A363905 -id:A363905 - cp's OEIS frontend

A363927 Numbers N such that in the concatenation of N^2 and N^3, each of the 10 decimal digits appears equally often.

69, 6534, 497375, 539019, 543447, 586476, 589629, 601575, 646479, 858609, 895688, 959097, 46839081, 47469378, 47693199, 47760623, 47841576, 48038964, 48527792, 48733506, 48886836, 48965892, 49229103, 49397283, 49594832, 49670616, 50013116, 50247423, 50359157

Offset: 1

Charles R Greathouse IV and M. F. Hasler, Jun 28 2023

a(3) = 497375 and a(11) = 895688 are the only terms < 10^6 that are not divisible by 3.
Each term has an even number of decimal digits, k, and a corresponding value between 10^(k-1)*100^(1/3) and 10^k. - Michael S. Branicky, Jun 29 2023
Indeed, the number of digits of concat(N^2, N^3) is floor(2*L + 1) + floor(3*L + 1) where L = log_10(N). This is a multiple of 10 iff L mod 2 is in the interval [5/3, 2), which means that N is in the above range for some even k. - M. F. Hasler, Jul 02 2023

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A363905, A363909: concat(n^2, n^3) has each digit at least once / twice.
Cf. A171102: pandigital numbers.
Cf. A036744, A054038, A071519 and A156977 for "pandigital squares".
Cf. A119735: n^3 is pandigital.

fQ[n_] := Length@ Union[ Count[ Sort[ Join[ IntegerDigits[n^2], IntegerDigits[n^3]]], #] & /@ Range[0, 9]] == 1; Select[ Range@ 52000000, fQ] (* Robert G. Wilson v, Jul 01 2023 *)

is(n)={my(v=concat(digits(n^2),digits(n^3)), c=#v); c%10==0 && vecsort(v)==[0..c-1]\(c\10)}
for(n=1,1e6, is(n)&& print1(n","))

a(13) and beyond from Michael S. Branicky, Jun 28 2023

A363909 Numbers whose square and cube taken together contain each decimal digit at least twice.

Original entry on oeis.org

6534, 11027, 11994, 21906, 22178, 22195, 23317, 24567, 27019, 27963, 28354, 29099, 29309, 29339, 29375, 29558, 29621, 30184, 30552, 30584, 31578, 31727, 32447, 32633, 32793, 32912, 32923, 33087, 33257, 33527, 34284, 35717, 36943, 36958, 37697, 38463

Offset: 1

M. F. Hasler, Jun 27 2023

The first term, a(1) = 6534 is the only number of which the square and cube taken together contain each digit 0 to 9 exactly twice.

Presumably a(n) ~ A363905(n) ~ n. - Charles R Greathouse IV, Jul 03 2023

			6534^2 = 42693156, 6534^3 = 278957081304, which together contain each digit 0-9 exactly twice.

Harold Suarez, Interesting..., Number Theory group on LinkedIn, (capture of the original post), June 2023

Cf. A363905: square and cube together contain each digit at least once.
Cf. A036744, A054038, A071519 and A156977 for "pandigital" squares.
Cf. A119735: Numbers n such that every digit occurs at least once in n^3.

is(n)=#Set(n=concat(digits(n^2),digits(n^3)))>9&&(n=vecsort(n))[#n-1]==9&&!n[2]&&!for(i=3,#n-2,n[i]>n[i-1]&&n[i]

A364024 Least number whose square and cube, taken together, are pandigital n times.

Original entry on oeis.org

69, 6534, 497375, 46839081, 4641856941, 464162827242, 46415911449392, 4641588990290676, 464158883559510629

Offset: 1

Robert G. Wilson v, Jul 01 2023

Inspired by A363905.

Proper subset of A364023.

			a(1) is 69 since 69^2 = 4761 and 69^3 = 328509; together they include each of the digits {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} exactly once;
a(2) is 6534 since 6534^2 = 42693156 and 6534^3 = 278957081304; together they include each decimal digit exactly twice;
a(3) is 497375 since 497375^2 = 247381890625 and 497375^3 = 123041567849609375; together they include each decimal digit exactly three times; etc.

Cf. A363905, A363927, A364023.

f[n_] := Block[{k = Floor[10^(2n-1/3)]}, While[ !MemberQ[{0, 3, 6, 8}, Mod[k, 9]] || Length@ Union[ Count[ Sort[ Join[ IntegerDigits[k^2], IntegerDigits[k^3]]], #] & /@ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}] > 1, k++]; k]

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A363927 Numbers N such that in the concatenation of N^2 and N^3, each of the 10 decimal digits appears equally often.

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A363909 Numbers whose square and cube taken together contain each decimal digit at least twice.

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A364024 Least number whose square and cube, taken together, are pandigital n times.

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