A343921 -id:A343921 - cp's OEIS frontend

A338659 The smallest positive number that can be added to n the maximum number of times, see A343921(n), such that the digits in each resulting sum are distinct, or -1 if no such number exists.

27, 1, 34, 81, 15, 81, 48, 86, 150, 37, 355, 23, 37, 47, 56, 15, 37, 44, 55, 37, 43, 37, 14, 17, 27, 340, 811, 27, 37, 340, 15, 37, 37, 15, 23, 35, 14, 91, 22, 48, 44, 233, 63, 33, 53, 75, 37, 3, 75, 37, 14, 27, 811, 37, 27, 88, 37, 63, 37, 171, 22, 391, 74, 43, 44, 37, 43, 480, 37, 37, 478

Offset: 0

Scott R. Shannon, Apr 22 2021

			a(0) = 27 as 27 can be added to 0 a total of A343921(0) = 36 times with each sum containing distinct digits. The 36 sums are 27, 54, 81, 108, 135, ..., 918, 945, 972. No other positive number can be added 36 or more times to 0 to produce such sums.
a(1) = 1 as 1 can be added to 1 a total of A343921(1) = 9 times with each sum containing distinct digits. The sums are 2,3,4,5,6,7,8,9,10. There are fourteen positive numbers in all which can be added to 1 a total of 9 times producing sums with distinct digits, the largest being 7012 (see A343922).
a(2) = 34 as 34 can be added to 2 a total of A343921(2) = 12 times with each sum containing distinct digits. The sums are 36, 70, 104, 138, 172, 206, 240, 274, 308, 342, 376, 410. No other positive number can be added 12 or more times to 2 to produce such sums.

Cf. A343921, A343922, A010784, A043096, A029743.

a(n) = -1 for n >= 9876543210.

A343922 The largest positive number that can be added to n the maximum number of times, see A343921(n), such that the digits in each resulting sum are distinct, or -1 if no such number exists.

Original entry on oeis.org

27, 7012, 34, 81, 15, 781, 48, 86, 150, 37, 355, 23, 37, 47, 56, 15, 37, 931, 55, 355, 44, 37, 14, 17, 27, 340, 811, 27, 37, 340, 31, 37, 37, 15, 778, 61, 14, 91, 22, 48, 44, 233, 63, 299, 606, 75, 37, 9111, 75, 37, 14, 27, 7811, 37, 27, 91, 37, 63, 37, 171, 287, 391, 74, 43, 44, 37, 43, 480

Offset: 0

Scott R. Shannon, May 04 2021

			a(0) = 27 as 27 can be added to 0 a total of A343921(0) = 36 times with each sum containing distinct digits. The 36 sums are 27, 54, 81, 108, 135, ..., 918, 945, 972. No other positive number can be added 36 or more times to 0 to produce such sums.
a(1) = 7012 as 7012 can be added to 1 a total of A343921(1) = 9 times with each sum containing distinct digits. The sums are 7013, 14025, 21037, 28049, 35061, 42073, 49085, 56097, 63109. There are fourteen positive numbers in all which can be added to 1 a total of 9 times producing sums with distinct digits, the smallest being 1 (see A338659).
a(47) = 9111 as 9111 can be added to 47 a total of A343921(47) = 9 times with each sum containing distinct digits. The sums are 9158, 18269, 27380, 36491, 45602, 54713, 63824, 72935, 82046. There are five positive numbers in all which can be added to 47 a total of 9 times producing sums with distinct digits, the smallest being 3 (see A338659).

Cf. A343921, A338659, A010784, A043096, A029743.

a(n) = -1 for n >= 9876543210.

