A114065 -id:A114065

Search: a114065 -id:a114065

Displaying 1-4 of 4 results found.

page 1

A115920

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

+10
14

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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

LINKS

V. Raman, Table of n, a(n) for n = 1..10000

EXAMPLE

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

MATHEMATICA

Select[Range[35000], Sort[IntegerDigits[#]]==Sort[ IntegerDigits[ DivisorSigma[ 1, #]]]&] (* Harvey P. Dale, May 09 2013 *)

PROG

(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

(PARI) isok(n) = vecsort(digits(n)) == vecsort(digits(sigma(n))); \\ Michel Marcus, Dec 13 2015 and May 27 2018

CROSSREFS

Cf. A000203, A114065, A115921, A175795.

KEYWORD

nonn,base

AUTHOR

Giovanni Resta, Feb 06 2006

STATUS

approved

A115921

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

+10
14

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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

LINKS

V. Raman, Table of n, a(n) for n = 1..10000

EXAMPLE

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

MATHEMATICA

Select[Range[45000], Sort[IntegerDigits[EulerPhi[#]]]==Sort[IntegerDigits[#]]&] (* Harvey P. Dale, Jul 25 2018 *)

PROG

(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

CROSSREFS

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

KEYWORD

nonn,base

AUTHOR

Giovanni Resta, Feb 06 2006

EXTENSIONS

Edited by M. F. Hasler, Nov 28 2007

STATUS

approved

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.

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Donovan Johnson, Table of n, a(n) for n = 1..1000

EXAMPLE

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

MATHEMATICA

okQ[n_] := Module[{idn = IntegerDigits[DivisorSigma[1, n]]}, Sort[idn] == Sort[IntegerDigits[EulerPhi[n]]]]; Select[Range[40000], okQ]

PROG

(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

(PARI) isok(n) = (de = digits(eulerphi(n))) && (ds = digits(sigma(n))) && (vecsort(de) == vecsort(ds)); \\ Michel Marcus, Dec 13 2015

CROSSREFS

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

KEYWORD

nonn,base

AUTHOR

Michel Lagneau, Sep 06 2010

STATUS

approved

A258786

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

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

A073930 is a subset of this sequence.

LINKS

Paolo P. Lava, Table of n, a(n) for n = 1..200

EXAMPLE

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.

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 *)

PROG

(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

CROSSREFS

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

KEYWORD

nonn,base

AUTHOR

Paolo P. Lava, Jun 10 2015

STATUS

approved

page 1

Search completed in 0.007 seconds