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 11 results. Next

A138829 Bisection of imperfect numbers A132999.

Original entry on oeis.org

1, 3, 5, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117
Offset: 1

Views

Author

Omar E. Pol, Apr 06 2008

Keywords

Links

Crossrefs

Cf. A000396, A099057, A099058, A132999.

Programs

  • Mathematica
    CoefficientList[Series[(-x^14 + x^13 - x^4 + x^3 + x + 1)/(x^2 - 2*x + 1), {x, 0, 50}], x] (* G. C. Greubel, Feb 21 2017 *)
  • PARI
    x='x+O('x^50); Vec((-x^14 + x^13 - x^4 + x^3 + x + 1)/(x^2 - 2*x + 1)) \\ G. C. Greubel, Feb 21 2017

Formula

G.f.: (-x^14 + x^13 - x^4 + x^3 + x + 1)/(x^2 - 2*x + 1). - Alexander R. Povolotsky, Apr 06 2008

A138830 Bisection of imperfect numbers A132999.

Original entry on oeis.org

2, 4, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118
Offset: 1

Views

Author

Omar E. Pol, Apr 06 2008

Keywords

Comments

The conjectured g.f. (-x^14+x^13-x^3+x^2+2)/(x^2-2*x+1) that a correspondent suggested is wrong and fails after roughly 240 terms [R. J. Mathar, Jun 15 2009]

Crossrefs

Cf. A000396, A099057, A099058, A132999, A138829.

A054027 Numbers that do not divide their sum of divisors.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74
Offset: 1

Views

Author

Asher Auel, Jan 19 2000

Keywords

Comments

Does not contain numbers like 1, 120, 672, 30240, 32760, 523776,.. which are in A132999. - R. J. Mathar, Jun 13 2025

Crossrefs

Complement of A007691. Cf. A000203, A054024.

Programs

  • Maple
    with(numtheory): [seq(`if`(sigma(i) mod i <> 0,i,print( )),i=1..90)];
  • Mathematica
    Select[Range[100],!Divisible[DivisorSigma[1,#],#]&] (* Harvey P. Dale, May 29 2019 *)
  • PARI
    isok(m) = (sigma(m) % m) != 0; \\ Michel Marcus, Jun 20 2021

A132943 Concatenation of first n imperfect numbers.

Original entry on oeis.org

1, 12, 123, 1234, 12345, 123457, 1234578, 12345789, 1234578910, 123457891011, 12345789101112, 1234578910111213, 123457891011121314, 12345789101112131415, 1234578910111213141516, 123457891011121314151617
Offset: 1

Views

Author

Omar E. Pol, Oct 16 2007

Keywords

Links

Crossrefs

Imperfect numbers: A132999. Cf. A000396, A007908, A019518, A132928.

A133017 Decimal expansion of the constant formed by concatenating the imperfect numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 9, 3, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 4, 0, 4, 1, 4, 2, 4, 3, 4, 4, 4, 5
Offset: 0

Views

Author

Omar E. Pol, Oct 20 2007

Keywords

Comments

A theorem of Copeland & Erdős proves that this constant is 10-normal. - Charles R Greathouse IV, Feb 06 2015

Examples

			0.12345789101112131415161718192021222324252627293031...

Links

Crossrefs

Cf. A007376, A033307, A033308, A079718. Imperfect numbers: A132999.

Programs

  • Mathematica
    Flatten[IntegerDigits/@Table[If[PerfectNumberQ[n],Nothing,n],{n,70}]] (* Harvey P. Dale, Dec 16 2021 *)
  • PARI
    print1(1); for(n=2, 45, if(sigma(n,-1)!=2, d=digits(n); for(i=1, #d, print1(", "d[i])))) \\ Charles R Greathouse IV, Feb 06 2015

A134733 Concatenation of next n imperfect numbers.

Original entry on oeis.org

1, 23, 457, 891011, 1213141516, 171819202122, 23242526272930, 3132333435363738, 394041424344454647, 48495051525354555657, 5859606162636465666768, 697071727374757677787980
Offset: 1

Views

Author

Omar E. Pol, Nov 12 2007

Keywords

Links

Crossrefs

Cf. A053067, A132943, A133013. Imperfect numbers: A132999.

Programs

  • Mathematica
    Module[{nn=120,in,len},in=Select[Range[nn],DivisorSigma[1,#]!=2#&];len= Floor[x/.Last[Solve[(x(x+1))/2==Length[in]]]];FromDigits[Flatten[ IntegerDigits/@ #]]&/@TakeList[in,Range[len]]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 01 2020 *)

A133012 Even imperfect numbers.

Original entry on oeis.org

2, 4, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118
Offset: 1

Views

Author

Omar E. Pol, Oct 20 2007

Keywords

Crossrefs

Cf. A005843, A014445, A014494. Imperfect numbers: A132999.

A133016 Even imperfect numbers, divided by 2.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64
Offset: 1

Views

Author

Omar E. Pol, Oct 20 2007

Keywords

Crossrefs

Cf. A005843, A014335, A014445, A014494, A026503, A028334. Imperfect numbers: A132999.

A138884 Numbers that are not even superperfect numbers.

Original entry on oeis.org

1, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65
Offset: 1

Views

Author

Omar E. Pol, Apr 06 2008

Keywords

Comments

Also, numbers that are not superperfect numbers A019279, if there are no odd superperfect numbers.

Crossrefs

Even superperfect numbers: A061652. Cf. A019279, A132999, A133398.

A272978 Numbers not in the range of the sum of perfect divisors function.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80
Offset: 1

Views

Author

Timothy L. Tiffin, Jul 13 2016

Keywords

Comments

Numbers which do not appear in A185351 or A187794; that is, there is no integer N whose sum of perfect divisors is equal to a(n) for any n.

Crossrefs

Cf. A000396, subsequence of A132999, A185351 (complement), A187794.
Showing 1-10 of 11 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.