A134844 -id:A134844 - cp's OEIS frontend

A328781 Nonnegative integers k such that k and k^2 contain the same number of zero digits in their decimal expansion.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 46, 54, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 72, 73, 74, 75, 76, 77, 79, 81, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 96, 104, 105

Offset: 1

Bernard Schott, Oct 27 2019

Inspired by A328780.
This sequence is not a duplicate of A052040. The first 72 terms until 96 are exactly the same but a(73) = 104 belongs to this sequence because 104^2 = 10816, but 104 doesn't belong to A052040 because there is one zero digit in the decimal expansion of 104^2.
The nonnegative integers that do not belong to this sequence are divided into three sequences:
1) A104315 = A052040 \ {this sequence}: Numbers k such that k contains at least one zero, but k^2 contains no zero (e.g., 106 with 106^2 = 11236).
2) A134844 = Numbers k such that k contains no zero but k^2 contains at least one zero (e.g., 32 with 32^2 = 1024).
3) A328783 = Numbers k such that k and k^2 contain at least one zero but not the same number of zeros (e.g., 101 with 101^2 = 10201).
Another sequence is A328782 = {this sequence} \ A052040 which lists the positive integers that have the same positive number of zeros in their decimal expansions as in their squares. The first two examples > 0 are 104 with 104^2 = 10816 and 105 with 105^2 = 11025.

			12 and 144 = 12^2 have no digit zero in their decimal representation, so 12 is a term.
203 and 41209 = 203^2 both have one digit zero in their decimal representation, so 203 is also a term.

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

Cf. A052040, A104315, A134844.

Cf. A323780, A328782, A328783.

select(t -> numboccur(0, convert(t^2,base,10))=numboccur(0, convert(t,base,10)), [$0..200]); # Robert Israel, Oct 27 2019

Select[Range[0, 105], Equal @@ Total /@ (1 - Sign@ IntegerDigits[{#, #^2}]) &] (* Giovanni Resta, Feb 27 2020 *)

A328782 Integers k such that k and k^2 contain the same number > 0 of digits zero in their decimal expansion.

Original entry on oeis.org

0, 104, 105, 203, 205, 302, 303, 305, 402, 403, 405, 504, 505, 506, 507, 508, 509, 601, 602, 603, 605, 609, 701, 702, 703, 705, 708, 709, 801, 802, 803, 805, 901, 902, 903, 905, 906, 1006, 1007, 1008, 1009, 1011, 1012, 1013, 1014, 1016, 1017, 1018, 1019, 1021

Offset: 1

Bernard Schott, Oct 28 2019

			703 and 494209 = 703^2 both have one zero digit in their decimal expansion.

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

Equals A328781 \ A052040.

Cf. A104315, A134844, A328780, A328783.

f:= n-> numboccur(0, convert(n, base, 10)):
q:= n-> ((x, y)-> x>0 and x=y)(f(n), f(n^2)):
select(q, [$0..1030])[];  # Alois P. Heinz, Oct 28 2019

Select[Range[0, 1100], DigitCount[#, 10, 0] == DigitCount[#^2, 10, 0] > 0 &] (* Giovanni Resta, Feb 27 2020 *)

More terms from Alois P. Heinz, Oct 28 2019

A328783 Numbers k such that k and k^2 contain at least one zero but not the same number of 0's.

Original entry on oeis.org

10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 101, 102, 103, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200, 201, 202, 210, 220, 230, 240, 250, 260, 270, 280, 290, 300, 301, 310, 320, 330, 340, 350, 360, 370, 380, 390, 400, 401, 410, 420, 430, 440, 450, 460, 470

Offset: 1

Bernard Schott, Oct 28 2019

This sequence is one of the three sequences whose numbers k and k^2 don't contain the same number of 0, the two others are A104315 and A134844.

			201 and 40401 = 201^2 have both at least one zero but not the same number of 0 in their decimal expansion, hence, 201 is a term.

