A048559 -id:A048559 - cp's OEIS frontend

A048561 a(n+1) is next smallest nontrivial square beginning with a(n), initial square is 9.

9, 961, 9616201, 961620162020001, 9616201620200016202000200000001, 96162016202000162020002000000018891984632653500321758097698025

Offset: 0

Patrick De Geest, May 15 1999

Next terms are 125, 251, 502 and 1003 digits long.

Cf. A048562.

Cf. A048557, A249893, A249894, A048559, A249895, A249896, A249897, A249898.

A050636 a(n+1) is next smallest square ending with a(n), initial term is 4.

Original entry on oeis.org

4, 64, 1764, 6041764, 96126041764, 8584885596126041764, 1094573934406914368584885596126041764, 164840278246153785356947805564195221094573934406914368584885596126041764

Offset: 1

Patrick De Geest, Jul 15 1999

			a(6) = 8584885596126041764 = 2929997542^2.

Cf. A050637, A048559, A050630, A061361.

More terms from Ulrich Schimke (ulrschimke(AT)aol.com), Nov 06 2001

A051671 Smallest positive number whose digits concatenated with all previous terms forms a perfect square.

Original entry on oeis.org

4, 9, 284, 61209, 14204828164, 4440027571600000000001, 7667716927194638935286982918857620344809476

Offset: 0

Felice Russo

			284 belongs to the sequence because 49284=222^2.

A. Murthy, Smar. Notions J. Vol. 11 N. 1-2-3 Spring 2000.

Cf. A048559, A048560, A051670.

More terms from Patrick De Geest, Dec 15 1999

A048557 a(1) = 1; a(n+1) is next smallest nontrivial square beginning with a(n).

Original entry on oeis.org

1, 16, 169, 169744, 16974441796, 1697444179660172924176, 16974441796601729241768240010144800000000001, 1697444179660172924176824001014480000000000182400101448000000000020000000000000000000001

Offset: 1

Patrick De Geest, May 15 1999

Next terms are 176, 351 and 702 digits long.

Robert G. Wilson v, Table of n, a(n) for n = 1..11

Cf. A048558, A050628, A061110. See A048558 for the square roots.

Cf. A249893, A249894, A048559, A249895, A249896, A249897, A249898, A048561.

$MaxExtraPrecision = 2^10; a[n_] := a[n] = Block[{k = 10, m = a[n - 1]}, id = IntegerDigits@ m; lng = Length@ id; While[c = Ceiling[ Sqrt[ m*k]]^2; Take[ IntegerDigits@ c, lng] != id, k *= 100]; Min[c, (Sqrt[m]*10^(lng/2) + 1)^2]]; a[1] = 1; Array[a, 11] (* Robert G. Wilson v, Dec 18 2014 *)

A050630 a(n+1) is next smallest nontrivial square containing a(n) as a substring, initial term is 4.

Original entry on oeis.org

4, 49, 1849, 18496, 103184964, 7383508103184964, 544680602158273835081031849649

Offset: 1

Patrick De Geest, Jul 15 1999

Starting with a(2)=49, the sequence is identical A050632.

Cf. A050631, A048559, A050636.

a(n) = A050631(n)^2.

a(7) from Max Alekseyev, Feb 15 2012

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.

Original entry on oeis.org

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

A249799 a(n+1) is the next smallest nontrivial cube beginning with a(n), initial term is 3.

Original entry on oeis.org

3, 343, 34328125, 343281258441879199985152, 3432812584418791999851522633552283890505384030397984598615707004422070464

Offset: 1

Derek Orr, Dec 03 2014

Terms a(6) and a(7) are 219 and 658 digits long, respectively. - Jon E. Schoenfield, Aug 06 2017

Jon E. Schoenfield, Table of n, a(n) for n = 1..7

Cf. A048559, A048561, A249894.

a(n)=k=n;s=1;while(s<10^7,if(s%10,if(s^3\(10^(#Str(s^3)-#Str(k)))==k,print1(s^3,", ");k=s^3));s++)
a(3)

def f(x):
    n = x
    s = 1
    while s < 10**7:
        if s % 10:
            S = str(s**3)
            if S.startswith(str(n)):
                print(s**3,end=', ')
                n = s**3
        s += 1
f(3)

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

A249894 a(n+1) is next smallest square not divisible by 10 beginning with a(n), initial term is 3.

Original entry on oeis.org

3, 36, 361, 361201, 36120142809, 3612014280924268428225, 361201428092426842822525044572369542546150009

Offset: 1

Derek Orr, Nov 08 2014

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..10

Cf. A048559, A048561.

a(n)=k=n; s=1; while(s<5*10^7, if(s%10, if(s^2\(10^(#Str(s^2)-#Str(k)))==k, print1(s^2, ", "); k=s^2)); s++)
a(3)

def f(x):
  print(x, end=', ')
  n = x
  s = 1
  while s < 10**7:
    if s % 10:
      S = str(s**2)
      if S.startswith(str(n)):
        print(s**2, end=', ')
        n = s**2
    s += 1
f(3)

a(7) corrected by Hiroaki Yamanouchi, Nov 17 2014

A249895 a(n+1) is next smallest square not divisible by 10 beginning with a(n), initial term is 5.

Original entry on oeis.org

5, 529, 529984, 52998485796, 5299848579629072273476, 529984857962907227347600150092608188876380736

Offset: 1

Derek Orr, Nov 08 2014

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..10

Cf. A048559, A048561.

a(n)=k=n; s=1; while(s<5*10^7, if(s%10, if(s^2\(10^(#Str(s^2)-#Str(k)))==k, print1(s^2, ", "); k=s^2)); s++)
a(5)

def f(x):
  print(x, end=', ')
  n = x
  s = 1
  while s < 10**7:
    if s % 10:
      S = str(s**2)
      if S.startswith(str(n)):
        print(s**2, end=', ')
        n = s**2
    s += 1
f(5)

cp's OEIS Frontend

A048561 a(n+1) is next smallest nontrivial square beginning with a(n), initial square is 9.

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A050636 a(n+1) is next smallest square ending with a(n), initial term is 4.

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A051671 Smallest positive number whose digits concatenated with all previous terms forms a perfect square.

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A048557 a(1) = 1; a(n+1) is next smallest nontrivial square beginning with a(n).

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Mathematica

A050630 a(n+1) is next smallest nontrivial square containing a(n) as a substring, initial term is 4.

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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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Haskell

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A249799 a(n+1) is the next smallest nontrivial cube beginning with a(n), initial term is 3.

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Python

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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A249894 a(n+1) is next smallest square not divisible by 10 beginning with a(n), initial term is 3.

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PARI

Python

Extensions

A249895 a(n+1) is next smallest square not divisible by 10 beginning with a(n), initial term is 5.

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