A277058 -id:A277058 - cp's OEIS frontend

A277055 Irregular array by rows: A(n,m) is the least number which gives a pandigital product when multiplied by the m-th repunit in base n; each row is truncated when it reaches its stationary point.

2, 11, 8, 5, 75, 15, 7, 694, 119, 34, 8345, 1505, 195, 123717, 105803, 2217, 2134, 727, 2177399, 241934, 37303, 3724, 44317196, 4431858, 487068, 54771, 9124, 1023456789, 93125079, 9222117, 1110789, 92115, 26432593615

Offset: 2

Andrey Zabolotskiy and Altug Alkan, Sep 26 2016

For row n, the initial number is A049363(n) and the trailing number is A277056(n). Row 10 is A277054(1)-A277054(5) [note that the initial row is row 2].

			The first rows of the array are:
2, (2, 2...)
11, 8, 5, (5, 5...)
75, 15, 7, (7, 7...)
694, 119, 34,
8345, 1505, 195,
123717, 105803, 2217, 2134, 727,
2177399, 241934, 37303, 3724,
44317196, 4431858, 487068, 54771, 9124,
1023456789, 93125079, 9222117, 1110789, 92115

Cf. A002275, A171102, A049363, A277054, A277056, A277058.

A277057 Least k such that n-th repunit times k contains all digits from 1 to 9.

Original entry on oeis.org

123456789, 11225079, 1113198, 210789, 11115, 11115, 11115, 11115, 11115, 11115, 11115, 11115, 11115, 11115, 11115, 11115, 11115, 11115, 11115, 11115, 11115, 11115, 11115, 11115, 11115, 11115, 11115, 11115, 11115, 11115, 11115, 11115, 11115, 11115, 11115

Offset: 1

Andrey Zabolotskiy and Altug Alkan, Sep 26 2016

Starting from n=5, a(n)=A277059(10)=11115, and the corresponding pandigital numbers are 123499...98765 (n-4 nines). Actually, for all n, the resulting numbers are zeroless pandigitals.
a(1)-a(5) constitute row 10 of A277058.
a(n)*A002275(n) is 123456789, 123475869, 123564978, 234186579, 123498765, ...

			a(2) = 11225079 because A002275(2)*11225079 = 11*11225079 = 123475869 that contains all digits from 1 to 9 and 11225079 is the least number with this property.

Cf. A002275, A050289, A277054, A277058, A277059.

isok(n) = my(d=digits(n)); vecmin(d) && (#Set(digits(n)) == 9);
a(n) = {if (n==1, return(123456789)); my(k=1); while(! isok(k*(10^n - 1)/9), k++); k;} \\ Michel Marcus, Sep 26 2019

A277059 Least k such that any sufficiently long repunit multiplied by k contains all nonzero digits in base n.

Original entry on oeis.org

1, 4, 6, 14, 45, 370, 588, 3364, 11115, 168496, 271458, 2442138

Offset: 2

Andrey Zabolotskiy and Altug Alkan, Sep 26 2016

Trailing terms of rows of A277058.

Written in base n, the terms read: 1, 11, 12, 24, 113, 1036, 1114, 4547, 11115, 105659, 111116, 67676A, ...

			Any binary repunit itself contains a 1, so a(2)=1.
k-th decimal repunit for k>4 multiplied by 11115 contains all nonzero decimal digits (see A277057) with no number less than 11115 having the same property, so a(10)=11115.

Cf. A002275, A277056, A277057, A277058.

Conjecture:
for n=2m, a(n) = (n^m-1)/(n-1) + m - 1;
for n=4m+1, a(n) = (n^(2m)-1)(n^2+1) / (2(n^2-1)) + m;
for n=4m-1, a(n) = (n^(2m-2)-1)(n^2+1) / (2(n^2-1)) + m + n^(2m-1).

cp's OEIS Frontend

A277055 Irregular array by rows: A(n,m) is the least number which gives a pandigital product when multiplied by the m-th repunit in base n; each row is truncated when it reaches its stationary point.

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A277057 Least k such that n-th repunit times k contains all digits from 1 to 9.

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A277059 Least k such that any sufficiently long repunit multiplied by k contains all nonzero digits in base n.

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