A067634 -id:A067634 - 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

A383214 a(n) = A067434(n) - A383213(n).

Original entry on oeis.org

1, 1, 0, 1, -1, 1, 0, 1, 0, 0, 0, 0, 0, -1, 1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 0, 0, -1, 1, 1, -1, 0, 1, 0, 0, 0, 0, 0, 1, -2, 1, -1, 1, 0, -1, 1, 0, -1, 1, 0, 0, 0, 0, -1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 1, 0, 0, 0, 0, -1, 1, 0, -1, 1, -1, 1, 0, -1, 1

Offset: 1

Clark Kimberling, Apr 19 2025

sign

Least n such that a(n) = -3 is 7055; least n such than a(n) = 3 is 740.

Cf. A000984, A067634, A383213.

u = Table[PrimeNu[Binomial[2 n, n]], {n,  200}]     (* A067434 *)
v = Table[PrimeNu[Binomial[2 n, n + 1]], {n,  200}] (* A383213 *)
u - v

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

A383214 a(n) = A067434(n) - A383213(n).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica