A014563 -id:A014563 - cp's OEIS frontend

A065297 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)=1.

1, 4, 13, 36, 113, 487, 1036, 3214, 10456, 36786, 100963, 319656, 1001964, 3165969, 10001786, 31626854, 100013919, 316256807, 1000029656, 3162322481, 10000115537

Offset: 0

Floor van Lamoen, Oct 29 2001

Probably infinite and at least O(10^(n/2)). - David W. Wilson

			13^2 = 169 and 36 is the next smallest number whose square (in this case 1296) properly contains the digits 1,6,9.

Cf. A050630, A050631, A048559, A050636, A065298, A014563, A066825.

Cf. A000290, A000196.

import Data.List ((\\), sort)
a065297 n = a065297_list !! n
a065297_list = 1 : f 1 (drop 2 a000290_list) where
   f x (q:qs) | null (xs \\ sq) && sort xs /= sort sq = y : f y qs
              | otherwise                             = f x qs
              where y = a000196 q; sq = show q; xs = show (x * x)
-- Reinhard Zumkeller, Nov 22 2012

More terms from Marc Paulhus, Jan 29 2002
More terms from David W. Wilson and Marc Paulhus, Feb 05 2002
a(19)-a(20) from Sean A. Irvine, Aug 26 2023

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

A067633 a(1) = 1; sequence of digits of a(n)^2 is a subsequence of the sequence of digits of a(n+1)^2.

Original entry on oeis.org

1, 4, 13, 37, 117, 367, 3667, 36667, 366667, 3666667, 36666667, 366666667, 3666666667, 36666666667, 366666666667, 3666666666667, 36666666666667, 366666666666667, 3666666666666667, 36666666666666667, 366666666666666667, 3666666666666666667, 36666666666666666667

Offset: 1

David W. Wilson, Feb 05 2002

Probably infinite. Does the obvious pattern continue?

The pattern continues until at least n=100 and is unlikely to change because insertion of 2 digits into a(n)^2 is sufficient to maintain the pattern. - Sean A. Irvine, Dec 28 2023

Sean A. Irvine, Table of n, a(n) for n = 1..100
Sean A. Irvine, Java program (github)

Cf. A014563, A066825, A068175.

a(n) = sqrt(A068175(n)). - Sean A. Irvine, Dec 28 2023

Revised with data from A091874 by Sean A. Irvine, Dec 28 2023

A067634 a(1) = 1; string of digits of a(n)^2 is a substring of the string of digits of a(n+1)^2.

Original entry on oeis.org

1, 4, 13, 130, 1300, 13000, 130000, 1300000, 13000000, 130000000, 1300000000, 13000000000, 130000000000, 1300000000000, 13000000000000, 130000000000000, 1300000000000000, 13000000000000000, 130000000000000000

Offset: 1

David W. Wilson, Feb 05 2002

Provably infinite.

The obvious pattern continues. Proof: By induction, assume that 13*10^k is the (k+2)nd element in the sequence for some k >= 1. Clearly 13*10^{k+1} satisfies the required condition; we need to show that no other number works. Equivalently, we need to show that 169*10^{2k+2} is the smallest square of one of the forms: 169*10^{2k+1}+a, a*10^{2k+3}+169*10^{2k}, 169*10^{2k+2}+a*10+b, a*10^{2k+4}+169*10^{2k+1}+b, a*10^{2k+4}+b*10^{2k+3}+169*10^{2k}, where 0 <= a,b <= 9. Insisting that the number be less than 169*10^{2k+2} and checking that it is a 2-adic, 3-adic and 5-adic square eliminates all but 169*10^{2k+1}+9 and 1169*10^{2k+1}+1. To eliminate these, reduce modulo the primes 101, 137=(10^4+1)/173 and 5882353=(10^8+1)/17; these all divide 10^16+1, so it suffices to check k=0,1,2,3,4,5,6,7. QED. - Eric Rains, Jan 29 2002

Cf. A014563, A066825.

Join[{1,4},NestList[10#&,13,20]] (* Harvey P. Dale, Jul 25 2024 *)

More terms from David W. Wilson, Feb 05 2002

A067635 a(1) = 1; set of digits of a(n)^2 is a proper subset of the set of digits of a(n+1)^2.

Original entry on oeis.org

1, 4, 13, 36, 113, 487, 1036, 3267, 10136, 32043

Offset: 1

David W. Wilson, Feb 05 2002

Cf. A014563, A066825.

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.

A067713 a(n+1) is the smallest square > a(n) such that the digits of a(n) are all (with multiplicity) contained in the digits of a(n+1), with a(0)=1.

Original entry on oeis.org

1, 16, 169, 196, 961, 1296, 2916, 9216, 12769, 96721, 237169, 729316, 1073296, 1630729, 2709316, 10329796, 69739201, 109327936, 130279396, 133079296, 133702969, 309971236, 317302969, 917302369, 1353209796, 1725903936

Offset: 0

Marc Paulhus, Feb 05 2002

Cf. A014563.

a(n) = A014563(n)^2.

cp's OEIS Frontend

A065297 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)=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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A067633 a(1) = 1; sequence of digits of a(n)^2 is a subsequence of the sequence of digits of a(n+1)^2.

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A067634 a(1) = 1; string of digits of a(n)^2 is a substring of the string of digits of a(n+1)^2.

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A067635 a(1) = 1; set of digits of a(n)^2 is a proper subset of the set of digits of a(n+1)^2.

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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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A067713 a(n+1) is the smallest square > a(n) such that the digits of a(n) are all (with multiplicity) contained in the digits of a(n+1), with a(0)=1.

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