A029783 -id:A029783 - cp's OEIS frontend

A360301 Smallest exclusionary square (A029783) with exactly n distinct prime factors.

2, 18, 84, 858, 31122, 3383898, 188841114, 68588585868, 440400004044, 7722272777722272

Offset: 1

Bernard Schott, Feb 02 2023

There is no 5 in the prime factorization of these terms.
No other terms less than 10^14. - Michael S. Branicky, Feb 02 2023
1.69 * 10^15 < a(10) <= 7722272777722272. - Daniel Suteu, Feb 05 2023

			84 = 2^2 * 3 * 7 is the smallest integer with 3 distinct prime factors that is also an exclusionary square, because 84^2 = 7056, so a(3) = 84.
858 = 2 * 3 * 11 * 13 is the smallest integer with 4 distinct prime factors that is also an exclusionary square, because 858^2 = 736164, so a(4) = 858.

Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 60.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, 1997, page 144, entry 567.

Cliff Pickover et al., Exclusionary Squares and Cubes, rec.puzzles topic on google groups, January 2002.
Eric Weisstein's World of Mathematics, Distinct Prime Factors.

Cf. A029783.

Similar: A060319 (Fibonacci), A083002 (oblong), A359960 (Niven), A359961 (Zuckerman).

omega_exclusionary_squares(A, B, n) = A=max(A, vecprod(primes(n))); (f(m, p, j) = my(list=List()); forprime(q=p, sqrtnint(B\m, j), if(q == 5, next); my(v=m*q); while(v <= B, if(j==1, if(v>=A && #setintersect(Set(digits(v)), Set(digits(v^2))) == 0, listput(list, v)), if(v*(q+1) <= B, list=concat(list, f(v, q+1, j-1)))); v *= q)); list); vecsort(Vec(f(1, 2, n)));
a(n) = my(x=vecprod(primes(n)), y=2*x); while(1, my(v=omega_exclusionary_squares(x, y, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Feb 05 2023

Assuming a(n) exists, a(n) >= A002110(n+1)/5 >> exp((1 + o(1)) * n * log(n)). (The inequality is presumably strict for all n; for n > 34 it seems that all A002110(n) are pandigital.) - Charles R Greathouse IV, Feb 05 2023

a(4)-a(7) from Amiram Eldar, Feb 02 2023
a(8)-a(9) from Michael S. Branicky, Feb 02 2023
a(10) from Michael S. Branicky, Feb 07 2023

A111116 Numbers n such that digits of n are not present in n^4.

Original entry on oeis.org

2, 3, 4, 7, 8, 9, 24, 27, 28, 32, 33, 42, 52, 53, 58, 59, 67, 77, 88, 89, 93, 202, 203, 258, 284, 303, 324, 329, 377, 383, 422, 669, 818, 832, 843, 878, 882, 887, 949, 2027, 2042, 2673, 3144, 3222, 3253, 3302, 3308, 3737, 3773, 3953, 3979, 3983, 4779, 5353, 5669

Offset: 1

Lekraj Beedassy, Oct 15 2005

The number of k-digit numbers for which this occurs is: 6,15,18,32,21,14,20,...

Giovanni Resta, Table of n, a(n) for n = 1..179 (terms < 10^19, first 136 terms from Chai Wah Wu)

Cf. A029783, A029785.

For the corresponding n^4, see A113316.

Select[Range[6000], Intersection[IntegerDigits[ # ], IntegerDigits[ #^4]] == {} &] (* Ray Chandler, Oct 17 2005 *)
fQ[n_] := Intersection[ Union[ IntegerDigits[n]], Union[ IntegerDigits[n^4]]] == {}; Select[ Range[ 5887], fQ[ # ] &] (* Robert G. Wilson v *)

A111116_list = [n for n in range(1,10**6) if set(str(n)) & set(str(n**4)) == set()]
# Chai Wah Wu, Jan 05 2015

Corrected and extended by Robert G. Wilson v and Ray Chandler, Oct 17 2005

A029785 Numbers k whose cube k^3 has no digit in common with k.

