A061147 -id:A061147 - cp's OEIS frontend

A045876 Sum of different permutations of digits of n (leading 0's allowed).

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 11, 33, 44, 55, 66, 77, 88, 99, 110, 22, 33, 22, 55, 66, 77, 88, 99, 110, 121, 33, 44, 55, 33, 77, 88, 99, 110, 121, 132, 44, 55, 66, 77, 44, 99, 110, 121, 132, 143, 55, 66, 77, 88, 99, 55, 121, 132, 143, 154, 66, 77, 88, 99, 110, 121, 66, 143

Offset: 1

Erich Friedman

Let the arithmetic mean of the digits of a 'D' digit number n be 'A', let 'N' = number of distinct numbers that can be formed by permuting the digits of n, and let 'I' = concatenation of 1 'D' times = (10^D-1)/9. then a(n) = A*N*I. E.g., let n = 324541, then A = (3+2+4+5+4+1)/6 = 19/6, N = 6!/(2!) = 360, I = 111111, and a(n) = A*N*I = (19/6)*(360)*(111111) = 126666540. - Amarnath Murthy, Jul 14 2003

It seems that the first person who studied the sum of different permutations of digits of a given number was the French scientist Eugène Aristide Marre (1823-1918). See links. - Bernard Schott, Dec 06 2012

Amarnath Murthy, An interesting result in combinatorics, Mathematics & Informatics Quarterly, Vol. 3, 1999, Bulgaria.

A. Dunigan AtLee, Table of n, a(n) for n = 1..100000.
A. Marre, Trouver la somme de toutes les permutations différentes d'un nombre donné., Nouvelles Annales de Mathématiques, 1ère série, tome 5 (1846), p. 57-60.
Norbert Verdier, QDV4 : Marre, Marre et Marre, page=1 (French mathematical forum les-mathematiques.net)

Same beginning as A033865. Cf. A061147.

f:= proc(x) local L,D,n,M,s,j;
  L:= convert(x,base,10);
  D:= [seq(numboccur(j,L),j=0..9)];
  n:= nops(L);
  M:= n!/mul(d!,d=D);
  s:= add(j*D[j+1],j=0..9);
  (10^n-1)*M/9/n*s
end proc:
map(f, [$1..100]); # Robert Israel, Jul 07 2015

Table[Total[FromDigits /@ Permutations[IntegerDigits[n]]], {n, 100}] (* T. D. Noe, Dec 06 2012 *)

A047726(n) = n=eval(Vec(Str(n))); (#n)!/prod(i=0, 9, sum(j=1, #n, n[j]==i)!);
A055642(n) = #Str(n);
A007953(n) = sumdigits(n);
a(n) = ((10^A055642(n)-1)/9)*(A047726(n)*A007953(n)/A055642(n)); \\ Altug Alkan, Aug 29 2016

A045876(n) = {my(d=digits(n), q=1, v, t=1); v = vecsort(d); for(i=1, #v-1, if(v[i]==v[i+1], t++, q*=binomial(i, t); t=1)); q*binomial(#v, t)*(10^#d-1)*vecsum(d)/9/#d} \\ David A. Corneth, Oct 06 2016

a(n) = ((10^A055642(n)-1)/9)*(A047726(n)*A007953(n)/A055642(n)). - Altug Alkan, Aug 29 2016

A061378 Product of all numbers formed by permuting the digits of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 121, 252, 403, 574, 765, 976, 1207, 1458, 1729, 40, 252, 484, 736, 1008, 1300, 1612, 1944, 2296, 2668, 90, 403, 736, 1089, 1462, 1855, 2268, 2701, 3154, 3627, 160, 574, 1008, 1462, 1936, 2430, 2944, 3478, 4032, 4606, 250

Offset: 0

Klaus Brockhaus, Jun 08 2001

Differs from A062003 at those n which have more than two digits.

			a(10) = 10*01 = 10, a(11) = 11*11 = 121, a(12) = 12*21 =252.

A061147, A062003.

