A067635 -id:A067635 - cp's OEIS frontend

A014563 a(n+1) is the smallest number > a(n) such that the digits of a(n)^2 are all (with multiplicity) contained in the digits of a(n+1)^2, with a(0)=1.

1, 4, 13, 14, 31, 36, 54, 96, 113, 311, 487, 854, 1036, 1277, 1646, 3214, 8351, 10456, 11414, 11536, 11563, 17606, 17813, 30287, 36786, 41544, 54927, 56547, 56586, 57363, 62469, 62634, 72813, 72897, 76944, 78345, 95061, 97944, 100963, 101944

Offset: 0

Marc Paulhus, Jan 29 2002

Probably infinite. - David W. Wilson, Jan 29 2002

			13^2 = 169 and 14 is the next smallest number whose square (in this case 196) contains the digits 1,6,9.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

If "contained in" is replaced by "properly contained in" we get A065297.
Cf. A066825, A067633, A067634, A067635, A065297.
Cf. A000290, A000196.

import Data.List ((\\))
a014563 n = a014563_list !! n
a014563_list = 1 : f 1 (drop 2 a000290_list) where
   f x (q:qs) | null $ xs \\ (show q) = y : f y qs
              | otherwise             = f x qs
              where y = a000196 q; xs = show (x * x)
-- Reinhard Zumkeller, Nov 22 2012

snd[n_]:=Module[{k=n+1},While[!AllTrue[Select[Transpose[{DigitCount[n^2],
DigitCount[k^2]}],#[[1]]>0&],#[[1]]<=#[[2]]&],k++];k]; NestList[ snd,1,40] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 21 2016 *)

A066825 a(1) = 1; set of digits of a(n)^2 is a subset of the set of digits of a(n+1)^2.

Original entry on oeis.org

1, 4, 13, 14, 31, 36, 54, 96, 113, 311, 487, 854, 1036, 1277, 1646, 3214, 3267, 3723, 4047, 4554, 4896, 5376, 10136, 13147, 13268, 16549, 20513, 21877, 25279, 26152, 27209, 28582, 31723, 32043, 32286, 33144, 35172, 35337, 35757, 35853

Offset: 1

David W. Wilson, Feb 05 2002

Probably infinite and dense over Z+.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A014563, A067633, A067634, A067635.

Cf. A000290, A000196.

import Data.List ((\\))
a066825 n = a066825_list !! (n-1)
a066825_list = 1 : f 1 (drop 2 a000290_list) where
   f x (q:qs) | all (`elem` show q) xs = y : f y qs
              | otherwise              = f x qs
              where y = a000196 q; xs = show (x * x)
-- Reinhard Zumkeller, Nov 22 2012

A217368 Smallest number having a power that in decimal has exactly n copies of all ten digits.

Original entry on oeis.org

32043, 69636, 643905, 421359, 320127, 3976581, 47745831, 15763347, 31064268, 44626422, 248967789, 85810806, 458764971, 500282265, 2068553967, 711974055, 2652652791, 901992825, 175536645, 3048377607, 3322858521, 1427472867, 3730866429, 9793730157

Offset: 1

James G. Merickel, Oct 01 2012

The exponents that produce the number with a fixed number of copies of each digit are listed in sequence A217378. See there for further comments.
Since we allow A217378(n)=1, the sequence is well defined, with the upper bound a(n) <= 100...99 ~ 10^(10n-1) (n copies of each digit, sorted in increasing order, except for one "1" permuted to the first position). - M. F. Hasler, Oct 05 2012
What is the minimum value of a(n)? Can it be proved that a(n) > 2 for all n? - Charles R Greathouse IV, Oct 16 2012

			The third term raised to the fifth power (A217378(3)=5), 643905^5 = 110690152879433875483274690625, has three copies of each digit (in its decimal representation), and no number smaller than 643905 has a power with this feature.

Cf. A020666, A054038, A054039, A066825, A067635, A074205, A156977, A217378.

f[n_] := Block[{k = 2, t = Table[n, {10}], r = Range[0, 9]}, While[c = Count[ IntegerDigits[k^Floor[ Log[k, 10^(10 n)]]], #] & /@ r; c != t, k++]; k] (* Robert G. Wilson v, Nov 28 2012 *)

is(n,k)=my(v);for(e=ceil((10*n-1)*log(10)/log(k)), 10*n*log(10)/log(k), v=vecsort(digits(k^e)); for(i=1,9,if(v[i*n]!=i-1 || v[i*n+1]!=i, return(0))); return(1)); 0
a(n)=my(k=2); while(!is(n,k),k++); k \\ Charles R Greathouse IV, Oct 16 2012

a(13)-a(14) from James G. Merickel, Oct 06 2012 and Oct 08 2012
a(15)-a(16) from Charles R Greathouse IV, Oct 17 2012
a(17)-a(19) from Charles R Greathouse IV, Oct 18 2012
a(20) from Charles R Greathouse IV, Oct 22 2012
a(21)-a(24) from Giovanni Resta, May 05 2017

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A014563 a(n+1) is the smallest number > a(n) such that the digits of a(n)^2 are all (with multiplicity) contained in the digits of a(n+1)^2, with a(0)=1.

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A066825 a(1) = 1; set of digits of a(n)^2 is a subset of the set of digits of a(n+1)^2.

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A217368 Smallest number having a power that in decimal has exactly n copies of all ten digits.

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