A241909 -id:A241909

A242378

Square array read by antidiagonals: to obtain A(i,j), replace each prime factor prime(k) in prime factorization of j with prime(k+i).

+10
17

0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 5, 5, 1, 0, 5, 9, 7, 7, 1, 0, 6, 7, 25, 11, 11, 1, 0, 7, 15, 11, 49, 13, 13, 1, 0, 8, 11, 35, 13, 121, 17, 17, 1, 0, 9, 27, 13, 77, 17, 169, 19, 19, 1, 0, 10, 25, 125, 17, 143, 19, 289, 23, 23, 1, 0, 11, 21, 49, 343, 19, 221, 23, 361, 29, 29, 1, 0

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CROSSREFS

Permutations whose formulas refer to this array: A122111, A241909, A242415, A242419, A246676, A246678, A246684.

A242424

Bulgarian solitaire operation on partition list A112798: a(1) = 1, a(n) = A000040(A001222(n)) * A064989(n).

+10
17

1, 2, 4, 3, 6, 6, 10, 5, 12, 9, 14, 10, 22, 15, 18, 7, 26, 20, 34, 15, 30, 21, 38, 14, 27, 33, 40, 25, 46, 30, 58, 11, 42, 39, 45, 28, 62, 51, 66, 21, 74, 50, 82, 35, 60, 57, 86, 22, 75, 45, 78, 55, 94, 56, 63, 35, 102, 69, 106, 42, 118, 87, 100, 13, 99, 70, 122, 65

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FORMULA

a(n) = A241909(A243051(A241909(n))).

A241917

If n is a prime with index i, p_i, a(n) = i, (with a(1)=0), otherwise difference (i-j) of the indices of the two largest primes p_i, p_j, i >= j in the prime factorization of n: a(n) = A061395(n) - A061395(A052126(n)).

+10
16

0, 1, 2, 0, 3, 1, 4, 0, 0, 2, 5, 1, 6, 3, 1, 0, 7, 0, 8, 2, 2, 4, 9, 1, 0, 5, 0, 3, 10, 1, 11, 0, 3, 6, 1, 0, 12, 7, 4, 2, 13, 2, 14, 4, 1, 8, 15, 1, 0, 0, 5, 5, 16, 0, 2, 3, 6, 9, 17, 1, 18, 10, 2, 0, 3, 3, 19, 6, 7, 1, 20, 0, 21, 11, 0, 7, 1, 4, 22, 2, 0, 12, 23

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CROSSREFS

Cf. A052126, A061395, A070003, A122111, A241909.

A241918

Table of partitions where the ordering is based on the modified partial sums of the exponents of primes in the prime factorization of n.

+10
15

0, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 2, 2, 4, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5

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CROSSREFS

Permutation A241909 maps between order of partitions employed here, and the order employed in A112798.

A243353

Permutation of natural numbers which maps between the partitions as encoded in A227739 (binary based system, zero-based) to A112798 (prime-index based system, one-based).

+10
15

1, 2, 4, 3, 9, 8, 6, 5, 25, 18, 16, 27, 15, 12, 10, 7, 49, 50, 36, 75, 81, 32, 54, 125, 35, 30, 24, 45, 21, 20, 14, 11, 121, 98, 100, 147, 225, 72, 150, 245, 625, 162, 64, 243, 375, 108, 250, 343, 77, 70, 60, 105, 135, 48, 90, 175, 55, 42, 40, 63, 33, 28, 22, 13, 169, 242, 196, 363, 441, 200, 294, 605, 1225, 450, 144

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FORMULA

a(n) = A241909(1+A075157(n)). [With A075157's original starting offset]

CROSSREFS

Cf. A227739, A112798, A075157, A241909, A005940, A003188, A226062, A242424.

A243503

Sums of parts of partitions (i.e., their sizes) as ordered in the table A241918: a(n) = Sum_{i=A203623(n-1)+2..A203623(n)+1} A241918(i).

+10
15

0, 1, 2, 2, 3, 4, 4, 3, 3, 6, 5, 6, 6, 8, 5, 4, 7, 5, 8, 9, 7, 10, 9, 8, 4, 12, 4, 12, 10, 8, 11, 5, 9, 14, 6, 7, 12, 16, 11, 12, 13, 11, 14, 15, 7, 18, 15, 10, 5, 7, 13, 18, 16, 6, 8, 16, 15, 20, 17, 11, 18, 22, 10, 6, 10, 14, 19, 21, 17, 10, 20, 9, 21, 24, 6, 24

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FORMULA

a(n) = A056239(A241909(n)).