: function permute(s: string): array; var i,k: integer; st1,st2: stack; ele,ws: string; begin stack_push(st2,s[0..0]); for i := 1 to length(s)-1 do while not stack_empty(st2) do stack_push(st1,stack_pop(st2)); end; ele := s[i]; while length(st1) > 0 do ws := stack_pop(st1); for k := 0 to length(ws)-1 do stack_push(st2,concat(ws[0..k-1],ele,ws[k..])); end; stack_push(st2,concat(ws[0..],ele)); end; end; return stack2array(st2); end; function int_permute(n: integer): array; var s: string; ar: array; i: integer; begin s := itoa(n); ar := permute(s); for i := 0 to length(ar)-1 do ar[i] := atoi(ar[i]); end; return ar; end; for k := 0 to 55 do write(product(int_permute(k))," "); end;

A062003 Product of the k numbers formed by cyclically permuting digits of n (where k = number of digits of n).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 252, 403, 574, 765, 976, 1207, 1458, 1729, 40, 252, 22, 736, 1008, 1300, 1612, 1944, 2296, 2668, 90, 403, 736, 33, 1462, 1855, 2268, 2701, 3154, 3627, 160, 574, 1008, 1462, 44, 2430, 2944, 3478, 4032, 4606, 250

Offset: 0

Amarnath Murthy, May 30 2001

Apparently some kind of distinctness is required, because n=11 (cyclically permuted again 11) is not translated to 11*11=121. - R. J. Mathar, Jun 13 2025

			a(14) = 14*41 = 574, a(100) = 100*010*001 = 1000, a(102) = 102*210*21 = 449820.

Cf. A061147, A061378.

for k := 0 to 60 do st := itoa(k); m := 1; for i := 1 to length(st) do st := concat(st[1..length(st)-1],st[0]); m := m*atoi(st); end; write(m," "); end;

Corrected and extended by Frank Ellermann and Klaus Brockhaus, Jun 03 2001

A179055 Numbers k such that the product of all numbers formed by cyclically permuting digits of k is a square.

Original entry on oeis.org

1, 4, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 243, 324, 432, 567, 675, 756, 1000, 1010, 1020, 1030, 1040, 1050, 1060, 1070, 1080, 1090, 1111, 1212, 1313, 1414, 1515, 1616, 1717, 1818, 1919, 2000, 2010, 2020, 2030, 2040, 2050, 2060, 2070, 2080, 2090, 2121, 2222, 2323, 2424

Offset: 1

Michel Lagneau, Jan 04 2011

			756 is in the sequence because 756 * 567 * 675 = 289340100 = 17010^2.

Cf. A061378 A062003 A061147.

with(numtheory):for n from 1 to 4000 do:pp:=1:n0:=n:l:=length(n0) :ind:=0:for
  j from 1 to l do:s:=0:for m from 1 to l do:q:=n0:u:=irem(q, 10):v:=iquo(q, 10):n0:=v
  :s:=s+ u*10^m:od:s:=floor(s-u*10^l+u):n0:=s: pp:=pp*s:od:x:=sqrt(pp) :y:=floor(x):if
  x=y then printf(`%d, `, n): else fi :od:

cycDigitPerms[n_Integer, b_:10] := Module[{list = {n}, digits = IntegerDigits[n, b], len, counter, holder, next}, len = Length[digits]; counter = 1; While[counter < len, holder = digits[[-1]]; digits = Drop[digits, -1]; digits = Insert[digits, holder, 1]; list = Append[list, FromDigits[digits, b]]; counter++]; Return[list]]; Select[Range[2000], IntegerQ[Sqrt[Times@@cycDigitPerms[#]]] &] (* Alonso del Arte, Jan 04 2011 *)

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A045876 Sum of different permutations of digits of n (leading 0's allowed).

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A061378 Product of all numbers formed by permuting the digits of n.

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A062003 Product of the k numbers formed by cyclically permuting digits of n (where k = number of digits of n).

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A179055 Numbers k such that the product of all numbers formed by cyclically permuting digits of k is a square.

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