A045876 -id:A045876 - cp's OEIS frontend

A276413 Non-repdigit numbers k that divide A045876(k).

370, 407, 481, 518, 592, 629, 2727, 13008, 14634, 16260, 19512, 22764, 29268, 39024, 87804, 101010, 102564, 103896, 104895, 105820, 108262, 109890, 113960, 115830, 116883, 124740, 125356, 125874, 126984, 128205, 129870, 132275, 134680, 135135, 136752

Offset: 1

Altug Alkan, Sep 05 2016

A161020 is a subsequence.

			2727 is a term because 2277 + 2727 + 2772 + 7227 + 7272 + 7722 = 29997 is divisible by 2727.

Robert Israel, Table of n, a(n) for n = 1..373

Cf. A161020, A045876.

filter:= proc(x) local L, D, n, M, s, j;
  L:= convert(x, base, 10);
  D:= [seq(numboccur(j, L), j=0..9)];
  if numboccur(0,D) = 9 then return false fi;
  n:= nops(L);
  M:= n!/mul(d!, d=D);
    s:= add(j*D[j+1], j=0..9);
  evalb(((10^n-1)*M/9/n*s) mod x = 0)
end proc:
select(filter, [$1..2*10^5]); # Robert Israel, Sep 12 2016

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);
A045876(n) = ((10^A055642(n)-1)/9)*(A047726(n)*A007953(n)/A055642(n));
isA010785(n) = {1==#Set(digits(n))}
lista(nn) = for(n=1, nn, if(A045876(n) % n == 0 && !isA010785(n), print1(n", ")));

A276502 Least k > 0 such that A045876(n) divides A045876(n*10^k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 6

Offset: 1

Altug Alkan, Sep 10 2016

Corresponding values of A045876(n*10^a(n))/A045876(n) are 11, 11, 11, 11, 11, 11, 11, 11, 11, 101, 303, 303, 303, 303, 303, 303, 303, 303, 303, 101, 303, 303, 303, 303, 303, 303, 303, 303, 303, 101, ...
From Charlie Neder, Jul 16 2018: (Start)
From the formula for A045876(n) we make the following modifications:
- A (the mean of the digits) becomes S/D (sum of digits / # of digits)
- N (# of arrangements of digits) becomes R*Z (# of arrangements of nonzero digits * # of ways to insert the proper number of zeros)
Appending zeros to n does not change S or R, so if (S*R*Z*I/D)(n) divides (S*R*Z*I/D)(n*10^k), then (Z*I/D)(n) divides (Z*I/D)(n*10^k). However, Z, I, and D are completely determined by the number of digits of n and the number of those digits which are zero, so a(n) = a(A136400(n)). (End)

			a(10) = 2 because A045876(10) = 1+10 = 11 does not divide A045876(100) = 1+10+100 = 111 and 11 divides A045876(1000) = 1+10+100+1000 = 1111.

Cf. A045876.

A045876[n_] := Total[FromDigits /@ Permutations[IntegerDigits[n]]]; a[n_] := For[k = 1, True, k++, If[Divisible[A045876[n*10^k], A045876[n]], Return[k] ] ]; Array[a, 101] (* Jean-François Alcover, Jul 26 2017 *)

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);
A045876(n) = ((10^A055642(n)-1)/9)*(A047726(n)*A007953(n)/A055642(n));
a(n) = {my(k = 1); while (A045876(n*(10^k)) % A045876(n), k++); k; }

A276510 Numbers k such that the sum of all the different permutations of the digits of k (A045876(k)) is a pandigital number (a term of A171102).

Original entry on oeis.org

10234567, 10234576, 10234579, 10234597, 10234657, 10234675, 10234678, 10234687, 10234756, 10234759, 10234765, 10234768, 10234786, 10234795, 10234867, 10234876, 10234957, 10234975, 10235467, 10235476, 10235479, 10235497, 10235647, 10235674, 10235746, 10235749

Offset: 1

Altug Alkan, Sep 06 2016

			10234759 is a term because A045876(10234759) = 1567999984320, which contains every digit from 0 to 9.

Cf. A045876, A171102.

