A156977 -id:A156977 - cp's OEIS frontend

A178960 Numbers n such that n! contains every digit at least once.

23, 27, 31, 33, 34, 35, 36, 37, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101

Offset: 1

Michel Lagneau, Dec 31 2010

			23 is in the sequence because 23! = 25852016738884976640000 contains every
  digit at least once.

Cf. A178929, A054038, A156977, A054038, A074205, A020666.

[n: n in [0..101] | Seqset(Intseq(Factorial(n))) eq {0..9}]; // Bruno Berselli, May 17 2011

with(numtheory):Digits:=200:B:={0,1,2,3,4,5,6,7,8,9}: T:=array(1..250) : for
  p from 1 to 200 do:ind:=0:n:=p!:l:=length(n):n0:=n:s:=0:for m from 1 to l do:q:=n0:u:=irem(q,10):v:=iquo(q,10):n0:=v : T[m]:=u:od: A:=convert(T,set):z:=nops(A):if A intersect B = B and ind=0 then ind:=1: printf(`%d, `,p):else fi:od:

Select[Range[101], Length[Union[IntegerDigits[#!]]] == 10 &]

A261213 Odd numbers n such that n^2 = m + (m+1), where both m and m+1 have no repeated digits.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 23, 27, 29, 31, 35, 37, 39, 41, 43, 57, 63, 69, 77, 81, 87, 89, 95, 109, 113, 121, 125, 127, 129, 137, 163, 193, 219, 239, 271, 273, 279, 281, 285, 305, 311, 315, 331, 339, 353, 357, 377, 381, 395, 403, 409, 435, 441, 443, 597

Offset: 1

Pieter Post, Aug 12 2015

This sequence is finite and a(146) = 40797 is the last term. 40797^2 = 1664395209 and 1664395209 = 832197604 + 832197605. These last two numbers both have no repeating digits.

			5 is in the sequence, because 5^2 = 25. 25 = 12 + 13. 12 and 13 both have no repeating digits.

Pieter Post, Table of n, a(n) for n = 1..146

Cf. A036745, A071519, A156977.

nr[n_] := 1 == Max@ DigitCount@ n; Select[ Range[1, 10^5, 2], nr[x= Floor[#^2 / 2]] && nr[x + 1] &] (* Giovanni Resta, Aug 12 2015 *)

A363909 Numbers whose square and cube taken together contain each decimal digit at least twice.

Original entry on oeis.org

6534, 11027, 11994, 21906, 22178, 22195, 23317, 24567, 27019, 27963, 28354, 29099, 29309, 29339, 29375, 29558, 29621, 30184, 30552, 30584, 31578, 31727, 32447, 32633, 32793, 32912, 32923, 33087, 33257, 33527, 34284, 35717, 36943, 36958, 37697, 38463

Offset: 1

M. F. Hasler, Jun 27 2023

The first term, a(1) = 6534 is the only number of which the square and cube taken together contain each digit 0 to 9 exactly twice.

Presumably a(n) ~ A363905(n) ~ n. - Charles R Greathouse IV, Jul 03 2023

			6534^2 = 42693156, 6534^3 = 278957081304, which together contain each digit 0-9 exactly twice.

Harold Suarez, Interesting..., Number Theory group on LinkedIn, (capture of the original post), June 2023

Cf. A363905: square and cube together contain each digit at least once.
Cf. A036744, A054038, A071519 and A156977 for "pandigital" squares.
Cf. A119735: Numbers n such that every digit occurs at least once in n^3.

is(n)=#Set(n=concat(digits(n^2),digits(n^3)))>9&&(n=vecsort(n))[#n-1]==9&&!n[2]&&!for(i=3,#n-2,n[i]>n[i-1]&&n[i]

A198863 Numbers whose squares are pandigital numbers with exactly two occurrences of each digit.

Original entry on oeis.org

3164252736, 3164326683, 3164389113, 3164391957, 3164406057, 3164416923, 3164421333, 3164454864, 3164466768, 3164482974, 3164528124, 3164547114, 3164689392, 3164695206, 3164735277, 3164770866, 3164789766, 3164863185, 3164867118, 3164907357, 3165009693

Offset: 1

Pablo Martínez, Oct 30 2011

Later terms include: 4000171725, 4000183233, 4000198443, 4000203567.

Because the sum of the digits of a(n)^2 is 90, 9 divides a(n)^2. Hence, 3 divides a(n). - T. D. Noe, Nov 08 2011

			4000171725^2 = 16001373829489475625.

Cf. A156977 (n^2 contains each digit once).

Select[Range[3164250000, 3164450000], Union[DigitCount[#^2]] == {2} &] (* Alonso del Arte, Oct 31 2011 *)
t = {}; n = 3164211348; nMax = 9994386752; While[n <= nMax && Length[t] < 21, While[n <= nMax && Union[DigitCount[n^2]] != {2}, n = n + 3]; If[n <= nMax, AppendTo[t, n]; Print[n]; n = n + 3]]; t (* T. D. Noe, Nov 08 2011 *)

All displayed terms are from Charles R Greathouse IV, Alonso del Arte and T. D. Noe, Nov 08 2011

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A178960 Numbers n such that n! contains every digit at least once.

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A261213 Odd numbers n such that n^2 = m + (m+1), where both m and m+1 have no repeated digits.

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A363909 Numbers whose square and cube taken together contain each decimal digit at least twice.

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A198863 Numbers whose squares are pandigital numbers with exactly two occurrences of each digit.

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