A029783 -id:A029783 - cp's OEIS frontend

A030086 Primes p whose digits do not appear in p^2.

2, 3, 7, 17, 29, 47, 53, 59, 67, 79, 157, 173, 257, 307, 313, 349, 353, 359, 409, 449, 467, 547, 557, 577, 659, 673, 677, 739, 787, 853, 929, 1447, 1607, 1747, 2029, 2087, 2113, 2237, 3257, 3359, 3467, 3533, 3559, 3767, 3779, 4253, 4337, 4787, 5333, 5557, 5659

Offset: 1

Patrick De Geest, Dec 11 1999

Primes of sequence A029783. - Michel Marcus, Jan 04 2015

			2 and 2^2 = 4 have no digits in common, hence 2 is in the sequence.

Michael S. Branicky, Table of n, a(n) for n = 1..13322 (terms < 10^16; terms 502..676 from Harvey P. Dale and Vincenzo Librandi, terms 1..501 from Harvey P. Dale)

Cf. A029783 (numbers n whose digits are not present in n^2).

Cf. similar sequences listed in A253574.

Select[Prime[Range[700]],Intersection[IntegerDigits[#],IntegerDigits[ #^2]] == {}&] (* Harvey P. Dale, Oct 02 2014 *)

lista(nn) = {forprime (n=1, nn, if (#setintersect(Set(vecsort(digits(n^2))), Set(vecsort(digits(n)))) == 0, print1(n, ", ")); ); } \\ Michel Marcus, Jan 04 2015

Offset changed from 0 to 1 from Vincenzo Librandi, Jan 04 2015

A258682 Remove from n^2 all digits of n from left to right, in decimal representation.

Original entry on oeis.org

0, 0, 4, 9, 16, 2, 3, 49, 64, 81, 0, 2, 44, 69, 96, 22, 25, 289, 324, 36, 40, 44, 484, 59, 576, 6, 76, 9, 74, 841, 90, 96, 104, 1089, 1156, 122, 129, 169, 1444, 1521, 160, 681, 176, 189, 1936, 202, 211, 2209, 230, 201, 20, 260, 704, 2809, 2916, 302, 313

Offset: 0

Reinhard Zumkeller, Jun 07 2015

a(A029783(n)) = A029783(n)^2; a(A189056(n)) < A189056(n)^2;

a(A178501(n)) = 0.

			.    n   n^2 |  a(n)
. -----------+------
.   20   400 |    40
.   21   441 |    44
.   22   484 |   484
.   23   529 |    59
.   24   576 |   576
.   25   625 |     6
.   26   676 |    76  not 67, as digits of n are to be removed
.   27   729 |     9                                  from left to right
.   28   784 |    74
.   29   841 |   841
.   30   900 |    90  .

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A000290, A029783, A189056, A178501.

import Data.List (delete)
a258682 n = read ('0' : minus (show (n ^ 2)) (show n)) :: Int  where
   minus [] _  = []
   minus us [] = us
   minus (u:us) vs | elem u vs = minus us $ delete u vs
                   | otherwise = u : minus us vs

a[n_]:=Module[{idn=IntegerDigits[n],idnn=IntegerDigits[n^2],idn1},
idn1=DeleteCases[idn,m_/;MemberQ[Complement[idn,idnn],m]];
Do[If[First[Position[idnn,idn1[[i]]]]!= {},idnn=Delete[idnn,First[Position[idnn,idn1[[i]]]]]],
{i,1,Length[idn1]}];FromDigits[idnn]]//Quiet;
a/@Range[0,56] (* Ivan N. Ianakiev, Jun 11 2015 *)

A103173 Numbers k such that the decimal digits of k are not present in k^2, k^3, or k^4.

Original entry on oeis.org

2, 3, 7, 8, 53, 77

Offset: 1

Labos Elemer, Feb 28 2005

No more terms exist below 10^100. The last five digits of any larger term must be 75557, 85557, 88787, 88188, 88988 or 98988. - Hagen von Eitzen, Jun 16 2009

			For k=77, the 1st through 4th powers are {77, 5929, 456533, 35153041}.
Digits of k appear first in the 5th powers {32, 243, 16807, 32768, 418195493, 2706784157}.

Cf. A029783, A029790.

Select[Range[10^3],ContainsNone[IntegerDigits[#],Union[IntegerDigits[#^2],IntegerDigits[#^3],IntegerDigits[#^4]]]&] (* James C. McMahon, Jan 17 2024 *)

A110815 Least n-digit number m whose square contains only digits not appearing in m.

