A029793 -id:A029793 - cp's OEIS frontend

A029795 Numbers k such that k and k^3 have the same set of digits.

0, 1, 10, 100, 1000, 10000, 100000, 107624, 109573, 132485, 138624, 159406, 165640, 192574, 205738, 215806, 251894, 281536, 318725, 419375, 427863, 568314, 642510, 713960, 953867, 954086, 963218, 965760, 1000000, 1008529, 1023479

Offset: 1

Patrick De Geest

Conjecture: there exists some m and N for which a(n) = m + n for all n >= N. - Charles R Greathouse IV, Jun 28 2011

			109573^3 = 1315559990715517. Since both numbers use the digits 0, 1, 3, 5, 7, 9, and no others, 109573 is in the sequence.

Nathaniel Johnston, Table of n, a(n) for n = 1..500

Cf. A029793, A029797, A232659-A232662.

[ n: n in [0..8*10^6] | Set(Intseq(n)) eq Set(Intseq(n^3)) ];  // Bruno Berselli, Jun 28 2011

seq(`if`(convert(convert(n, base, 10), set) = convert(convert(n^3, base, 10), set), n, NULL), n=0..500000); # Nathaniel Johnston, Jun 28 2011

Select[Range[0, 199999], Union[IntegerDigits[#]] == Union[IntegerDigits[#^3]] &] (* Alonso del Arte, Jan 12 2020 *)

isA029795(n)=Set(Vec(Str(n)))==Set(Vec(Str(n^3))) \\ Charles R Greathouse IV, Jun 28 2011

A178501 Zero followed by powers of ten.

Original entry on oeis.org

0, 1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000, 10000000000, 100000000000, 1000000000000, 10000000000000, 100000000000000, 1000000000000000, 10000000000000000, 100000000000000000, 1000000000000000000, 10000000000000000000, 100000000000000000000

Offset: 0

Reinhard Zumkeller, May 28 2010

The sequence S consisting of the nonnegative numbers arranged in lexicographic order according to their decimal expansion begins 0, 1, 10, 100, 1000, ..., 2, 20, 200, 2000, ..., 3, 30, ... does not have an OEIS entry, since there are uncountably many terms before 2 appears (or even before 100000010000 appears). However, S does begin with the present sequence. - N. J. A. Sloane, Dec 09 2024

a(n)^k + reverse(a(n))^k is a palindrome for any positive integer k. - Bui Quang Tuan, Mar 31 2015

Index entries for linear recurrences with constant coefficients, signature (10).

Cf. A093136, A131577, A140429, A178500; subsequence of A029793.

The powers of 10, A011557, is a subsequence.

Join[{0}, 10^Range[0, 19]] (* Alonso del Arte, Mar 31 2015 *)

a(n)=10^n\10 \\ Charles R Greathouse IV, Oct 07 2015

concat(0, Vec(-x/(10*x-1) + O(x^22))) \\ Elmo R. Oliveira, Jul 21 2025

a(n+1) = A011557(n).
a(n) = A178500(n)/10.
From Paul Barry, Jul 09 2003: (Start)
a(n) = (10^n - 0^n)/10.
E.g.f.: exp(5*x)*sinh(5*x)/5.
Binomial transform of A015577. (End)
G.f.: x/(1 - 10*x). - Chai Wah Wu, Jun 17 2020
From Elmo R. Oliveira, Jul 21 2025: (Start)
a(n) = 10*a(n-1) for n > 1.
a(n) = A093136(n)/2 for n >= 1. (End)

More terms from Elmo R. Oliveira, Jul 21 2025

A232659 Numbers n such that n and n^4 have the same set of digits.

Original entry on oeis.org

0, 1, 10, 100, 1000, 10000, 35641, 100000, 129486, 146980, 356410, 465780, 1000000, 1059281, 1083749, 1206794, 1239876, 1245890, 1265360, 1294860, 1297853, 1348970, 1469800, 1486920, 1495860, 1567038, 1572086, 1574689, 1956740, 2035817, 2084615, 2114760

Offset: 1

Arkadiusz Wesolowski, Nov 27 2013

			{1, 3, 4, 5, 6} - the set of digits of 35641 and of 35641^4, so 35641 is in the sequence.

Lars Blomberg, Table of n, a(n) for n = 1..10000

Cf. A029793, A029795, A232660-A232662.

[n : n in [0..2114760] | Set(Intseq(n)) eq Set(Intseq(n^4))];

Select[Range[0,22*10^5],Union[IntegerDigits[#]]== Union[ IntegerDigits[ #^4]]&] (* Harvey P. Dale, Aug 02 2016 *)

for(n=0, 2114760, if(Set(Vec(Str(n)))==Set(Vec(Str(n^4))), print1(n, ", ")));

A232662 Numbers n such that n and n^7 have the same set of digits.

