579667086, 907521650, 8582999958, 13876688358, 19244570848, 195485816050, 255701999358, 1038635009650, 1410759512050, 3308222326688, 6293446033554, 12859914783762, 15343909268584, 18359652610976, 19142664182226, 41584649258178, 45090324794034, 56293124233554
Offset: 1

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Author

Joseph L. Pe, Dec 17 2001

Keywords

Comments

I propose this generalization of perfect numbers: for an arithmetical function f, the "f-perfect numbers" are the n such that f(n) = sum of f(k) where k ranges over proper divisors of n. The usual perfect numbers are i-perfect numbers, where i is the identity function. This sequence lists the sigma-perfect numbers. It is not hard to see that the EulerPhi-perfect numbers are the powers of 2 and the d-perfect numbers are the squares of primes (d(n) denotes the number of divisors of n).
Problems: Find an expression generating sigma-perfect numbers. Are there infinitely many of these? Find other interesting sets generated by other f's. 3.
a(17) > 2*10^12. - Giovanni Resta, Jun 20 2013
Numbers k such that A296075(k) = 0. - Amiram Eldar, Apr 16 2024
No more terms < 10^14. - Jud McCranie, Nov 28 2024

Examples

			Proper divisors of 198 = {1, 2, 3, 6, 9, 11, 18, 22, 33, 66, 99}; sum of their sigma values = 1 + 3 + 4 + 12 + 13 + 12 + 39 + 36 + 48 + 144 + 156 = 468 = sigma(198).

Links

Crossrefs

Cf. A211779, A000203, A066229, A066230, A296075.

Programs

  • Mathematica
    f[ x_ ] := DivisorSigma[ 1, x ]; Select[ Range[ 1, 10^5 ], 2 * f[ # ] == Apply[ Plus, Map[ f, Divisors[ # ] ] ] & ]
  • PARI
    is(n)=sumdiv(n,d,sigma(d))==2*sigma(n) \\ Charles R Greathouse IV, Mar 09 2014

Formula

Integer n = p1^k1 * p2^k2 * ... * pm^km is in this sequence if and only if g(p1^k1)*g(p2^k2)*...*g(pm^km)=2, where g(p^k) = (p^(k+2)-(k+2)*p+k+1)/(p^(k+1)-1)/(p-1) for prime p and integer k>=1. - Max Alekseyev, Oct 23 2008

Extensions

More terms from Naohiro Nomoto, May 07 2002
a(7)-a(8) from Farideh Firoozbakht, Sep 18 2006
a(9)-a(13) from Donovan Johnson, Jun 25 2012
a(14)-a(16) from Giovanni Resta, Jun 20 2013
a(17)-a(25) from Jud McCranie, Nov 28 2024

A066230 f-perfect numbers, where f(m) = m - 1.

Original entry on oeis.org

1, 12, 196, 368, 1696, 30848, 437745, 2075648, 8341504, 33452032, 34355150848, 562949131337728, 2305842943715442688, 590295809173294678016
Offset: 1

Views

Author

Joseph L. Pe, Dec 18 2001

Keywords

Comments

f-perfect numbers are defined in A066218.
Equivalently, let g(n) = sigma(n)-n-d(n)+2, where d(n) is the number of divisors of n and sigma(n) is their sum. Then n is in the sequence if g(n)=n.
If 2^k - 2*k + 1 is prime (i.e. k in A301744), then 2^(k-1)*(2^k - 2*k + 1) is a term. The only known terms not of this form are 1, 196, and 437745. - Lambert Klasen (lambert.klasen(AT)gmx.net), Jul 31 2005; updated by Max Alekseyev, Jul 30 2025
If 2^(i + 1)-(2i + 1) is prime then n = 2^i*(2^(i + 1)-(2i + 1)) is in the sequence because sigma(n)-d(n) + 2 = (2^(i + 1)-1)*(2^(i + 1)-2i)-2(i + 1) + 2 = 2^(i + 1)*(2^(i + 1)-(2i + 1)) = 2n, so sigma(n)-n-d(n) + 2 = n. - Farideh Firoozbakht, Sep 18 2006

Examples

			f(12) = 11 = 0 + 1 + 2 + 3 + 5 = f(1) + f(2) + f(3) + f(4) + f(6), hence 12 is a term of the sequence.

Links

Crossrefs

Cf. A066218, A066511, A066229, A301744.

Programs

  • Mathematica
    g[ n_ ] := DivisorSigma[ 1, n ]-n-DivisorSigma[ 0, n ]+2; For[ n=1, True, n++, If[ g[ n ]==n, Print[ n ] ] ]

Extensions

Edited by Dean Hickerson, Jan 10 2002.
More terms from Jason Earls, May 14 2002
2 more terms from Farideh Firoozbakht, Sep 18 2006
a(11) from Donovan Johnson, Jun 25 2012
a(12)-a(14) from Max Alekseyev, Jul 11 2025

A066505 f-amicable numbers where f(n) = n+1.

Original entry on oeis.org

36, 62, 168, 326, 9936, 14056, 16198, 19862, 45304, 51910, 82662, 90152, 337688, 388102, 472902, 479672, 1970586, 2353756, 2969288, 3769942, 6319544, 8454886, 12276056, 13125574, 16783976, 17948854, 18818780, 20825882, 21738114, 22479040, 25960468, 31470614
Offset: 1

Views

Author

Joseph L. Pe, Jan 04 2002

Keywords

Comments

f-amicable pairs are defined similarly to f-perfect numbers in A066218. That is, a, b is a f-amicable pair if f(a) = D(b) and f(b) = D(a), where D(n) = sum_{k divides n, k
Pairs are (36,62), (14056,16198), (9936,19862), (45304,51910), (82662,90152) (337688,388102) and (472902,479672). The sequence shows them unbundled, then elements sorted according to size. - R. J. Mathar, Sep 07 2006, Dec 07 2006

Examples

			Proper divisors of 36 = {1, 2, 3, 4, 6, 9, 12, 18}. f applied to these divisors = {2, 3, 4, 5, 7, 10, 13, 19}; their sum = 63. So D(36) = f(62). proper divisors of 62 = {1, 2, 31}. f applied to these divisors = {2, 3, 32}; their sum = 37. So D(62) = f(36). Therefore 36, 62 is an f-amicable pair.

Links

Crossrefs

Cf. A066229, A066218.

Programs

  • Mathematica
    f[x_] := x + 1; d[x_] := Apply[ Plus, Map[ f, Divisors[ x] ] ] - f[ x]; m = Table[{x, y}, {x, 1, 1000}, {y, 1, 1000}]; Do[a = m[[i, j]]; If[ (a[[1]] < a[[2]]) && (f[a[[1]]] == d[a[[2]]]) && (f[a[[2]]] == d[a[[1]]]), Print[{i, j}]], {i, 1, 1000}, {j, 1, 1000}]

Extensions

More terms from John W. Layman, Nov 11 2002
More terms from R. J. Mathar, Sep 07 2006
a(17)-a(32) from Donovan Johnson, Jun 23 2012
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