Original entry on oeis.org

2, 17, 144, 1447, 14144, 141494, 1414414, 14144134, 141431114, 1414411113, 14143143413, 141431113114, 1414311131114, 14143111141813, 141431113114113, 1414311344113314, 14143111141141113, 141431113114331413

Offset: 1

Lekraj Beedassy, Aug 17 2005

floor(sqrt(2)*10^(n-1)) < a(n). - Robert G. Wilson v, Oct 04 2005

Cf. A110816 (corresponding squares), A112321. Subsequence of A029783.

f[n_] := Block[{k = Ceiling[10^n*(1/9 + .03032)]}, While[ Intersection[ IntegerDigits[k], IntegerDigits[k^2]] != {}, k++ ]; k]; Table[ f[n], {n, 18}] (* Robert G. Wilson v *)

from math import isqrt
def a(n):
  m = isqrt(int('2'+'0'*(2*n-2)))
  while set(str(m*m)) & set(str(m)) != set(): m += 1
  return m
print([a(n) for n in range(1, 12)]) # Michael S. Branicky, Feb 17 2021

a(3) to a(6) corrected, a(7) to a(13) from Klaus Brockhaus, Aug 31 2005; revised Sep 09 2005

a(14)-a(18) from Robert G. Wilson v, Oct 04 2005

A387070 Remove every digit that appears in n from the decimal representation of n^2. If no digits remain, set a(n) = 0.

Original entry on oeis.org

0, 0, 4, 9, 16, 2, 3, 49, 64, 81, 0, 2, 44, 69, 96, 22, 25, 289, 324, 36, 4, 44, 484, 59, 576, 6, 7, 9, 74, 841, 9, 96, 104, 1089, 1156, 122, 129, 169, 1444, 1521, 16, 68, 176, 189, 1936, 202, 211, 2209, 230, 201, 2, 260, 704, 2809, 2916, 302, 313, 3249, 3364, 3481, 3, 372, 3844, 99, 9, 422, 435

Offset: 0

Ali Sada, Aug 15 2025

			a(25) = 6 since 25^2 = 625 and once we remove the 2 and 5, we are left with 6.
a(26) = 7 since 26^2 = 676 and once we remove the 6, we are left with 7.

Cf. A029793, A258682.

a[n_] := FromDigits[Select[IntegerDigits[n^2], FreeQ[IntegerDigits[n], #] &]]; Array[a, 100, 0] (* Amiram Eldar, Aug 16 2025 *)

a(n)={my(S=Set(digits(n))); fromdigits(select(x->!setsearch(S,x), digits(n^2)))} \\ Andrew Howroyd, Aug 15 2025

def A387070(k):
    s = set(str(k))
    t = "".join(d for d in str(k**2) if d not in s)
    return int(t) if t != "" else 0
print([A387070(n) for n in range(67)]) # Michael S. Branicky, Aug 16 2025

a(n) = A258682(n) for n <= 19.
From David A. Corneth, Aug 16 2025: (Start)
a(A029793(k)) = 0.
a(A029783(k)) = A029783(k)^2. (End)

A059931 Numbers n such that n and n^(1/2) combined use different digits.

Original entry on oeis.org

4, 9, 16, 49, 64, 81, 289, 324, 576, 841, 2809, 2916, 3249, 3481, 5184, 6241, 7056, 43681, 67081, 321489, 651249, 729316

Offset: 1

Patrick De Geest, Feb 15 2001

There are exactly 22 solutions in base 10.

Cf. A059930, A029783.

A247843 Consecutive exclusionary squares: Numbers n such that n^2 does not contain digits of n and (n+1)^2 does not contain digits of n+1.

Original entry on oeis.org

2, 3, 7, 8, 17, 33, 38, 53, 57, 58, 157, 187, 237, 313, 333, 557, 558, 672, 688, 738, 787, 788, 812, 813, 853, 1557, 2087, 2107, 2112, 3112, 3113, 3157, 3333, 3357, 3358, 4453, 4553, 5598, 5857, 6672, 6688, 7017, 7287, 7772, 7888, 7908, 8087, 8337, 15787, 17157, 18557, 22112, 32358

Offset: 1

Derek Orr, Sep 25 2014

Derek Orr, Table of n, a(n) for n = 1..53

Cf. A029783.

for n in range(10**5):
  s,t = str(n),str(n+1)
  s2,t2 = str(n**2),str((n+1)**2)
  c = 0
  for i in s:
    if s2.count(i):
      c += 1
      break
  for j in t:
    if t2.count(j):
      c += 1
      break
  if not c:
    print(n,end=', ')

A285211 Numbers k that share no digits with the k-th triangular number.

Original entry on oeis.org

2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 24, 26, 28, 29, 30, 31, 33, 34, 38, 39, 42, 44, 46, 47, 48, 49, 52, 58, 59, 62, 64, 66, 69, 70, 71, 77, 79, 82, 84, 86, 88, 89, 114, 115, 117, 122, 124, 129, 131, 132, 246, 252, 258, 259, 262, 266, 270, 271, 277, 282, 286

Offset: 1

Colin Barker, Apr 13 2017

Robert Israel, Table of n, a(n) for n = 1..6434

Cf. A000217, A029783.

filter:= proc(n) convert(convert(n,base,10),set) intersect convert(convert(n*(n+1)/2,base,10),set) = {} end proc:
select(filter, [$1..1000]); # Robert Israel, Apr 21 2017

L=List(); for(n=1, 500, if(setintersect(vecsort(digits(n),,8), vecsort(digits(n*(n+1)/2),,8))==[], listput(L, n))); Vec(L)

A385515 Repdigit numbers whose square does not contain the repeated digit.

Original entry on oeis.org

2, 3, 4, 7, 8, 9, 22, 33, 44, 77, 88, 333, 444, 3333, 33333, 44444, 88888, 333333, 3333333, 33333333, 333333333, 3333333333, 33333333333, 333333333333, 3333333333333, 33333333333333, 333333333333333, 3333333333333333, 33333333333333333, 333333333333333333

Offset: 1

Gonzalo Martínez, Jul 01 2025

For n >= 18, all terms are of the form 33...3; that is, elements of A002277.
A002277(m) is a term, for m > 0. Proof: 3 is in a(n) because 3^2 = 9. If 33...3 is composed of k 3's, with k > 1, it is satisfied that 33...3^2 = 11...1088...89; i.e., (k - 1) 1's followed by a 0, then (k - 1) 8's and a 9, so that 3 is not among the digits of its square.
Let's see that there are no other terms of the form 33...3 besides 2, 4, 7, 8, 9, 22, 44, 77, 88, 444, 44444, 88888. In this sequence there are no repdigits of the form 11...1, 55...5, 66...6, since their squares end in 1, 5 and 6 respectively. On the other hand, 9 is the only number of the form 999...9, since if it has 2 or more 9's its square starts with 9. Suppose that dd...d contains 6 or more digits. We already saw that the cases d = 1, 5, 6 and 9 are discarded. Let us analyze what happens for d = 2, 4, 7 and 8:
For d = 2, we have that 22...2^2 == 284 (mod 10^3).
For d = 4, we have that 44...4^2 == 469136 (mod 10^6).
For d = 7, we have that 77...7^2 == 729 (mod 10^3).
For d = 8, we have that 88...8^2 == 876544 (mod 10^6).
Thus, we conclude that a(n) only consists of digits 3 for n >= 18. And, in fact, a(n) consists of (n - 12) 3's.

			22 is a term since 22^2 = 484 does not contain the digit 2.

Cf. A002277, A010785, A029783.

Intersection of A010785 and A029783.

Select[Union@ Flatten@ Table[k (10^n - 1)/9, {k, 0, 9}, {n, 18}] ,ContainsNone[IntegerDigits[#^2],IntegerDigits[#]]&] (* James C. McMahon, Jul 07 2025 *)

a(n) = A002277(n - 12), for n >= 18.

cp's OEIS Frontend

A030086 Primes p whose digits do not appear in p^2.

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A258682 Remove from n^2 all digits of n from left to right, in decimal representation.

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Haskell

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A103173 Numbers k such that the decimal digits of k are not present in k^2, k^3, or k^4.

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Mathematica

A110815 Least n-digit number m whose square contains only digits not appearing in m.

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A387070 Remove every digit that appears in n from the decimal representation of n^2. If no digits remain, set a(n) = 0.

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A059931 Numbers n such that n and n^(1/2) combined use different digits.

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A247843 Consecutive exclusionary squares: Numbers n such that n^2 does not contain digits of n and (n+1)^2 does not contain digits of n+1.

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Python

A285211 Numbers k that share no digits with the k-th triangular number.

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Maple

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A385515 Repdigit numbers whose square does not contain the repeated digit.

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