A212172 -id:A212172 - cp's OEIS frontend

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

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)).

A212641 Largest odd divisor of A212640(n) (number of divisors of n-th powerful number that is the first integer of its prime signature).

Original entry on oeis.org

1, 3, 1, 5, 3, 9, 7, 3, 1, 15, 1, 9, 9, 5, 5, 21, 3, 27, 11, 3, 25, 7, 9, 3, 27, 15, 1, 45, 13, 15, 35, 3, 9, 27, 9, 7, 33, 5, 15, 5, 63, 21, 15, 9, 45, 9, 1, 11, 9, 3, 75, 1, 39, 25, 21, 81, 49, 5, 3, 81, 27, 45, 17, 21, 55, 3, 27, 7, 3, 13, 45, 15, 105, 9, 45

Offset: 1

Matthew Vandermast, Jun 09 2012

nonn

The odd part of d(n), or largest odd divisor of d(n) (A212181(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 A212181 for each second signature in order of its first appearance.
Note: The odd part of d(n) is not the same as the number of odd divisors of n (A001227(n)).
Each odd integer appears an infinite number of times.

			A181800(5) = 32 has 6 divisors (1, 2, 4, 8, 16 and 32).  The largest odd divisor of 6 is 3. Hence, a(5) = 3.

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

Cf. A000005, A000265, A181800, A212172, A212181, A212640.

a(n) = A000265(A212640(n)) = A212181(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)).

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)).

A212646 a(n) = number of Abelian groups of order A181800(n) (n-th powerful number that is the first integer of its prime signature).

Original entry on oeis.org

1, 2, 3, 5, 7, 4, 11, 6, 15, 10, 9, 22, 14, 15, 30, 22, 21, 8, 42, 30, 25, 33, 12, 56, 44, 35, 45, 20, 77, 60, 55, 18, 66, 28, 49, 101, 84, 75, 30, 90, 44, 77, 135, 112, 110, 42, 27, 126, 60, 105, 50, 176, 154, 150, 66, 16, 121, 45, 168, 88, 154, 70, 231, 202

Offset: 1

Matthew Vandermast, Jun 09 2012

nonn

The number of Abelian groups of order n, or A000688(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 A000688 for each second signature in order of its first appearance.

			There are 6 Abelian groups of order 72, corresponding to the 6 factorizations of 72 into prime powers: 2^3*3^2, 2^3*3*3, 2^2*2*3^2, 2^2*2*3*3, 2*2*2*3^2, and 2*2*2*3*3. Since 72 = A181800(8), a(8) = 6.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Abelian Group

Cf. A000688, A046054, A046055, A046056, A050360, A181800, A212172.

a(n) = A000688(A181800(n)).

cp's OEIS Frontend

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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A212641 Largest odd divisor of A212640(n) (number of divisors of n-th powerful number that is the first integer of its prime signature).

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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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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.

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Formula

A212646 a(n) = number of Abelian groups of order A181800(n) (n-th powerful number that is the first integer of its prime signature).

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