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);
A045876(n) = ((10^A055642(n)-1)/9)*(A047726(n)*A007953(n)/A055642(n));
isA171102(n) = 9<#vecsort(Vecsmall(Str(n)), , 8);
is(n) = isA171102(A045876(n));

A276597 Least k such that n divides A045876(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 19, 10, 39, 49, 59, 69, 79, 89, 102, 199, 109, 106, 13, 599, 103, 799, 139, 108, 149, 2999, 104, 4999, 169, 12, 179, 8999, 105, 100, 289, 139, 389, 10000, 106, 79999, 13, 159, 689, 299999, 107, 100006, 1789, 179, 2789, 899999, 108, 14, 4789, 199, 5789, 5999999, 109

Offset: 1

Altug Alkan, Sep 08 2016

Corresponding values of A045876(a(n))/n are 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 11, 11, 11, 11, 11, 11, 37, 111, 111, 74, 2, 111, 37, 111, 111, 74, 111, 1111, 37, 1111, 111, 1, 111, 1111, 37, 3, 111, 74, 111, 271, 37, 11111, 1, 74, 111, 111111, 37, 79365, 3333, ...

			a(10) = 19 because 19+91 = 110 is divisible by 10.
a(18) = 102 because 12+21+102+120+201+210 is divisible by 18.

Giovanni Resta, Table of n, a(n) for n = 1..150

Cf. A045876.

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);
A045876(n) = ((10^A055642(n)-1)/9)*(A047726(n)*A007953(n)/A055642(n));
a(n) = {my(k = 1); while (A045876(k) % n, k++); k; }

A276802 Non-repdigit numbers n such that A045876(n) ends with n.

Original entry on oeis.org

554, 3328, 55553, 77764, 222221, 444442, 666663, 888865, 888884, 5555552, 6666595, 9999840, 33332680, 55555526, 66666557, 99998670, 333332176, 333333312, 555555551, 666665752, 666666624, 999997536, 999999936, 9999976480, 9999997844, 9999999668, 9999999923, 11111111110

Offset: 1

Altug Alkan, Sep 17 2016

			554 is a term because 455+545+554 = 1554 that ends with 554.
2338 is the least term having its digits. For all permutations p of digits of n, in this case 2338, (without leading zeros if any), A045876(n) = A045876(p). A045876(2338) = 53328. It contains the digits of 2338 and ends with its digits permuted. 2338 has 4 digits, as has 53328 mod 10^4 so 53328 mod 10^4 == 3328 is a term. - _David A. Corneth_, Oct 04 2016

David A. Corneth, Pari Program.

Cf. A045876, A139819 (non-repdigits), A179239.

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);
A045876(n) = ((10^A055642(n)-1)/9)*(A047726(n)*A007953(n)/A055642(n));
isA010785(n) = {1==#Set(digits(n))}
is(n) = (A045876(n)-n) % 10^A055642(n) == 0 && !isA010785(n)

More terms from David A. Corneth, Oct 06 2016

A282134 First differences of A045876.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 22, 11, 11, 11, 11, 11, 11, 11, -88, 11, -11, 33, 11, 11, 11, 11, 11, 11, -88, 11, 11, -22, 44, 11, 11, 11, 11, 11, -88, 11, 11, 11, -33, 55, 11, 11, 11, 11, -88, 11, 11, 11, 11, -44, 66, 11, 11, 11, -88, 11, 11, 11, 11, 11, -55, 77, 11, 11, -88, 11, 11, 11, 11, 11, 11, -66, 88, 11, -88, 11

Offset: 1

Altug Alkan, Feb 06 2017

A276739 Least k such that A045876(k) is divisible by 10^n.

Original entry on oeis.org

1, 19, 10699, 102589, 10000112389

Offset: 0

Altug Alkan, Sep 16 2016

Corresponding values of A045876(k) are 1, 110, 3333300, 333333000, ...
Sequence is infinite.
a(5) > 10^18. - Giovanni Resta, Sep 27 2016
Subsequence of A179239. - David A. Corneth, Oct 01 2016

			a(1) = 19 because 19+91 = 110 is divisible by 10.

Cf. A045876.

Table[k = 1; While[! Divisible[Total[FromDigits /@ Permutations@ IntegerDigits@ k], 10^n], k++]; k, {n, 0, 3}] (* Michael De Vlieger, Sep 16 2016 *)

A047726(n) = n=digits(n); (#n)!/prod(i=0, 9, sum(j=1, #n, n[j]==i)!);
A055642(n) = #Str(n);
A007953(n) = sumdigits(n);
A045876(n) = ((10^A055642(n)-1)/9)*(A047726(n)*A007953(n)/A055642(n));
a(n) = {my(k = 1); while (A045876(k) % (10^n), k++); k;}

a(4) from Giovanni Resta, Sep 27 2016

A276758 Numbers n such that A045876(n) = A045876(n+1).

