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.

A115920 Numbers k such that the digits of sigma(k) are a permutation of those of k, in base 10.

Original entry on oeis.org

1, 69, 258, 270, 276, 609, 639, 2391, 2556, 2931, 3409, 3678, 3679, 4291, 5092, 6937, 8251, 10231, 12087, 12931, 15480, 16387, 20850, 22644, 22893, 24369, 26145, 26442, 27846, 28764, 29880, 29958, 30823, 31812, 32658, 34207, 34758
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Comments

There is some m > 1 such that a(n) > m*n for all n > 1. This follows from the positive density of numbers k such that sigma(k)/k > 10. - Charles R Greathouse IV, Sep 07 2012

Examples

			sigma(10231) = 11032, sigma(31812) = 81312.

Links

Crossrefs

Cf. A000203, A114065, A115921, A175795.

Programs

  • Mathematica
    Select[Range[35000],Sort[IntegerDigits[#]]==Sort[ IntegerDigits[ DivisorSigma[ 1,#]]]&] (* Harvey P. Dale, May 09 2013 *)
  • PARI
    isok(n) = vecsort(digits(n)) == vecsort(digits(sigma(n))); \\ Michel Marcus, Dec 13 2015 and May 27 2018
  • Python
    from sympy import divisor_sigma
A115920_list = [n for n in range(1,10**4) if sorted(str(divisor_sigma(n))) == sorted(str(n))] # Chai Wah Wu, Dec 13 2015

A115921 Numbers k such that the decimal digits of phi(k) are a permutation of those of k.

Original entry on oeis.org

1, 21, 63, 291, 502, 2518, 2817, 2991, 4435, 5229, 5367, 5637, 6102, 6174, 6543, 6822, 7236, 7422, 8022, 8541, 8982, 17631, 18231, 18261, 20301, 20518, 20617, 21058, 22471, 22851, 25196, 25918, 27615, 29817, 34816, 35683, 43218, 44305
Offset: 1

Views

Author

Giovanni Resta, Feb 06 2006

Keywords

Comments

Contains A069215 and A113781; is itself a subsequence of A082060. - M. F. Hasler, Nov 28 2007
There is some m > 1 such that a(n) > m*n for all n > 1. This follows from the positive density of numbers n such that n/phi(n) > 10. - Charles R Greathouse IV, Sep 07 2012

Examples

			phi(20301) = 13200, phi(6543) = 4356.

Links

Crossrefs

Cf. A069215, A113781, A082060, A115920, A114065, A175795.

Programs

  • Mathematica
    Select[Range[45000],Sort[IntegerDigits[EulerPhi[#]]]==Sort[IntegerDigits[#]]&] (* Harvey P. Dale, Jul 25 2018 *)
  • PARI
    for(n=1,10^5,if(vecsort(Vecsmall(Str(n)))==vecsort(Vecsmall(Str(eulerphi(n)))),print1(n", "))) \\ M. F. Hasler, Nov 28 2007
  • Python
    from sympy import totient
A115921_list = [n for n in range(1,10**4) if sorted(str(totient(n))) == sorted(str(n))] # Chai Wah Wu, Dec 13 2015

Extensions

Edited by M. F. Hasler, Nov 28 2007

A175795 Numbers n such that the digits of sigma(n) are exactly the same (albeit in different order) as the digits of phi(n), in base 10.

Original entry on oeis.org

1, 65, 207, 1769, 2066, 2771, 3197, 4330, 4587, 4769, 4946, 5067, 6443, 6623, 6989, 7133, 8201, 9263, 11951, 12331, 13243, 16403, 17429, 17441, 21416, 22083, 23161, 24746, 27058, 27945, 28049, 28185, 28451, 29111, 30551, 31439, 32554, 32566, 32849, 33715
Offset: 1

Views

Author

Michel Lagneau, Sep 06 2010

Keywords

Examples

			2771 is in the sequence because sigma(2771) = 2952, phi(2771) = 2592

Links

Crossrefs

Cf. A000010 (Euler totient function), A000203 (sigma function), A115920, A115921, A114065.

Programs

  • Mathematica
    okQ[n_] := Module[{idn = IntegerDigits[DivisorSigma[1,n]]}, Sort[idn] == Sort[IntegerDigits[EulerPhi[n]]]]; Select[Range[40000], okQ]
  • PARI
    isok(n) = (de = digits(eulerphi(n))) && (ds = digits(sigma(n))) && (vecsort(de) == vecsort(ds)); \\ Michel Marcus, Dec 13 2015
  • Python
    from sympy import totient, divisor_sigma
A175795_list = [n for n in range(1,10**4) if sorted(str(divisor_sigma(n))) == sorted(str(totient(n)))] # Chai Wah Wu, Dec 13 2015

A258786 Numbers n whose sum of anti-divisors is a permutation of their digits.

Original entry on oeis.org

5, 8, 41, 56, 64, 358, 614, 946, 1092, 1382, 1683, 2430, 2683, 2734, 2834, 2945, 3045, 3067, 3602, 4056, 4286, 5186, 5784, 6874, 7251, 8104, 8546, 9264, 12881, 14028, 14384, 15258, 17386, 21103, 22044, 23331, 24434, 24603, 25346, 26420, 26822, 26845, 27024, 27232
Offset: 1

Views

Author

Paolo P. Lava, Jun 10 2015

Keywords

Comments

A073930 is a subset of this sequence.

Examples

			Anti-divisors of 5 are 2, 3 whose sum is 5.
Anti-divisors of 41 are 2, 3, 9, 27 whose sum is 41.
Anti-divisors of 64 are 3, 43 whose sum is 46 that is a permutation of the digit of 64.

Links

Crossrefs

Cf. A066417, A073930, A114065, A115920, A115921, A175795, A225902.

Programs

  • Maple
    with(numtheory):P:=proc(q) local a,b,j,k,ok,n,p;
for n from 1 to q do k:=0; j:=n;
while j mod 2 <> 1 do k:=k+1; j:=j/2; od;
a:=sigma(2*n+1)+sigma(2*n-1)+sigma(n/2^k)*2^(k+1)-6*n-2;
if ilog10(n)=ilog10(a) then j:=sort(convert(n,base,10)); a:=sort(convert(a,base,10)); ok:=1;
for k from 1 to nops(a) do if j[k]<>a[k] then ok:=0; break;
fi; od; if ok=1 then print(n); fi; fi; od; end: P(10^9);
  • Mathematica
    ad[n_] := Cases[Range[2, n - 1], ?(Abs[Mod[n, #] - #/2] < 1 &)]; Select[Range@ 5000, SameQ[DigitCount@ #, DigitCount[Total[ad@ #]]] &] (* _Michael De Vlieger, Jun 10 2015 *)
  • Python
    from sympy.ntheory.factor_ import antidivisors
A258786_list = [n for n in range(1,10**5) if sorted(str(n)) == sorted(str(sum(antidivisors(n))))] # Chai Wah Wu, Jun 11 2015
Showing 1-4 of 4 results.

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

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