Original entry on oeis.org

2, 3, 7, 8, 22, 27, 43, 47, 48, 52, 53, 63, 68, 77, 92, 157, 172, 177, 187, 188, 192, 222, 223, 252, 263, 303, 378, 408, 423, 442, 458, 468, 477, 478, 487, 527, 552, 558, 577, 587, 588, 608, 648, 692, 707, 772, 808, 818, 823, 843, 888, 918, 922

Offset: 1

Patrick De Geest

Original name: Digits of n are not present in n^3.

Might be called "Exclusionary Cubes", although this might be reserved for terms having no duplicate digits, cf. link to rec.puzzles discussion group. In that case the largest term would be 7658 = A113951(3). - M. F. Hasler, Oct 17 2018; corrected thanks to David Radcliffe and Michel Marcus, Apr 30 2020

			k = 80800000008880080808880080088 is in the sequence because the 87-digit number k^3 has only digits 1, ..., 7 and 9. - _M. F. Hasler_, Oct 16 2018

Giovanni Resta, Table of n, a(n) for n = 1..1699 (terms < 10^19, first 528 terms from Charles R Greathouse IV)
Cliff Pickover et al, Exclusionary Squares and Cubes, rec.puzzles topic on google groups, January 2002

Cf. A029786, A029783, A029784, A111116, A113316.

Select[Range[5000], Intersection[IntegerDigits[#], IntegerDigits[#^3]]=={}&] (* Vincenzo Librandi, Oct 04 2013 *)

is(n)=my(d=Set(digits(n))); setminus(d,Set(digits(n^3)))==d \\ Charles R Greathouse IV, Oct 02 2013

is_A029785(n)=setintersect(Set(digits(n)),Set(digits(n^3)))==[] \\ M. F. Hasler, Oct 16 2018

Name reworded by Jon E. Schoenfield and M. F. Hasler, Oct 16 2018

A076493 Number of common (distinct) decimal digits of n and n^2.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 2, 1, 2, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 1, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 2, 2, 1, 1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 2, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 3, 2, 1, 1, 1

Offset: 0

Labos Elemer, Oct 21 2002

a(A029783(n)) = 0, a(A189056(n)) > 0; 0 <= a(n) <= 10, see example for first occurrences. [Reinhard Zumkeller, Apr 16 2011]

			a(2) = #{} = 0: 2^2 = 4;
a(0) = #{0} = 1: 0^2 = 0;
a(10) = #{0,1} = 2: 10^2 = 100;
a(105) = #{0,1,5} = 3: 105^2 = 11025;
a(1025) = #{0,1,2,5} = 4: 1025^2 = 1050625;
a(10245) = #{0,1,2,4,5} = 5: 10245^2 = 104960025;
a(102384) = #{0,1,2,3,4,8} = 6: 102384^2 = 10482483456;
a(1023456789) = #{0 .. 9} = 10: 1023456789^2 = 1047463798950190521.

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

Cf. A006880, A076489-A076491.

Cf. A000290, A043537.

import Data.List (intersect, nub)
import Data.Function (on)
a076493 n = length $ (intersect `on` nub . show) n (n^2)
-- Reinhard Zumkeller, Apr 16 2011

Table[Length[Intersection[IntegerDigits[n], IntegerDigits[n^2]]], {n, 1, 100}]

Initial 1 added and offset adjusted by Reinhard Zumkeller, Apr 16 2011

A189056 Numbers having in decimal representation at least one common digit with their squares.

Original entry on oeis.org

0, 1, 5, 6, 10, 11, 12, 13, 14, 15, 16, 19, 20, 21, 23, 25, 26, 27, 28, 30, 31, 32, 35, 36, 37, 40, 41, 42, 43, 45, 46, 48, 49, 50, 51, 52, 55, 56, 60, 61, 63, 64, 65, 66, 68, 69, 70, 71, 73, 74, 75, 76, 78, 80, 81, 82, 83, 85, 86, 87, 89, 90, 91, 93, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120

Offset: 1

Reinhard Zumkeller, Apr 16 2011

Complement of A029783; A076493(a(n)) > 0.

