A029793 -id:A029793 - cp's OEIS frontend

A257774 Numbers n such that the products of the decimal digits of n^2 and n^3 coincide, n^2 and n^3 are zeroless.

1, 5, 7, 6057, 292839, 1295314, 4897814, 4967471, 5097886, 6010324, 6919146, 7068165, 7189558, 9465077, 15347958, 22842108, 24463917, 26754863, 43378366, 48810128, 48885128, 50833026, 54588458, 54649688, 68093171, 69925865, 69980346, 73390374, 74357144

Offset: 1

Pieter Post, May 08 2015

This sequence is more sporadic than A257760. It appears there is no sequence for zeroless numbers n and n^3 such that the products of the decimal digits coincide, except for the trivial 1.

			5 is in the sequence since 5^2 = 25 and 5^3 = 125 and we have 2*5 = 1*2*5 = 10 > 0.
6057 is in the sequence since 6057^2 = 36687249 and 6057^3 = 222214667193 and we have 3*6*6*8*7*2*4*9 = 2*2*2*2*1*4*6*6*7*1*9*3 = 435456 > 0.

Giovanni Resta, Table of n, a(n) for n = 1..544 (terms < 4*10^10)

Cf. A000290, A000578, A029793, A052382, A257760, A257763.

pod[n_] := Times@@IntegerDigits@n; Select[Range[10^7], pod[#^3] == pod[#^2] > 0 &] (* Giovanni Resta, May 08 2015 *)

Corrected and extended by Giovanni Resta, May 08 2015

A345391 a(n) is the least proper multiple of n with the same set of decimal digits as n.

Original entry on oeis.org

11, 22, 33, 44, 55, 66, 77, 88, 99, 100, 1111, 1212, 1131, 1414, 1155, 1616, 1717, 1188, 1919, 200, 2121, 2222, 322, 2424, 225, 2262, 2727, 2828, 2929, 300, 1333, 3232, 3333, 3434, 3535, 3636, 3737, 3838, 3393, 400, 4141, 4242, 344, 4444, 4455, 644, 4747

Offset: 1

Rémy Sigrist, Jun 17 2021

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A020338, A029793, A276769.

a(n) = { my (d=Set(digits(n))); forstep (m=2*n, oo, n, if (Set(digits(m))==d, return (m))) }

def a(n):
    kn, ss = 2*n, set(str(n))
    while set(str(kn)) != ss: kn += n
    return kn
print([a(n) for n in range(1, 49)]) # Michael S. Branicky, Jun 17 2021

a(n) <= A020338(n) for any n > 0.

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)

A029794 Squares n such that sqrt(n) and n have the same set of digits.

Original entry on oeis.org

0, 1, 100, 10000, 1000000, 22676644, 23348224, 100000000, 107661376, 110103049, 125552025, 153388225, 160022500, 219899241, 504002500, 552532036, 605406025, 696643236, 1169366416, 1311526225, 2267664400

Offset: 1

Patrick De Geest

Only perfect squares are considered. Otherwise, this sequence would have to include numbers like 1234567890, since sqrt(1234567890) = 35136.41828644462161665823116758077037159... - Alonso del Arte, Jan 12 2020

Shawn A. Broyles and David A. Corneth, Table of n, a(n) for n = 1..10000 (First 1001 terms by Shawn A. Broyles)

Cf. A029793.

Select[Range[0, 199999], Union[IntegerDigits[#]] == Union[IntegerDigits[#^2]] &]^2 (* Alonso del Arte, Jan 12 2020 *)

lista(nn) = {for (n=0, nn, if (Set(digits(n^2)) == Set(digits(n)), print1(n^2, ", ")););} \\ Michel Marcus, Apr 29 2018

a(n) = A029793(n)^2. - Michel Marcus, Apr 29 2018

Offset corrected by Michel Marcus, Apr 29 2018

A232712 Least positive k (not a power of 10) such that k and k^n have the same set of digits.

Original entry on oeis.org

2, 4762, 107624, 35641, 39568, 1380796, 12635940, 40837596, 102349857, 102567384, 106342987, 129046873, 107623945, 231940678, 239607415, 368709154, 1023456789, 164758903, 176384592, 1023456789, 1023456789, 1023456789, 1023456789, 1023456789, 1023456789

Offset: 1

Arkadiusz Wesolowski, Nov 28 2013

a(17) and a(20)-a(40) = A050278(1) = 1023456789, the smallest pandigital number. [Lars Blomberg, Dec 10 2013]

Lars Blomberg, Table of n, a(n) for n = 1..40

Cf. A171102, A029793, A029795, A232659-A232662.

for(n=1, 6, k=1; until(Set(Vec(Str(k)))==Set(Vec(Str(k^n)))&&!(sumdigits(k)==1), k++); print1(k, ", "));

a(14)-a(25) from Lars Blomberg, Dec 10 2013

A257968 Zeroless numbers n such that the product of digits of n, the product of digits of n^2 and the product of digits of n^3 form a geometric progression.

