A212179 -id:A212179 - cp's OEIS frontend

A212176 Row n of table lists exponents in canonical prime factorization of A181800(n) (n-th powerful number that is the first integer of its prime signature), in nonincreasing order.

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

Offset: 2

Matthew Vandermast, Jun 03 2012

A212179(n) gives length of row n.

Table represents prime signature (cf. A212171) and second signature (cf. A212172) of A181800.

			Since 72 is a member of A181800, all positive exponents in its prime factorization (2^3*3^2) equal or exceed 2. Therefore, its second signature is the same as its prime signature, namely, {3,2} (nonincreasing version).  Since 72 = A181800 (8), row 8 represents the prime signature and second signature {3,2}.

Will Nicholes, Prime Signatures

Cf. A181800, A212171, A212172, A212179.

Row n is identical to row A181800(n) of tables A212171 and A212172.

A212639 Number of prime factors of A181800(n) (n-th powerful number that is the first integer of its prime signature), counted with multiplicity.

Original entry on oeis.org

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

Offset: 1

Matthew Vandermast, Jun 09 2012

nonn

Every nonnegative integer n appears A002865(n) times.

			72 (2^3*3^2, or 2*2*2*3*3) has a total of 5 prime factors when repetitions are counted.  Since 72 = A181800(8), a(8) = 5.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Will Nicholes, List of prime signatures

Cf. A181800, A001222, A212172, A212176, A212179, A212645.

a(n) = A001222(A181800(n)).

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

Original entry on oeis.org

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)).

cp's OEIS Frontend

A212176 Row n of table lists exponents in canonical prime factorization of A181800(n) (n-th powerful number that is the first integer of its prime signature), in nonincreasing order.

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A212639 Number of prime factors of A181800(n) (n-th powerful number that is the first integer of its prime signature), counted with multiplicity.

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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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