Original entry on oeis.org

0, 1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 12635940, 26875130, 29851046, 31572460, 36082794, 38625410, 39756810, 42675139, 47025831, 50748936, 58291760, 65279801, 68249735, 76942451, 78952160, 80572614, 100000000, 102359784, 102374865

Offset: 1

Arkadiusz Wesolowski, Nov 27 2013

			{0, 1, 2, 3, 4, 5, 6, 9} - the set of digits of 12635940 and of 12635940^7, so 12635940 is in the sequence.

Lars Blomberg, Table of n, a(n) for n = 1..10000

Cf. A029793, A029795, A232659-A232661.

for(n=0, 102374865, if(Set(Vec(Str(n)))==Set(Vec(Str(n^7))), print1(n, ", ")));

A029797 Numbers k such that k^2 and k^3 have the same set of digits.

Original entry on oeis.org

0, 1, 10, 100, 146, 1000, 1203, 1460, 7652, 8077, 8751, 8965, 10000, 10406, 11914, 12030, 12057, 12586, 12768, 12961, 13055, 14202, 14600, 14625, 16221, 19350, 20450, 21539, 22040, 22175, 23682, 24071, 25089, 25201, 25708, 26653, 26981

Offset: 1

Patrick De Geest

Conjecture: there exists some m and N for which a(n) = m + n for all n >= N. [Charles R Greathouse IV, Jun 28 2011]

This conjecture is false. If the conjecture is true then for some N we would have k is in the sequence if k >= n. But 10^e + 1 (A062397) is not in the sequence for any integer e >= 0. - David A. Corneth, Nov 13 2023

			146 is in the sequence as 146^2 = 21316 has digits {1, 2, 3, 6} and 146^3 = 3112136 has digits {1, 2, 3, 6} as well. - _David A. Corneth_, Nov 13 2023

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A029793, A029795, A029800, A062397.

Cf. A011557 (a subsequence).

[ n: n in [0..34*10^4] | Set(Intseq(n^2)) eq Set(Intseq(n^3)) ];  // Bruno Berselli, Jun 28 2011

isA029797(n)=Set(Vec(Str(n^2)))==Set(Vec(Str(n^3))) \\ Charles R Greathouse IV, Jun 28 2011

A257763 Zeroless numbers n such that n and n^2 have the same set of decimal digits.

Original entry on oeis.org

1, 4762, 4832, 12385, 14829, 26394, 34196, 36215, 49827, 68474, 72576, 74528, 79286, 79836, 94583, 94867, 96123, 98376, 123385, 123546, 124235, 124365, 124579, 124589, 125476, 125478, 126969, 129685, 135438, 139256, 139261, 139756, 149382, 152385, 156242

Offset: 1

Alois P. Heinz, May 07 2015

			4762 is in the sequence because it is zeroless and 4762^2 = 22676644 has the same set of decimal digits as 4762: {2,4,6,7}.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A029793, A052382, A257760.

a:= proc(n) option remember; local k, s;
      for k from 1+`if`(n=1, 0, a(n-1)) do
        s:= {convert(k, base, 10)[]};
        if not 0 in s and s={convert(k^2, base, 10)[]}
           then return k fi
      od
    end:
seq(a(n), n=1..10);

sameQ[n_]:=Union[IntegerDigits[n]]==Union[IntegerDigits[n^2]];Select[Range@156242,And[FreeQ[IntegerDigits[#],0],sameQ[#]]&] (* Ivan N. Ianakiev, May 08 2015 *)

isok(n) = vecmin(d=digits(n)) && Set(d) == Set(digits(n^2)); \\ Michel Marcus, May 31 2015

A257763_list = [n for n in range(1,10**6) if not '0' in str(n) and set(str(n)) == set(str(n**2))] # Chai Wah Wu, May 31 2015

{A029793} intersect {A052382}.

A257760 Zeroless numbers n such that the products of the decimal digits of n and n^2 coincide.

Original entry on oeis.org

1, 1488, 3381, 14889, 18489, 181965, 262989, 338646, 358489, 367589, 437189, 438329, 479285, 781839, 964941, 1456589, 1763954, 2579285, 2868489, 3365285, 3419389, 3451988, 3584889, 3625619, 4378829, 4653989, 6868877, 7295986, 9548479, 14529839, 14534488

Offset: 1

Pieter Post, May 07 2015

It is unknown if this sequence is infinite.
Number of terms < 10^n: 1, 1, 1, 3, 5, 15, 29, 75, 211, 583, 1694, ..., . - Robert G. Wilson v, May 25 2015
Also nontrivial numbers n such that the products of the decimal digits of n and n^2 are equal. Trivial solutions are any number which contains a zero in its decimal expansion. - Robert G. Wilson v, May 11 2015

			1488 is in the sequence since 1488^2 = 2214144 and we have 256 = 1*4*8*8 = 2*2*1*4*1*4*4.
3381 is in the sequence because 3381^2 = 11431161 and 72 = 3*3*8*1 = 1*1*4*3*1*1*6*1.

