A115921 -id:A115921 - cp's OEIS frontend

A216381 Phi(n) values in A115921.

1, 12, 36, 192, 250, 1258, 1872, 1992, 3544, 2952, 3576, 3756, 2016, 1764, 4356, 2268, 2376, 2472, 2280, 5184, 2988, 11736, 11832, 12168, 13200, 10258, 20176, 10528, 21472, 15228, 12596, 12958, 12576, 19872, 16384, 33568, 12348, 35440, 14256, 44512

Offset: 1

V. Raman, Sep 06 2012

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

Cf. A000010, A115921.

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

Giovanni Resta, Feb 06 2006

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

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

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

Cf. A000203, A114065, A115921, A175795.

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

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

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

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

Giovanni Resta, Feb 13 2006

Intersection of A115920 and A115921.

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

Giovanni Resta, Table of n, a(n) for n = 1..520 (terms < 10^12)

Cf. A115920, A115921, A175795.

Cf. A000010, A000203.

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

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

Michel Lagneau, Sep 06 2010

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

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

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

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

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

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

A082060 Numbers n such that n and phi(n) have the same distinct decimal digits.

Original entry on oeis.org

1, 21, 63, 101, 233, 291, 502, 677, 1021, 1031, 1051, 1061, 1091, 1201, 1226, 1301, 1601, 1801, 1901, 2011, 2201, 2333, 2383, 2393, 2518, 2633, 2677, 2700, 2767, 2817, 2833, 2991, 3011, 3023, 3122, 3203, 3253, 3323, 3623, 3677, 3767, 3823, 3923, 3989

Offset: 1

Labos Elemer, Apr 04 2003

Contains A113781 as a subsequence. - M. F. Hasler, Nov 28 2007

Numbers n such that n and phi(n) have the same decimal digits = A115921. - Jaroslav Krizek, Nov 13 2014

			n=502 is a member since phi[502]=250

Harvey P. Dale, Table of n, a(n) for n = 1..2500

Cf. A000005, A000010, A082059, A115921, A113781, A113622, A069215, A072395, A113781.

Select[Range[4000],Union[IntegerDigits[#]]==Union[IntegerDigits[ EulerPhi[ #]]]&] (* Harvey P. Dale, Jan 31 2022 *)

for(n=1,10^4,if(Set(Vec(Str(n)))==Set(Vec(Str(eulerphi(n)))),print1(n", "))) \\ M. F. Hasler, Nov 28 2007

from sympy import totient
A082060_list = [n for n in range(1,10**4) if set(str(totient(n))) == set(str(n))] # Chai Wah Wu, Dec 13 2015

Definition and comment corrected by Jaroslav Krizek, Nov 13 2014

A113781 Numbers k such that the representation of phi(k) is a cyclic permutation of that of k, in base 10.

Original entry on oeis.org

1, 21, 63, 502, 4435, 5229, 5637, 6822, 8022, 35683, 98802, 176481, 210526, 421052, 442881, 480249, 529443, 544435, 640170, 842104, 920262, 976482, 7390422, 21251221, 28934019, 36174255, 36563587, 51804709, 59963997, 60550457

Offset: 1

Giovanni Resta, Jan 26 2006

This is a subsequence of A115921, which in turn is a subsequence of A082060. - M. F. Hasler, Nov 28 2007

			phi(442881) = 288144.

Cf. A000010, A115921, A082060.

lst = {}; Do[s = ToString@n; d = ToString@EulerPhi@n; If[StringLength@d == StringLength@n && {}!= StringPosition[s<>s, d], AppendTo[lst, n]], {n, 10^6}]; lst
lst = {}; Do[s = ToString(AT)n; d = ToString(AT)EulerPhi(AT)n; If[StringLength(AT)d == StringLength(AT)n && {}!= StringPosition[s<>s, d], AppendTo[lst, n]], {n, 10^6}]; lst (* M. F. Hasler, Nov 28 2007 *)

a(24)-a(30) from Donovan Johnson, Aug 27 2010

A216394 Number of values of k for which phi(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, 1, 1, 0, 0, 2, 0, 0, 3, 11, 2, 13, 21, 26, 49, 91, 186, 108, 335, 937, 500, 1681, 4208, 4156