A258682(a(n)) < a(n)^2. - Reinhard Zumkeller, Jun 07 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A029783, A076493, A258682.

a189056 n = a189056_list !! (n-1)
a189056_list = 0 : filter (\x -> a258682 x /= x ^ 2) [1..]
-- Reinhard Zumkeller, Jun 07 2015, Apr 16 2011

Select[Range[0,120],Length[Intersection[IntegerDigits[#], IntegerDigits[ #^2]]]>0&] (* Harvey P. Dale, Aug 28 2012 *)

isok(n) = (n==0) || #setintersect(Set(digits(n)), Set(digits(n^2))); \\ Michel Marcus, Jun 13 2015

A113316 Fourth powers m^4 none of whose digits are present in their corresponding roots m.

Original entry on oeis.org

16, 81, 256, 2401, 4096, 6561, 331776, 531441, 614656, 1048576, 1185921, 3111696, 7311616, 7890481, 11316496, 12117361, 20151121, 35153041, 59969536, 62742241, 74805201, 1664966416, 1698181681, 4430766096, 6505390336, 8428892481

Offset: 1

Lekraj Beedassy, Oct 26 2005

For the associated root m, see A111116.

Giovanni Resta, Table of n, a(n) for n = 1..179 (terms < 10^76, first 100 terms from Harvey P. Dale)

Cf. A111116, A029783, A029784, A029785, A029786.

Select[Range[350]^4,Intersection[IntegerDigits[#], IntegerDigits[ Surd[ #,4]]] =={}&] (* Harvey P. Dale, Dec 19 2015 *)

A029784 Squares such that digits of sqrt(n) are not present in n.

Original entry on oeis.org

4, 9, 16, 49, 64, 81, 289, 324, 484, 576, 841, 1089, 1156, 1444, 1521, 1936, 2209, 2809, 2916, 3249, 3364, 3481, 3844, 4489, 5184, 5929, 6241, 7056, 7744, 8464, 8836, 20736, 24649, 24964, 29929, 34969, 35344, 36864, 37636, 43681, 44944, 50176, 56169, 56644

Offset: 1

Patrick De Geest

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A029783, A029785, A029786, A111116, A113316.

Select[Range[500], {} == Intersection @@ IntegerDigits[{#, #^2}] &]^2 (* Giovanni Resta, Aug 08 2018 *)

a(n) = A029783(n)^2. - Michel Marcus, Aug 08 2018

More terms from Giovanni Resta, Aug 08 2018

A112736 Numbers whose square is exclusionary.

Original entry on oeis.org

2, 3, 4, 7, 8, 9, 17, 18, 24, 29, 34, 38, 39, 47, 53, 54, 57, 58, 59, 62, 67, 72, 79, 84, 92, 94, 157, 158, 173, 187, 192, 194, 209, 237, 238, 247, 253, 257, 259, 307, 314, 349, 359, 409, 437, 459, 467, 547, 567, 612, 638, 659, 672, 673, 689, 712, 729, 738, 739, 749

Offset: 1

Lekraj Beedassy, Sep 16 2005

The number m with no repeated digits has an exclusionary square m^2 if the latter is made up of digits not appearing in m. For the corresponding exclusionary squares see A112735.

a(49) = 567 and a(68) = 854 are the only two numbers k such that the equation k^2 = m uses only once each of the digits 1 to 9 (reference David Wells). Exactly: 567^2 = 321489, and, 854^2 = 729316. - Bernard Schott, Dec 20 2021

			409^2 = 167281 and the square 167281 is made up of digits not appearing in 409, hence 409 is a term.

H. Ibstedt, Solution to Problem 2623, "Exclusionary Powers", pp. 346-9, Journal of Recreational Mathematics, Vol. 32 No.4 2003-4 Baywood NY.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, 1997, page 144, entry 567.

Giovanni Resta, Table of n, a(n) for n = 1..142 (full sequence)

Cf. A000290, A112321, A112735.