CROSSREFS

Cf. A243504 (the products of parts), A241918, A000041, A227183, A075158, A056239, A241909.

A243070

Square array read by antidiagonals: rows are successively recursivized versions of Bulgarian solitaire operation (starting from the usual "first order" version, A242424), as applied to the partitions listed in A112798.

+10
10

1, 2, 1, 4, 2, 1, 3, 4, 2, 1, 6, 3, 4, 2, 1, 6, 8, 3, 4, 2, 1, 10, 6, 8, 3, 4, 2, 1, 5, 12, 6, 8, 3, 4, 2, 1, 12, 5, 16, 6, 8, 3, 4, 2, 1, 9, 9, 5, 16, 6, 8, 3, 4, 2, 1, 14, 12, 9, 5, 16, 6, 8, 3, 4, 2, 1, 10, 20, 12, 9, 5, 16, 6, 8, 3, 4, 2, 1, 22, 10, 24, 12, 9, 5, 16, 6, 8, 3, 4, 2, 1, 15, 28, 10, 32, 12, 9, 5, 16, 6, 8, 3, 4, 2, 1, 18, 18, 40, 10, 32, 12, 9, 5, 16, 6, 8, 3, 4, 2, 1

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COMMENTS

Each row is a A241909-conjugate of the corresponding row in A243060.

CROSSREFS

Cf. A243060, A241909, A112798, A105560, A000040, A001222, A064989.

A243354

Permutation of natural numbers which maps between the partitions as encoded in A112798 (prime-index based system, one-based) to A227739 (binary based system, zero-based).

+10
10

0, 1, 3, 2, 7, 6, 15, 5, 4, 14, 31, 13, 63, 30, 12, 10, 127, 9, 255, 29, 28, 62, 511, 26, 8, 126, 11, 61, 1023, 25, 2047, 21, 60, 254, 24, 18, 4095, 510, 124, 58, 8191, 57, 16383, 125, 27, 1022, 32767, 53, 16, 17, 252, 253, 65535, 22, 56, 122, 508, 2046, 131071

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FORMULA

a(n) = A075158(A241909(n)-1). [With A075158's original starting offset].

CROSSREFS

Cf. A227739, A112798, A006068, A156552, A075158, A241909.

A246684

"Caves of prime shift" permutation: a(1) = 1, a(n) = A242378(A007814(n), 2*a(A003602(n))) - 1.

+10
9

1, 2, 3, 4, 5, 8, 7, 6, 9, 14, 15, 24, 13, 26, 11, 10, 17, 20, 27, 34, 29, 80, 47, 48, 25, 32, 51, 124, 21, 44, 19, 12, 33, 74, 39, 54, 53, 98, 67, 76, 57, 104, 159, 624, 93, 404, 95, 120, 49, 50, 63, 64, 101, 152, 247, 342, 41, 38, 87, 174, 37, 62, 23, 16, 65, 56, 147, 244, 77, 188, 107, 90, 105, 374, 195, 324, 133, 170, 151, 142, 113, 92

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CROSSREFS

Similar or related permutations: A005940, A163511, A241909, A245606, A246278, A246375, A249814, A250243.

A165199

a(n) is obtained by flipping every second bit in the binary representation of n starting at the second-most significant bit and on downwards.

+10
8

0, 1, 3, 2, 6, 7, 4, 5, 13, 12, 15, 14, 9, 8, 11, 10, 26, 27, 24, 25, 30, 31, 28, 29, 18, 19, 16, 17, 22, 23, 20, 21, 53, 52, 55, 54, 49, 48, 51, 50, 61, 60, 63, 62, 57, 56, 59, 58, 37, 36, 39, 38, 33, 32, 35, 34, 45, 44, 47, 46, 41, 40, 43, 42, 106, 107, 104, 105, 110, 111, 108

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FORMULA

a(n) = A075158(A243353(n)-1) = A075158((A241909(1+A075157(n))) - 1).

CROSSREFS

Cf. A000035, A000523, A075157, A075158, A241909, A243353, A245443, A245444.