Offset: 1

V. Raman, Sep 06 2012

			a(14) = 2 because the values of k satisfying the condition for 2^13 < k < 2^14 are {8541, 8982}. - _V. Raman_, Feb 18 2014

Cf. A115921, A000010.

a(n)=sum(k=2^(n-1), 2^n, vecsort(digits(k)) == vecsort(digits(eulerphi(k)))) \\ V. Raman, Feb 18 2014, based on edits by M. F. Hasler

from sympy import totient
def A216394(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(totient(i)))
        if len(s1) == len(s2) and s1 == s2:
            c += 1
    return c # Chai Wah Wu, Jul 23 2015

a(n) = # { k in A115921 | 2^(n-1) < k < 2^n }. - M. F. Hasler, Feb 24 2014

A216391 Number of values of k for which phi(k) is a permutation of decimal digits of k, for k < 2^n.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 3, 3, 5, 5, 5, 8, 19, 21, 34, 55, 81, 130, 221, 407, 515, 850, 1787, 2287, 3968, 8176, 12332, 18560, 36832, 71375, 94015, 173754

Offset: 1

V. Raman, Sep 06 2012

Partial sums of A216394.

			a(12) = 8 because the values of k satisfying the condition for k < 2^12 are {1, 21, 63, 291, 502, 2518, 2817, 2991}. - _V. Raman_, Feb 18 2014

Cf. A000010, A115921, A216394.

A216391 := proc(n)
    local a,k,kdgs,pdgs ;
    a := 0 ;
    for k from 1 to 2^n do
        kdgs := sort(convert(k,base,10)) ;
        numtheory[phi](k) ;
        pdgs := sort(convert(%,base,10)) ;
        if pdgs = kdgs then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
for n from 1 do
    print(A216391(n)) ;
end do: # R. J. Mathar, Mar 04 2014

a(n)=sum(k=1, 2^n, vecsort(digits(k)) == vecsort(digits(eulerphi(k)))) \\ V. Raman, Feb 18 2014, edited by M. F. Hasler, Mar 04 2014

a(n) = # { k in A115921 | k < 2^n }. - M. F. Hasler, Feb 24 2014

a(28)-a(32) from Amiram Eldar, Nov 09 2024

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

Paolo P. Lava, Jun 10 2015

A073930 is a subset of this sequence.

			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.

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

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

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);

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 *)

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

A273799 Numbers n such that the decimal digits of n-phi(n) are a permutation of those of n.

Original entry on oeis.org

91, 342, 351, 627, 684, 874, 950, 1824, 2114, 2318, 2715, 3051, 3171, 3172, 3402, 3411, 4228, 6344, 6804, 7346, 8414, 8456, 8474, 9093, 9125, 9191, 9254, 9500, 9531, 9548, 9672, 9824, 9950, 15804, 18010, 18312, 18508, 18930, 19190, 21028, 22126, 23218, 24180, 24716

Offset: 1

Paolo P. Lava, May 31 2016

			91 - phi(91) = 91 - 72 = 19;
342 - phi(342) = 342 - 108 = 234.

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A051953, A115921.

with(numtheory): P:=proc(q) local a,b,c,d,k,n; for n from 1 to q do a:=n; b:=n-phi(n);
if ilog10(a)=ilog10(b) then c:=[]; d:=[]; for k from 1 to ilog10(n)+1 do
c:=[op(c),(a mod 10)]; a:=trunc(a/10); d:=[op(d),(b mod 10)]; b:=trunc(b/10); od;
c:=sort(c); d:=sort(d); if c=d then print(n); fi; fi; od; end: P(10^25);

Select[Range[25000], Sort@ IntegerDigits@ # == Sort@ IntegerDigits[# - EulerPhi@ #] &] (* Michael De Vlieger, Jun 01 2016 *)

cp's OEIS Frontend

A216381 Phi(n) values in A115921.

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

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

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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.

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A082060 Numbers n such that n and phi(n) have the same distinct decimal digits.

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A113781 Numbers k such that the representation of phi(k) is a cyclic permutation of that of k, in base 10.

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A216394 Number of values of k for which phi(k) is a permutation of decimal digits of k, for 2^(n-1) < k < 2^n.

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A216391 Number of values of k for which phi(k) is a permutation of decimal digits of k, for k < 2^n.

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