This is a subsequence of A029783 (Digits of n are not present in n^2) of numbers with all different digits. The sequence A059930 (Numbers n such that n and n^2 combined use different digits) is a subsequence of this sequence.

Select[Range[1000], Intersection[IntegerDigits[ # ], IntegerDigits[ #^2]] == {} && Length[Union[IntegerDigits[ # ]]] == Length[IntegerDigits[ # ]] &] (* Tanya Khovanova, Dec 25 2006 *)

More terms from Tanya Khovanova, Dec 25 2006

A029786 Cubes such that digits of cube root of n are not present in n.

Original entry on oeis.org

8, 27, 343, 512, 10648, 19683, 79507, 103823, 110592, 140608, 148877, 250047, 314432, 456533, 778688, 3869893, 5088448, 5545233, 6539203, 6644672, 7077888, 10941048, 11089567, 16003008, 18191447, 27818127, 54010152, 67917312, 75686967, 86350888, 96071912

Offset: 1

Patrick De Geest

Giovanni Resta, Table of n, a(n) for n = 1..1699 (terms < 10^57)

Cf. A029785, A029783, A029784, A111116, A113316.

Select[Range[1000], {} == Intersection @@ IntegerDigits[{#, #^3}] &]^3 (* Giovanni Resta, Aug 08 2018 *)

More terms from Giovanni Resta, Aug 08 2018

A059930 Numbers n such that n and n^2 combined use different digits.

Original entry on oeis.org

2, 3, 4, 7, 8, 9, 17, 18, 24, 29, 53, 54, 57, 59, 72, 79, 84, 209, 259, 567, 807, 854

Offset: 1

Patrick De Geest, Feb 15 2001

There are exactly 22 solutions in base 10.
More precisely: the concatenation of n and n^2 does not contain any digit twice. - M. F. Hasler, Oct 16 2018
a(20) = 567 and a(22) = 854 are the only two numbers k such that k and k^2 combined use each of the digits 1 to 9 exactly once (reference David Wells): 567^2 = 321489 and 854^2 = 729316. - Bernard Schott, Mar 23 2021

M. Kraitchik, Mathematical Recreations, p. 48, Problem 12. - From N. J. A. Sloane, Mar 15 2013
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, 1997, page 144, entry 567.

Cf. A059931, A029783 (digits of n are not present in n^2), A112736 (numbers whose squares are exclusionary).

M:=1000;
a1:=[]; a2:=[];
for n from 1 to M do
# are digits of n and n^2 distinct?
t1:=convert(n,base,10);
t2:=convert(n^2,base,10);
s3:={op(t1),op(t2)};
if nops(t1)+nops(t2) = nops(s3) then a1:=[op(a1),n]; a2:=[op(a2),n^2]; fi;
od:
a1; a2;
# N. J. A. Sloane, Mar 15 2013

Select[Range[10000], Intersection[IntegerDigits[ # ], IntegerDigits[ #^2]] == {} && Length[Union[IntegerDigits[ # ], IntegerDigits[ #^2]]] == Length[IntegerDigits[ # ]] + Length[IntegerDigits[ #^2]] &] (* Tanya Khovanova, Dec 25 2006 *)
Select[Range[10^3], Union@ Tally[Flatten@ IntegerDigits@ {#, #^2}][[All, -1]] == {1} &] (* Michael De Vlieger, Oct 17 2018 *)

select( is(n)=#Set(Vecsmall(n=Str(n,n^2)))==#n, [1..999]) \\ M. F. Hasler, Oct 16 2018

cp's OEIS Frontend

A360301 Smallest exclusionary square (A029783) with exactly n distinct prime factors.

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A111116 Numbers n such that digits of n are not present in n^4.

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A029785 Numbers k whose cube k^3 has no digit in common with k.

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A076493 Number of common (distinct) decimal digits of n and n^2.

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A189056 Numbers having in decimal representation at least one common digit with their squares.

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A113316 Fourth powers m^4 none of whose digits are present in their corresponding roots m.

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A029784 Squares such that digits of sqrt(n) are not present in n.

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