Robert G. Wilson v, Table of n, a(n) for n = 1..1747 (first 75 terms from Pieter Post)

Subsequence of A052040.

Cf. A000290, A007954, A029793, A052382, A257763.

fQ[n_] := Times @@ IntegerDigits[n] == Times @@ IntegerDigits[n^2] > 0; Select[ Range@ 10000000, fQ] (* Robert G. Wilson v, May 07 2015 *)

isok(n) = (d = digits(n)) && vecmin(d) && (dd = digits(n^2)) && (prod(k=1, #d, d[k]) == prod(k=1, #dd, dd[k])); \\ Michel Marcus, May 07 2015

A232660 Numbers n such that n and n^5 have the same set of digits.

Original entry on oeis.org

0, 1, 10, 100, 1000, 10000, 39568, 100000, 395680, 1000000, 2114325, 2751490, 3246105, 3956800, 4356891, 4768209, 4926051, 6274019, 8021439, 10000000, 10267394, 10352849, 10368279, 10456932, 10478632, 10489723, 10489725, 10527934, 10567293, 10639428, 10827439

Offset: 1

Arkadiusz Wesolowski, Nov 27 2013

			{3, 5, 6, 8, 9} - the set of digits of 39568 and of 39568^5, so 39568 is in the sequence.

Lars Blomberg, Table of n, a(n) for n = 1..10000
B. Avila and T. Khovanova, Free Fibonacci Sequences, arXiv preprint arXiv:1403.4614, 2014

Cf. A029793, A029795, A232659, A232661, A232662.

[n : n in [0..10827439] | Set(Intseq(n)) eq Set(Intseq(n^5))];

for(n=0, 10827439, if(Set(Vec(Str(n)))==Set(Vec(Str(n^5))), print1(n, ", ")));

A232661 Numbers n such that n and n^6 have the same set of digits.

Original entry on oeis.org

0, 1, 10, 100, 1000, 10000, 100000, 1000000, 1380796, 10000000, 10423786, 10489362, 10532689, 10689247, 10743958, 12645980, 13042697, 13674925, 13807960, 14205893, 14857690, 16892043, 17284360, 17983256, 19046537, 19754203, 20634971, 20637451, 21865409

Offset: 1

Arkadiusz Wesolowski, Nov 27 2013

			{0, 1, 3, 6, 7, 8, 9} - the set of digits of 1380796 and of 1380796^6, so 1380796 is in the sequence.

Lars Blomberg, Table of n, a(n) for n = 1..10000

Cf. A029793, A029795, A232659, A232660, A232662.

for(n=0, 21865409, if(Set(Vec(Str(n)))==Set(Vec(Str(n^6))), print1(n, ", ")));

A030091 Primes such that p and p^2 have same set of digits.

Original entry on oeis.org

94583, 100469, 102953, 107251, 110923, 184903, 279863, 285101, 406951, 459521, 493621, 499423, 504821, 684581, 752681, 758141, 758941, 786431, 836291, 843701, 928637, 976513, 980261, 1008947, 1009859, 1024399, 1029647

Offset: 1

Patrick De Geest

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A000040, A029793.

import Data.List (nub, sort)
import Data.Function (on)
a030091 n = a030091_list !! (n-1)
a030091_list =
   filter (\x -> ((==) `on` (nub . sort . show)) x (x^2)) a000040_list
-- Reinhard Zumkeller, Aug 11 2011

Select[Prime[Range[82000]],Union[IntegerDigits[#]]== Union[ IntegerDigits [#^2]]&] (* Harvey P. Dale, Aug 12 2011 *)

isA030091(n)=isprime(n)&&Set(Vec(Str(n)))==Set(Vec(Str(n^2))) \\ Charles R Greathouse IV, Jun 28 2011

Equals A000040 INTERSECTION A029793. - Jonathan Vos Post, Jul 06 2008

cp's OEIS Frontend

A029795 Numbers k such that k and k^3 have the same set of digits.

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A178501 Zero followed by powers of ten.

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A232659 Numbers n such that n and n^4 have the same set of digits.

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A232662 Numbers n such that n and n^7 have the same set of digits.

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A029797 Numbers k such that k^2 and k^3 have the same set of digits.

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A257763 Zeroless numbers n such that n and n^2 have the same set of decimal digits.

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A257760 Zeroless numbers n such that the products of the decimal digits of n and n^2 coincide.

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