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

Cf. A052040, A104315, A134844, A328780, A328781, A328782.

f:= n-> numboccur(0, convert(n, base, 10)):
q:= n-> ((x, y)-> x>0 and y>0 and x<>y)(f(n), f(n^2)):
select(q, [$0..500])[];  # Alois P. Heinz, Oct 28 2019

Select[Range[0, 470], (x = DigitCount[#, 10, 0]) > 0 && (y = DigitCount[ #^2, 10, 0]) > 0 && x != y &] (* Giovanni Resta, Feb 27 2020 *)

More terms from Alois P. Heinz, Oct 28 2019

A135216 a(n)= number of numbers with n+1 digits and without zero digits whose squares have maximal number of zero digits = A135215(n+1).

Original entry on oeis.org

18, 3, 13, 1, 7, 1

Offset: 1

Artur Jasinski, Nov 23 2007

			a(1)=18 because we have 18 numbers with 2 digits and without zero digit whose square have maximal possible value 1 zero: 32, 33, 45, 47, 48, 49, 51, 52, 53, 55, 64, 71, 78, 84, 95, 97, 98, 99.
a(2)=3 because we have 3 numbers with 3 digits and without zero digit whose square have maximal possible value 3 zeros: 448, 548, 949.
a(3)=13 because we have 13 numbers with 4 digits and without zero digit whose square have maximal possible value 4 zeros: 3747, 3751, 4899, 6245, 6249, 6253, 7746, 7747, 7749, 7751, 7753, 9747, 9798.
a(4)=1 because we have only one number with 5 digits and without zero digit whose square have maximal possible value 6 zeros: 62498.
a(5)=7 because we have 7 numbers with 6 digits and without zero digit whose square have maximal possible value 7 zeros: 248998, 316245, 489898, 498999, 781249, 948951, 997998.
a(6)=1 because we have only one number with 7 digits and without zero digit whose square have maximal possible value 10 zeros: 6244998.

Cf. A134843, A134844, A134845, A134846, A134847, A134848, A134849, A135215, A135217, A135219.

(*For a(7) *) c = 0; mx = 10; Do[Do[Do[Do[Do[Do[Do[k = 10^6b + 10^5q + 10^4r + 10^3p + 10^2s + 10n + m; w = IntegerDigits[k^2]; ile = 0; Do[If[w[[t]] == 0, ile = ile + 1], {t, 1, Length[w]}]; If[ile == mx, c = c + 1], {m, 1, 9}], {n, 1, 9}], {s, 1, 9}], {p, 1, 9}], {r, 1, 9}], {q, 1, 9}], {b, 1, 9}]; c (*Artur Jasinski*)

A351807 Integers m such that pod(m) divides pod(m^2) where pod = product of digits = A007954.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 11, 12, 13, 15, 16, 18, 19, 21, 22, 23, 25, 26, 27, 28, 31, 32, 33, 36, 41, 42, 43, 45, 47, 48, 49, 51, 52, 53, 55, 61, 62, 63, 64, 66, 68, 71, 74, 76, 78, 82, 83, 84, 93, 94, 95, 96, 97, 98, 99, 111, 112, 113, 114, 115, 116, 118, 121, 122, 123

Offset: 1

Bernard Schott, Feb 19 2022

Inspired by A351650 where pod is replaced by sod.
All terms are zeroless (A052382).
Repunits form a subsequence (A002275).
Integers m without 0 and such that m^2 has a 0 form a subsequence (A134844).
The smallest term k such that the corresponding quotient = n is A351809(n).

			Product of digits of 27 = 2*7 = 14; then 27^2 = 729, product of digits of 729 = 7*2*9 = 81; as 81 divides 729, 27 is a term.

Cf. A007954, A002473, A351808 (corresponding quotients), A351809.

