A257763 -id:A257763 - cp's OEIS frontend

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

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

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.

Original entry on oeis.org

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

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

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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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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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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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