Original entry on oeis.org

1, 2, 38296, 151373, 398293, 422558, 733381, 971973, 2797318, 3833215, 6988327, 7271256, 8174876, 8732657, 9872323, 9981181, 11617988, 11798921, 14791421, 15376465, 15487926, 15625186, 16549885, 18543639, 21316582, 21492828, 22346329, 22867986, 23373644

Offset: 1

Pieter Post, May 15 2015

This sequence appears to be infinite.

			38296 is in the sequence because the pod equals 2592 (=3*8*2*9*6), pod(38296^2) is 622080, pod(38296^3) is 149299200. 2592*240 = 622080 => 622080*240 = 149299200.

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

Cf. A052382 (zeroless numbers), A007954 (product of digits).

Cf. A029793, A257760, A257763, A257774.

pod[n_]:=Times@@IntegerDigits@n; Select[Range[10^8], pod[#^3] pod[#] == pod[#^2]^2 >0 &] (* Vincenzo Librandi, May 16 2015 *)

pod(n) = my(d = digits(n)); prod(k=1, #d, d[k]);
isok(n) = (pd = pod(n)) && (pdd = pod(n^2)) && (pdd/pd == pod(n^3)/pdd); \\ Michel Marcus, May 30 2015

def pod(n):
    kk = 1
    while n > 0:
        kk= kk*(n%10)
        n =int(n//10)
    return kk
for i in range (1,10**7):
    if pod(i**3)*pod(i)==pod(i**2)**2 and pod(i**2)!=0:
        print (i, pod(i),pod(i**2),pod(i**3),pod(i**2)//pod(i))

pod(n^3)/pod(n^2)=pod(n^2)/pod(n), where pod(n) = A007954(n).

a(17)-a(29) from Giovanni Resta, May 15 2015

A258231 Numbers n such that both n and n squared contain exactly the same digits, and n is not divisible by 10.

Original entry on oeis.org

1, 4762, 4832, 10376, 10493, 11205, 12385, 14829, 23506, 24605, 26394, 34196, 36215, 48302, 49827, 68474, 71205, 72576, 74528, 79286, 79603, 79836, 94583, 94867, 96123, 98376, 100469, 100496, 100498, 100499, 100946, 102245, 102953, 103265, 103479, 103756

Offset: 1

Harvey P. Dale, Apr 23 2016

If n is in this sequence, then n*10^k also satisfies the first portion of the definition for all k >= 0.

			4832 is a term because 4832 squared = 23348224 which contains exactly the same digits as 4832.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A029774, A029793, A257763.

Select[Select[Range[200000],ContainsExactly[IntegerDigits[ #], IntegerDigits[ #^2]]&], !Divisible[#,10]&]

A258231_list = [n for n in range(10**6) if n % 10 and set(str(n)) == set(str(n**2))] # Chai Wah Wu, Apr 23 2016

A383328 Numbers that have the same set of digits as the sum of the squares of their digits.

Original entry on oeis.org

0, 1, 155, 224, 242, 334, 343, 422, 433, 505, 515, 550, 551, 1388, 1788, 1838, 1878, 1883, 1887, 3188, 3334, 3336, 3343, 3363, 3433, 3633, 3818, 3881, 4333, 5005, 5050, 5500, 6333, 7188, 7818, 7881, 8138, 8178, 8183, 8187, 8318, 8381, 8718, 8781, 8813, 8817, 8831

Offset: 1

Jean-Marc Rebert, Apr 23 2025

			155 and 1^2 + 5^2 + 5^2 = 51 have the same set of digits {1,5}, so 155 is a term.

Cf. A003132, A029793, A249515.

q[k_] := Module[{d = IntegerDigits[k]}, Union[d] == Union[IntegerDigits[Total[d^2]]]]; Select[Range[0, 10000], q] (* Amiram Eldar, Apr 23 2025 *)

isok(k) = my(d=digits(k)); Set(d) == Set(digits(sum(i=1, #d, d[i]^2))); \\ Michel Marcus, May 13 2025

def ok(n): return set(s:=str(n)) == set(str(sum(int(d)**2 for d in s)))
print([k for k in range(10**4) if ok(k)]) # Michael S. Branicky, Apr 23 2025

cp's OEIS Frontend

A257774 Numbers n such that the products of the decimal digits of n^2 and n^3 coincide, n^2 and n^3 are zeroless.

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A345391 a(n) is the least proper multiple of n with the same set of decimal digits as n.

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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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A029794 Squares n such that sqrt(n) and n have the same set of digits.

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A232712 Least positive k (not a power of 10) such that k and k^n have the same set of digits.

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A257968 Zeroless numbers n such that the product of digits of n, the product of digits of n^2 and the product of digits of n^3 form a geometric progression.

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A258231 Numbers n such that both n and n squared contain exactly the same digits, and n is not divisible by 10.

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