Original entry on oeis.org

10, 1010, 1100, 1119, 1339, 1519, 3139, 5119, 8899, 27799, 46699, 48499, 50559, 55059, 64699, 72799, 84499, 100110, 101010, 101100, 110010, 110100, 111000, 111229, 112129, 117799, 121129, 136699, 147499, 163699, 168199, 171799, 174499, 177199, 186199

Offset: 1

Altug Alkan, Sep 17 2016

A138147 is a subsequence. Therefore, the sequence is infinite. - David A. Corneth, Sep 17 2016
Suppose a term is of the form SDN, where S is a sequence of digits without leading zeros, D is a digit less than 9 and N is a sequence of digits 9 (possibly 0 nines; terms from A002283) and SDN is a concatenation of S, D and N. Let S' be a permutation of digits of S without leading zeros. Then S'DN is also in the sequence. To search terms one may choose S from A179239. - David A. Corneth, Sep 18 2016
Since (n + 8*k) = (n - k + 1)*(n - k) has solutions that are n = k + 3*sqrt(k) and n = k - 3*sqrt(k), for square values of k there are infinitely many terms such that: 1119, 1111119999, 111111111999999999, ...

			1339 is a term because A045876(1339) = A045876(1340).
See 2nd comment. As 27799 is in the sequence, we can see S = 27, D = 7 and N = 99. Now all permutations S' (distinct) of S without leading zeros give terms. They are 72, giving term 72799. - _David A. Corneth_, Sep 18 2016

Cf. A045876, A179239.

A047726(n) = n=digits(n); (#n)!/prod(i=0, 9, sum(j=1, #n, n[j]==i)!);
A007953(n) = sumdigits(n);
lista(nn) = for(n=1, nn, if(A047726(n)*A007953(n) == A047726(n+1)*A007953(n+1), print1(n, ", ")))

A276810 Numbers n such that A045876(n) has distinct decimal digits.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 39, 48, 49, 57, 58, 59, 67, 68, 69, 75, 76, 78, 79, 84, 85, 86, 87, 89, 93, 94, 95, 96, 97, 98, 149, 158, 167, 176, 185, 194, 199, 239, 248, 257, 275, 284, 289, 293, 298, 329, 347, 356, 365, 374, 379, 388, 392, 397, 419, 428, 437, 469, 473, 478, 482

Offset: 1

Altug Alkan, Sep 18 2016

This sequence contains 146 elements. The largest is 991. No more terms below 10^10. As A045876(n) >= n, for all n >= 10^10, A045876(n) will have at least one digit not distinct. - David A. Corneth, Sep 19 2016

			289 is a term because 289+298+829+892+928+982 = 4218 has distinct decimal digits.

Cf. A010784, A045876.

Select[Range[10^3], Max@ DigitCount@ Total@ Map[FromDigits, Permutations@ IntegerDigits@ #] == 1 &] (* Michael De Vlieger, Sep 19 2016 *)

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);
A045876(n) = ((10^A055642(n)-1)/9)*(A047726(n)*A007953(n)/A055642(n));
isA010784(n) = my(v=vecsort(digits(n))); v==vecsort(v, , 8);
is(n) = isA010784(A045876(n));

Clarified comment. - Harvey P. Dale, Apr 30 2022

A061147 Product of all distinct numbers formed by permuting digits of n.

Original entry on oeis.org

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, 765, 1300

Offset: 1

N. J. A. Sloane, May 30 2001

			a(12) = 12*21 = 252.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A061378, A062003, A045876.

Table[Times@@Union[FromDigits/@Permutations[IntegerDigits[n]]],{n,60}] (* Harvey P. Dale, Apr 06 2023 *)

More terms from Larry Reeves (larryr(AT)acm.org), May 30 2001

cp's OEIS Frontend

A276413 Non-repdigit numbers k that divide A045876(k).

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A276502 Least k > 0 such that A045876(n) divides A045876(n*10^k).

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A276510 Numbers k such that the sum of all the different permutations of the digits of k (A045876(k)) is a pandigital number (a term of A171102).

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A276597 Least k such that n divides A045876(k).

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A276802 Non-repdigit numbers n such that A045876(n) ends with n.

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A282134 First differences of A045876.

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Formula

A276739 Least k such that A045876(k) is divisible by 10^n.

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A276758 Numbers n such that A045876(n) = A045876(n+1).

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