A065297 -id:A065297 - 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 *)

A065298 a(n+1) is the smallest number > a(n) such that the digits of a(n)^2 are all (with multiplicity) properly contained in the digits of a(n+1)^2, with a(0)=2.

Original entry on oeis.org

2, 7, 43, 136, 367, 1157, 3658, 10183, 32193, 101407, 320537, 1001842, 3166463, 10001923, 31627114, 100017313, 316599084, 1000104687, 3162331407, 10000483663

Offset: 0

Floor van Lamoen, Oct 29 2001

a(n) for n>0 remains the same when a(0)=3.

			43^2 = 1849 and 136 is the next smallest number whose square (in this case 18496) properly contains the digits 1,4,8,9.

Cf. A050630, A050631, A048559, A050636, A065297.

More terms from Marc Paulhus, Feb 04 2002

a(18)-a(19) from Sean A. Irvine, Aug 26 2023

A067971 a(n+1) is the smallest number > a(n) such that the digits of a(n)^3 are all (with multiplicity) properly contained in the digits of a(n+1)^3, with a(0)=1.

Original entry on oeis.org

1, 5, 25, 105, 678, 2685, 6631, 17248, 48056, 109016, 255085, 576468, 1026598, 2161819, 4793623, 10107506, 21624889, 46584891, 100042239, 215671305, 464309988, 1000305876, 2156229901, 4642492109, 10000266599

Offset: 0

Marc Paulhus, Feb 05 2002

			5^3 = 125 and 25 is the next smallest number whose cube (in this case 15625) properly contains the digits 1,2,5.

Cf. A000578, A065297, A065298, A067972.

a(22)-a(24) from Sean A. Irvine, Jan 16 2024

A067973 a(n+1) is the smallest number > a(n) such that the digits of a(n)^3 are all (with multiplicity) contained in the digits of a(n+1)^3, with a(0)=1.

Original entry on oeis.org

1, 5, 8, 25, 105, 678, 2685, 6631, 17248, 48056, 109016, 255085, 576468, 993996, 1026598, 1029697, 1061509, 1089598, 1231339, 1374358, 1753291, 1832389, 1896319, 2161819, 2162446, 2163199, 2231416, 2263201, 2264893, 2674003

Offset: 0

Marc Paulhus, Feb 05 2002

If "contained in" is replaced by "properly contained in" we get A067971.

			5^3 = 125 and 8 is the next smallest number whose cube (in this case 512) that contains the digits 1,2,5.

Cf. A067971, A065298, A067974, A014563, A065297.

A067975 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)=2.

Original entry on oeis.org

2, 7, 43, 136, 367, 1157, 1822, 3658, 5558, 6196, 9679, 10183, 11794, 17852, 20813, 28354, 32193, 42852, 53787, 55044, 55707, 55983, 57636, 58464, 61719, 70209, 95232, 96354, 96921, 96963, 101407, 114223, 114323, 133564, 162293, 170843

Offset: 0

Marc Paulhus, Feb 05 2002

a(n) for n>0 remains the same when a(0)=3. If "contained in" is replaced by "properly contained in" we get A065298.

			1157^2 = 1338649 and 1822 is the next smallest number whose square (in this case 3319684) contains the digits 1,3,3,8,6,4,9.

Cf. A065297, A014563, A067971, A065298, A067973.

A067711 a(n+1) is the smallest square > a(n) such that the digits of a(n) are all (with multiplicity) properly contained in the digits of a(n+1), with a(0)=1.

Original entry on oeis.org

1, 16, 169, 1296, 12769, 237169, 1073296, 10329796, 109327936, 1353209796, 10193527369, 102179958336, 1003931857296, 10023359708961, 100035723189796, 1000257893937316, 10002783993738561, 100018367973835249

Offset: 0

Marc Paulhus, Feb 05 2002

Cf. A065297.

a(n) = A065297(n)^2.

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.

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A065298 a(n+1) is the smallest number > a(n) such that the digits of a(n)^2 are all (with multiplicity) properly contained in the digits of a(n+1)^2, with a(0)=2.

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A067971 a(n+1) is the smallest number > a(n) such that the digits of a(n)^3 are all (with multiplicity) properly contained in the digits of a(n+1)^3, with a(0)=1.

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A067973 a(n+1) is the smallest number > a(n) such that the digits of a(n)^3 are all (with multiplicity) contained in the digits of a(n+1)^3, with a(0)=1.

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A067975 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)=2.

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A067711 a(n+1) is the smallest square > a(n) such that the digits of a(n) are all (with multiplicity) properly contained in the digits of a(n+1), with a(0)=1.

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