A368313 -id:A368313 - cp's OEIS frontend

A368194 Irregular table T(n, k), n > 0, k = 1..A368195(n), read by rows: the n-th row lists the numbers that can be obtained by replacing any positive number without leading zeros, say m, appearing in the decimal expansion of n by one of the divisors of m.

1, 1, 2, 1, 3, 1, 2, 4, 1, 5, 1, 2, 3, 6, 1, 7, 1, 2, 4, 8, 1, 3, 9, 1, 2, 5, 10, 1, 11, 1, 2, 3, 4, 6, 11, 12, 1, 11, 13, 1, 2, 7, 11, 12, 14, 1, 3, 5, 11, 15, 1, 2, 4, 8, 11, 12, 13, 16, 1, 11, 17, 1, 2, 3, 6, 9, 11, 12, 14, 18, 1, 11, 13, 19, 1, 2, 4, 5, 10, 20

Offset: 1

Rémy Sigrist, Dec 16 2023

The n-th row starts with 1, ends with n, and contains the divisors of n (A027750).

			Table T(n, k) begins:
    1;
    1, 2;
    1, 3;
    1, 2, 4;
    1, 5;
    1, 2, 3, 6;
    1, 7;
    1, 2, 4, 8;
    1, 3, 9;
    1, 2, 5, 10;
    1, 11;
    1, 2, 3, 4, 6, 11, 12;
    1, 11, 13;
    1, 2, 7, 11, 12, 14;
    1, 3, 5, 11, 15;
    1, 2, 4, 8, 11, 12, 13, 16;
    1, 11, 17;
    1, 2, 3, 6, 9, 11, 12, 14, 18;
      ...

Rémy Sigrist, Table of n, a(n) for n = 1..16938 (rows for n = 1..1000 flattened)
Rémy Sigrist, PARI program
Index entries for sequences related to decimal expansion of n

Cf. A027750, A323286, A342072, A368195, A368313 (binary variant).

PARI
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See Links section.
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T(n, 1) = 1.

T(n, A368195(n)) = n.

A368314 a(n) is the number of numbers that can be obtained by replacing any positive number without leading zeros, say m, appearing in the binary expansion of n by one of the divisors of m.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 3, 4, 4, 5, 5, 6, 5, 6, 5, 5, 5, 7, 6, 7, 7, 8, 7, 8, 7, 8, 8, 9, 7, 10, 7, 6, 7, 7, 8, 10, 9, 9, 9, 9, 8, 12, 10, 11, 11, 11, 10, 10, 9, 12, 10, 11, 10, 13, 12, 12, 11, 11, 10, 15, 11, 13, 11, 7, 8, 11, 9, 9, 10, 13, 10, 13, 11, 12, 14, 12

Offset: 1

Rémy Sigrist, Dec 21 2023

a(n) gives the number of terms in the n-th row of A368313.

			For n = 42: the 42nd row of A368313 contains 12 terms (1, 2, 3, 5, 6, 7, 10, 14, 21, 22, 26, 42), so a(42) = 12.

Rémy Sigrist, Table of n, a(n) for n = 1..8192
Rémy Sigrist, PARI program
Index entries for sequences related to binary expansion of n

Cf. A000005, A368195 (decimal variant), A368313.

PARI
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See Links section.
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a(n) >= A000005(n).

a(2^k) = k + 1 for any k >= 0.

A368315 a(n) gives the number of ways to go from n to 1 with steps consisting of replacing a positive number without leading zero, say m, appearing in the binary expansion of a number, by a proper divisor of m.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 2, 4, 4, 7, 6, 8, 6, 8, 6, 8, 8, 17, 14, 21, 18, 28, 18, 20, 16, 27, 26, 26, 18, 31, 22, 16, 22, 37, 34, 58, 48, 76, 52, 58, 48, 98, 80, 102, 80, 105, 76, 48, 40, 85, 80, 96, 80, 153, 104, 76, 70, 99, 98, 119, 82, 136, 116, 32, 44, 123, 98

Offset: 1

Rémy Sigrist, Dec 21 2023

			a(10) = 7 for we have seven ways to go from 10 to 1:
    10 -> 1,
    10 -> 2 -> 1,
    10 -> 5 -> 1,
    10 -> 5 -> 3 -> 1,
    10 -> 6 -> 1,
    10 -> 6 -> 2 -> 1,
    10 -> 6 -> 3 -> 1.

Rémy Sigrist, Table of n, a(n) for n = 1..8192
Rémy Sigrist, PARI program
Index entries for sequences related to binary expansion of n

Cf. A011782, A368198 (decimal variant), A368313, A368314.

PARI
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See Links section.
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a(1) = 1.
a(n) = Sum_{k = A368314(n)-1} a(A368313(k)) for any n > 1.
a(2^k) = A011782(k) for any k >= 0.

cp's OEIS Frontend

A368194 Irregular table T(n, k), n > 0, k = 1..A368195(n), read by rows: the n-th row lists the numbers that can be obtained by replacing any positive number without leading zeros, say m, appearing in the decimal expansion of n by one of the divisors of m.

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A368314 a(n) is the number of numbers that can be obtained by replacing any positive number without leading zeros, say m, appearing in the binary expansion of n by one of the divisors of m.

Views

Author

Keywords

Comments

Examples

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Crossrefs

Programs

PARI

Formula

A368315 a(n) gives the number of ways to go from n to 1 with steps consisting of replacing a positive number without leading zero, say m, appearing in the binary expansion of a number, by a proper divisor of m.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula