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 13 results. Next

A216372 Sigma(n) values in A115920.

Original entry on oeis.org

1, 96, 528, 720, 672, 960, 936, 3192, 6552, 3912, 3904, 7368, 3976, 4912, 9520, 7936, 8512, 11032, 18720, 13192, 51480, 18736, 52080, 62244, 32928, 32496, 52416, 62244, 78624, 78624, 98280, 59928, 33208, 81312, 65328, 34720, 75348, 39904, 65340, 49632
Offset: 1

Views

Author

V. Raman, Sep 06 2012

Keywords

Links

Crossrefs

Cf. A115920, A000203.

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

A114065 Numbers k such that the digits of phi(k) and sigma(k) are permutations of those of k.

Original entry on oeis.org

1, 3014685, 21638943, 170726121, 207380169, 215341083, 233559801, 234511083, 321634251, 1620475083, 1982243007, 2019804093, 2084013063, 2185499607, 2410658685, 2653713819, 2741018409, 2859457041, 3018792645, 3075268041, 3148920504, 3701484126, 4071408255
Offset: 1

Views

Author

Giovanni Resta, Feb 13 2006

Keywords

Comments

Intersection of A115920 and A115921.

Examples

			sigma(3014685) = 5431680 and phi(3014685) = 1586304.

Links

Crossrefs

Cf. A115920, A115921, A175795.
Cf. A000010, A000203.

Programs

  • PARI
    isok(n) = (d = vecsort(digits(n))) && (ds = vecsort(digits(sigma(n)))) && (d == ds) && (de = vecsort(digits(eulerphi(n)))) && (ds == de); \\ Michel Marcus, Dec 13 2015
  • Python
    from sympy import totient, divisor_sigma
A114065_list = [n for n in range(1,10**7) if sorted(str(divisor_sigma(n))) == sorted(str(totient(n))) == sorted(str(n))] # Chai Wah Wu, Dec 13 2015

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

A307498 Numbers k such that the digits of k in base 10 are a permutation of those of k in some other base.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 21, 23, 31, 41, 42, 43, 46, 51, 53, 61, 62, 63, 71, 73, 81, 82, 83, 84, 86, 91, 93, 158, 191, 196, 227, 261, 265, 283, 316, 370, 371, 441, 445, 511, 518, 551, 774, 782, 825, 834, 882, 910, 911, 912, 913, 914, 915, 916, 917, 918
Offset: 1

Views

Author

Jinyuan Wang, Aug 05 2019

Keywords

Comments

Supersequence of A034294 and subsequence of A307498.
If the digits of k in base 10 is a permutation of m = (k in base b), 10^j < k < 10^(j+1), then 10^(j/(j+1)) < b < 10^((j+1)/j).
If k > 10, other base can only be 4, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 25, 26, 28, 37, 46, 55, 64, 73, 82.
The digits of k in base 10 is a permutation of k in base 82 iff k = 91.
The largest term is less than 10^25. See proof in A034294.

Examples

			13 in base 4 is 31, 227 in base 9 is 272.

Crossrefs

Cf. A034294, A115920, A308493.

Programs

  • PARI
    isok(k) = {my(v = vecsort(digits(k))); k < 10 || sum(j = 3, 82, vecsort(digits(k, j)) == v) > 1;}

A249899 Numbers k such that sigma(k) contains the same digits as k in base 10.

Original entry on oeis.org

1, 69, 211, 258, 270, 276, 433, 609, 639, 787, 877, 1021, 1201, 1231, 1255, 1291, 1321, 1433, 1621, 1721, 1787, 1877, 2011, 2111, 2131, 2141, 2161, 2204, 2311, 2391, 2411, 2556, 2676, 2711, 2931, 3121, 3343, 3409, 3413, 3433, 3463, 3554, 3643, 3678, 3679, 3877
Offset: 1

Views

Author

Jaroslav Krizek, Jan 05 2015

Keywords

Comments

Supersequence of A115920 and A069216.

Examples

			211 is in the sequence because the set of digits of n {1, 2} equals the set of digits of sigma(211) = 212.

Links

Crossrefs

Cf. A000203, A069216, A115920.

Programs

  • Magma
    [n: n in [1..10^5] | Set(Intseq(n)) eq Set(Intseq(SumOfDivisors(n)))];
  • Mathematica
    Select[Range[4000],Union[IntegerDigits[DivisorSigma[1,#]]] == Union[ IntegerDigits[#]]&] (* Harvey P. Dale, Dec 29 2015 *)
  • PARI
    isok(n) = Set(digits(n)) == Set(digits(sigma(n))); \\ Michel Marcus, May 27 2018

A216395 Number of values of k for which sigma(k) is a permutation of decimal digits of k, for 2^(n-1) < k < 2^n.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 0, 3, 2, 0, 6, 3, 5, 14, 22, 26, 60, 64, 71, 179, 333, 274, 751, 1653, 1726, 3032
Offset: 1

Views

Author

V. Raman, Sep 06 2012

Keywords

Examples

			a(12) = 6 because the values of k satisfying the condition for 2^11 < k < 2^12 are {2391, 2556, 2931, 3409, 3678, 3679}. - _V. Raman_, Feb 19 2014

Crossrefs

Cf. A115920, A000203.

Programs

  • PARI
    a(n)=sum(k=2^(n-1), 2^n, vecsort(digits(k)) == vecsort(digits(sigma(k)))) \\ V. Raman, Feb 19 2014, based on edits by M. F. Hasler
  • Python
    from sympy import divisor_sigma
def A216395(n):
    if n == 1:
        return 1
    c = 0
    for i in range(2**(n-1)+1, 2**n):
        s1, s2 = sorted(str(i)), sorted(str(divisor_sigma(i)))
        if len(s1) == len(s2) and s1 == s2:
            c += 1
    return c # Chai Wah Wu, Jul 23 2015

Formula

a(n) = # { k in A115920 | 2^(n-1) < k < 2^n }. - M. F. Hasler, Feb 24 2014

A230541 Numbers n such that the digits of sigma(n) are a permutation of those of sigma*(n), where sigma*(n) is the sum of anti-divisors of n (A066417).

Original entry on oeis.org

11, 20, 22, 26, 33, 65, 82, 117, 209, 218, 376, 417, 483, 508, 537, 561, 675, 758, 910, 1186, 1208, 1317, 1350, 1828, 2039, 2192, 2347, 2471, 2840, 2889, 4129, 4369, 4389, 4495, 4893, 5007, 6430, 7276, 7690, 8246, 8777, 9289, 10651, 11727, 11797, 12048, 12099
Offset: 1

Views

Author

Paolo P. Lava, Oct 23 2013

Keywords

Examples

			Divisors of 376 are 1, 2, 4, 8, 47, 94, 376, 188 and sigma(376) = 720; anti-divisors of 376 are 3, 16, 251 and sigma*(376) = 270.
Therefore 376 is part of the sequence because the digits of 720 are a permutation of the digits of 270.

Links

Crossrefs

Cf. A000203, A066417, A115920.

Programs

  • Maple
    with(numtheory); P:= proc(i) local a,b,c,j,k,n,ok,p;
for n from 3 to i do b:=[]; c:=[];
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;
while a>0 do b:=[op(b),a mod 10]; a:=trunc(a/10); od; a:=sigma(n);
while a>0 do c:=[op(c),a mod 10]; a:=trunc(a/10); od;
if nops(b)=nops(c) then b:=sort(b); c:=sort(c); b:=b-c; ok:=1;
for j from 1 to nops(b) do if b[j]<>0  then ok:=0; break; fi; od;
if ok=1 then print(n); fi; fi; od; end; P(10^6);

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

A277114 Numbers such that the decimal digits of sigma(n) are a permutation of those of sigma(n)-n.

Original entry on oeis.org

504, 1350, 3267, 3834, 6291, 6597, 7920, 12024, 13680, 19800, 19872, 19980, 25047, 26604, 30645, 46251, 49491, 53091, 54000, 55647, 57951, 60309, 60741, 62019, 62991, 64530, 65997, 70020, 72522, 80982, 86940, 86994, 95580, 99990, 100908, 103374, 103950, 117702
Offset: 1

Views

Author

Paolo P. Lava, Sep 30 2016

Keywords

Examples

			sigma(504) = 1560 and sigma(504) - 504 = 1056. - Typo fixed by _Ivan N. Ianakiev_, Oct 04 2016

Links

Crossrefs

Cf. A000203, A001065, A115920,

Programs

  • Maple
    with(numtheory): P:= proc(q) local n; for n from 1 to q do
if sort(convert(sigma(n),base,10))=sort(convert(sigma(n)-n,base,10)) then print(n); fi; od; end: P(10^9);
  • Mathematica
    Select[Range[10^5],Sort[IntegerDigits[DivisorSigma[1,#]]]==Sort[IntegerDigits[DivisorSigma[1,#]-#]]&] (* Ivan N. Ianakiev, Oct 04 2016 *)
Showing 1-10 of 13 results. Next

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