Subsequences: A002275, A134844.

pod[n_] := Times @@ IntegerDigits[n]; Select[Range[120], FreeQ[IntegerDigits[#], 0] && Divisible[pod[#^2], pod[#]] &] (* Amiram Eldar, Feb 19 2022 *)

isok(m) = my(d=digits(m)); vecmin(d) && denominator(vecprod(digits(m^2))/vecprod(d)) == 1; \\ Michel Marcus, Feb 19 2022

from math import prod
def pod(n): return prod(map(int, str(n)))
def ok(m): pdm = pod(m); return pdm > 0 and pod(m*m)%pdm == 0
print([m for m in range(124) if ok(m)]) # Michael S. Branicky, Feb 19 2022

More terms from Amiram Eldar, Feb 19 2022

A351808 a(n) is the quotient obtained when pod(m) divides pod(m^2), with pod = product of digits = A007954 and m = A351807(n).

Original entry on oeis.org

1, 2, 3, 2, 3, 3, 2, 8, 18, 4, 10, 3, 2, 8, 32, 15, 6, 21, 9, 14, 18, 0, 0, 6, 12, 21, 24, 0, 0, 0, 0, 0, 0, 0, 0, 7, 32, 81, 0, 10, 4, 0, 30, 35, 0, 21, 144, 0, 64, 32, 0, 2, 0, 0, 0, 12, 80, 252, 243, 12, 60, 27, 48, 256, 15, 30, 140, 36, 8, 14, 336, 96, 144

Offset: 1

Bernard Schott, Feb 20 2022

a(n) = 0 iff m = A351807(n) is a term of A134844.

As pod(m) is 7-smooth number and pod(m^2) can be 0 (see example), all terms of the sequence are in {0} union A002473. The smallest term k such that the corresponding quotient = 0 or A002473(n) is A351809(n).

			A351807(9) = 13, then pod(13) = 1*3 = 3 while pod(13^2) = pod(169) = 1*6*9 = 54; hence, a(9) = 54/3 = 18.
A351807(23) = 33, then pod(33) = 3*3 = 9 while pod(33^2) = pod(1089) = 1*0*8*9 = 0; hence, a(23) = 0.

Cf. A002473, A007954, A134844, A351807, A351809.

pod[n_] := Times @@ IntegerDigits[n]; r[n_] := If[(p = pod[n]) > 0, pod[n^2]/p, 1/2]; Select[r /@ Range[200], IntegerQ] (* Amiram Eldar, Feb 21 2022 *)

lista(nn) = {my(list=List()); for (m=1, nn, my(d=digits(m), q); if (vecmin(d) && denominator(q = vecprod(digits(m^2))/vecprod(d)) == 1, listput(list, q);); ); Vec(list);} \\ Michel Marcus, Feb 21 2022

from math import prod
from itertools import count, islice
def A351808_gen(): # generator of terms
    return (q for q, r in (divmod(prod(int(d) for d in str(m**2)),prod(int(d) for d in str(m))) for m in count(1) if '0' not in str(m)) if r == 0)
A351808_list = list(islice(A351808_gen(),20)) # Chai Wah Wu, Feb 25 2022

More terms from Amiram Eldar, Feb 21 2022

A135251 Maximal number of zero digits in square of number with n digits not divisible by 10.

Original entry on oeis.org

0, 1, 3, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124

Offset: 1

Artur Jasinski, Nov 24 2007

Cf. A134843, A134844, A134845, A134846, A134847, A134848, A134849, A135215, A135217, A135219, A135252, A135253.

(*For a(7)*) mx = 0; Do[Do[Do[Do[Do[Do[Do[k = 10^6b + 10^5q + 10^4r + 10^3p + 10^2s + 10n + m; w = IntegerDigits[k^2]; ile = 0; Do[If[w[[t]] == 0, ile = ile + 1; If[ile > mx, mx = ile]], {t, 1, Length[w]}], {m, 1, 9}], {n, 0, 9}], {s, 0, 9}], {p, 0, 9}], {r, 0, 9}], {q, 0, 9}], {b, 1, 9}]; mx

2*n-4 <= a(n) <= 2*n-2 since, if k is an n-digit number not divisible by 10, then k^2 has at most 2*n digits of which the first and last are nonzero; and for n >= 2, the square of the n-digit number 10^(n-1)+1 contains 2*n-4 zeros. It seems likely that a(n) = 2*n-4 for all n >= 4. - Pontus von Brömssen, Jun 09 2025

a(8)-a(64) from Pontus von Brömssen, Jun 09 2025

A135252 a(n) = number of numbers with n+1 digits and not divisible by 10 whose squares have maximal number of zero digits = A135251(n+1).