A343925 Irregular triangle read by rows: n-th row gives the numbers > 1 that can be multiplied by n the maximum number of times, see A343924, such that each product has distinct digits.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 5, 2, 3, 2, 13, 14, 15, 18, 35, 72, 77, 2, 2, 5, 2, 2, 2, 3, 6, 17, 3, 13, 2, 2, 7, 12, 2, 2, 5, 39, 3, 2, 7, 17, 78, 3, 2, 2, 4, 2, 5, 9, 12, 18, 93, 3, 17, 5, 3, 2, 4, 2, 5, 2, 2, 2, 9, 2, 3, 5, 6, 7, 11, 12, 15, 21, 24, 25, 34, 59, 74, 87, 107, 113, 118, 127, 158, 173, 207

Offset: 1

Scott R. Shannon, May 04 2021

See A343924 for the maximum number of times the numbers in each row can multiply n to produce a series of products with distinct digits.
The number of terms in each row is extremely variable. For n below 1000 the numbers 556, 748, 813, 818, 848 can only be multiplied one time before a product with non-distinct digits is produced. For 556, for example, there are 7002 different numbers which satisfy this condition, the list starting with 5, 7, 15, 17, 19, ... . In comparison the next row for 557 has one term, 25, which can be multiplied by 557 the maximum of three times.
All rows correspond to numbers ending in two or more zeros, for example 100, have no terms as any product will also end in at least that many zeros.

			row(1) = 2 as 1 can be multiplied by 2 the maximum of 15 times producing products with distinct digits. The products are: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16348, 32768.
row(11) = 13, 14, 15, 18, 35, 72, 77 as these numbers can be multiplied by 11 the maximum of 3 times producing products with distinct digits. For example choosing 13 the products are 143, 1859, 24167.
The table begins:
.
2;
2;
2;
2;
2;
2;
5;
2;
3;
2;
13, 14, 15, 18, 35, 72, 77;
2;
2;
5;
2;
2;
2, 3, 6, 17;
3, 13;
...

Scott R. Shannon, Table for n = 1..1000.

Cf. A343924, A343921 (addition), A010784, A003991, A043537.

row(n) has no terms for n > 4938271605 or for any number n ending in two or more 0's.

A343924 a(n) = the maximum number of times n can be multiplied by a number > 1 such that each product has distinct digits.

Original entry on oeis.org

15, 14, 10, 13, 10, 9, 6, 12, 5, 9, 3, 8, 6, 6, 10, 11, 3, 4, 5, 8, 5, 4, 6, 7, 3, 6, 10, 3, 4, 9, 4, 10, 3, 4, 5, 4, 4, 4, 6, 7, 3, 3, 2, 4, 4, 5, 2, 6, 5, 4, 8, 4, 3, 9, 3, 3, 3, 4, 4, 8, 3, 3, 3, 9, 4, 5, 4, 3, 4, 5, 3, 3, 4, 4, 4, 3, 4, 5, 5, 6, 4, 2, 3, 2, 4, 3, 3, 7, 5, 5, 4, 4, 2, 2, 4, 5

Offset: 1

Scott R. Shannon, May 04 2021

See A343925 for the list of numbers for each n which can multiply n to produce the maximum length series of products with distinct digits.

			a(1) = 15 as 1 can be multiplied by 2 a total of fifteen times with each product containing distinct digits. The products are 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16348, 32768. No other number can multiply 1 to produce a longer series.
a(7) = 6 as 7 can be multiplied by 5 a total of six times with each product containing distinct digits. The products are 35, 175, 875, 4375, 21875, 109375. No other number can multiply 7 to produce a longer series.
a(17) = 3 as 17 can be multiplied by 2, 3, 6, or 17 a total of three times with each product containing distinct digits. For example for 17 the products are 289, 4913, 83521. No other numbers can multiply 17 to produce a longer series.

Cf. A343925, A343921 (addition), A010784, A003991, A043537.

a(n) = 0 for n > 4938271605 or for any number n ending in two or more 0's.

cp's OEIS Frontend

A338659 The smallest positive number that can be added to n the maximum number of times, see A343921(n), such that the digits in each resulting sum are distinct, or -1 if no such number exists.

Views

Author

Keywords

Examples

Crossrefs

Formula

A343922 The largest positive number that can be added to n the maximum number of times, see A343921(n), such that the digits in each resulting sum are distinct, or -1 if no such number exists.

Views

Author

Keywords

Examples

Crossrefs

Formula

A343925 Irregular triangle read by rows: n-th row gives the numbers > 1 that can be multiplied by n the maximum number of times, see A343924, such that each product has distinct digits.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A343924 a(n) = the maximum number of times n can be multiplied by a number > 1 such that each product has distinct digits.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula