A212638 -id:A212638 - cp's OEIS frontend

A212645 a(n) = number of excess prime divisors of A181800(n) (n-th powerful number that is the first integer of its prime signature).

0, 1, 2, 3, 4, 2, 5, 3, 6, 4, 4, 7, 5, 5, 8, 6, 6, 3, 9, 7, 6, 7, 4, 10, 8, 7, 8, 5, 11, 9, 8, 5, 9, 6, 8, 12, 10, 9, 6, 10, 7, 9, 13, 11, 10, 7, 6, 11, 8, 10, 7, 14, 12, 11, 8, 4, 10, 7, 12, 9, 11, 8, 15, 13, 12, 9, 5, 11, 8, 13, 10, 12, 9, 16, 14, 8, 13, 10

Offset: 1

Matthew Vandermast, Jun 09 2012

nonn

The excess of n, or A046660(n), is a function of the second signature of n (cf. A212172). Since A181800 gives the first integer of each second signature, this sequence gives the value of A046660 for each second signature in order of its first appearance. Each nonnegative integer n occurs A000041(n) times in the sequence.

a(n) is also the number of prime factors of A212638(n), counted with multiplicity.

			36 (2^2*3^2, or 2*2*3*3) has 4 prime factors when repetitions are counted, but only 2 distinct prime factors.  Therefore, its "excess" as defined in A046660 is (4-2) = 2.  Since 36 = A181800(6), a(6) = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A046660, A181800, A212172, A212176, A212179, A212638, A212639, A212647.

A rearrangement of A036042.

a(n) = A046660(A181800(n)) = A212639(n)-A212179(n).

a(n) = A001222(A212638(n)).

A212647 a(n) = product of exponents in canonical prime factorization of A181800(n) (n-th powerful number that is the first integer of its prime signature); a(1) = 1 by convention.

Original entry on oeis.org

1, 2, 3, 4, 5, 4, 6, 6, 7, 8, 9, 8, 10, 12, 9, 12, 15, 8, 10, 14, 16, 18, 12, 11, 16, 20, 21, 16, 12, 18, 24, 18, 24, 20, 25, 13, 20, 28, 24, 27, 24, 30, 14, 22, 32, 30, 27, 30, 28, 35, 32, 15, 24, 36, 36, 16, 36, 36, 33, 32, 40, 40, 16, 26, 40, 42, 24, 42, 45

Offset: 1

Matthew Vandermast, Jun 09 2012

nonn

The product of exponents in the canonical prime factorization of n, or A005361(n), is a function of the second signature of n (cf. A212172). Since A181800 consists of the first integer of each second signature, this sequence gives the value of A005361 for each second signature in order of its first appearance.

a(n) also gives the number of divisors of A212638(n), a permutation of A025487. Each positive integer n appears A001055(n) times in this sequence.

			The product of the exponents in the prime factorization of 144 (2^4*3^2) is 4*2 = 8.  Since 144 = A181800(10), a(10) = 8.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A005361, A181800, A212172, A212638.

a(n) = A005361(A181800(n)).

a(n) = A000005(A212638(n)).

cp's OEIS Frontend

A212645 a(n) = number of excess prime divisors of A181800(n) (n-th powerful number that is the first integer of its prime signature).

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