Original entry on oeis.org

18, 3, 24, 11, 10, 11

Offset: 1

Artur Jasinski, Nov 24 2007

			a(1)=18 because we have 18 numbers with 2 digits not divisible by 10 whose squares have maximal possible number of zero digits, namely 1 zero: 32, 33, 45, 47, 48, 49, 51, 52, 53, 55, 64, 71, 78, 84, 95, 97, 98, 99
a(2)=3 because we have 3 numbers with 3 digits not divisible by 10 whose square have maximal possible number of zero digits, namely 3 zeros: 448, 548, 949
a(3)=24 because we have 24 numbers with 4 digits not divisible by 10 whose square have maximal possible number of zero digits, namely 4 zeros: 1001, 1002, 1003, 2001, 2002, 3001, 3747, 3751, 4001, 4899, 5001, 5002, 5003, 6245, 6249, 6253, 7746, 7747, 7749, 7751, 7753, 9503, 9747, 9798
a(4)=11 because we have 11 numbers with 5 digits not divisible by 10 whose square have maximal possible number of zero digits, namely 6 zeros: 10001, 10002, 10003, 20001, 20002, 30001, 40001, 50001, 50002, 50003, 62498
a(5)=10 because we have 10 numbers with 6 digits not divisible by 10 whose square have maximal possible number of zero digits, namely 8 zeros: 100001, 100002, 100003, 200001, 200002, 300001, 400001, 500001, 500002, 500003
a(6)=11 because we have 11 numbers with 7 digits not divisible by 10 whose square have maximal possible number of zero digits, namely 10 zeros: 1000001, 1000002, 1000003, 2000001, 2000002, 3000001, 4000001, 5000001, 5000002, 5000003, 6244998

Cf. A134843, A134844, A134845, A134846, A134847, A134848, A134849, A135215, A135217, A135219, A135251, A135253.

(* For a(6) *) a = {}; c = 0; mx = 10; Do[Do[Do[Do[Do[Do[Do[k = 10^6b + 10^5q + 10^4r + 10^3p + 10^2s + 10n + m; w = IntegerDigits[k^2]; ile = 0; Do[If[w[[t]] == 0, ile = ile + 1], {t, 1, Length[w]}]; If[ile == mx, c = c + 1; AppendTo[a, k]], {m, 1, 9}], {n, 0, 9}], {s, 0, 9}], {p, 0, 9}], {r, 0, 9}], {q, 0, 9}], {b, 1, 9}]; c

cp's OEIS Frontend

A328781 Nonnegative integers k such that k and k^2 contain the same number of zero digits in their decimal expansion.

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A328782 Integers k such that k and k^2 contain the same number > 0 of digits zero in their decimal expansion.

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A328783 Numbers k such that k and k^2 contain at least one zero but not the same number of 0's.

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A135216 a(n)= number of numbers with n+1 digits and without zero digits whose squares have maximal number of zero digits = A135215(n+1).

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A351807 Integers m such that pod(m) divides pod(m^2) where pod = product of digits = A007954.

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A351808 a(n) is the quotient obtained when pod(m) divides pod(m^2), with pod = product of digits = A007954 and m = A351807(n).

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A135251 Maximal number of zero digits in square of number with n digits not divisible by 10.

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A135252 a(n) = number of numbers with n+1 digits and not divisible by 10 whose squares have maximal number of zero digits = A135251(n+1).

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