A181800 -id:A181800 - 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.

A212642 a(n) = number of distinct prime signatures represented among divisors of A181800(n) (n-th powerful number that is the first integer of its prime signature).

Original entry on oeis.org

1, 3, 4, 5, 6, 6, 7, 9, 8, 12, 10, 9, 15, 14, 10, 18, 18, 10, 11, 21, 15, 22, 16, 12, 24, 20, 26, 22, 13, 27, 25, 19, 30, 28, 21, 14, 30, 30, 28, 34, 34, 27, 15, 33, 35, 37, 20, 38, 40, 33, 31, 16, 36, 40, 46, 15, 28, 30, 42, 46, 39, 43, 17, 39, 45, 55, 25, 35

Offset: 1

Matthew Vandermast, Jun 05 2012

nonn

Also, number of divisors of A181800 that are members of A025487.

Consider a member of A181800 with second signature {S} whose divisors represent a total of k distinct second signatures and a total of (j+k) distinct prime signatures. Let n be any integer with second signature {S}. Then A212180(n) = k and A085082(n) is congruent to j modulo k. Cf. A212643, A212644.

			The divisors of 36 represent a total of 6 distinct prime signatures (cf. A085082), as can be seen from the positive exponents, if any, in the canonical prime factorization of each divisor:
{ }: 1 (multiset of positive exponents is the empty multiset)
{1}: 2 (2^1), 3 (3^1)
{1,1}: 6 (2^1*3^1)
{2}: 4 (2^2), 9 (3^2),
{2,1}: 12 (2^2*3^1), 18 (2^1*3^2)
{2,2}: 36 (2^2*3^2)
Since 36 = A181800(6), a(6) = 6.

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

Cf. A181800, A085082, A212171, A212172, A212643, A212644.

a(n) = A085082(A181800(n)).

A212179 Number of distinct prime factors of A181800(n) (n-th powerful number that is the first integer of its prime signature).

Original entry on oeis.org

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

Offset: 1

Matthew Vandermast, Jun 04 2012

nonn

Since each prime factor of A181800(n) divides A181800(n) at least twice, this is also the number of exponents > 2 in prime factorization of A181800(n).

Length of row A181800(n) of table A212171 equals a(n) for n > 1. Row A181800(n) of table A212172 has the same length when n > 1 (length = 1 if n = 1).

			72 (2^3*3^2) has 2 distinct prime factors. Since 72 = A181800(8), a(8) = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Will Nicholes, Prime Signatures

Cf. A181800, A001694, A025487, A212171, A212172, A212176.

a(n) = A001221(A181800(n)) = A056170(A181800(n)).

A212643 Let b(n) and c(n) be the total numbers of distinct prime signatures and second signatures, respectively, represented among divisors of A181800(n) (first integers of each second signature; cf. A212172). b(n) mod c(n) = a(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 0, 1, 4, 1, 5, 4, 1, 6, 5, 1, 7, 6, 2, 1, 8, 5, 7, 2, 1, 9, 6, 8, 2, 1, 10, 7, 1, 9, 2, 6, 1, 11, 8, 0, 10, 2, 7, 1, 12, 9, 18, 0, 11, 2, 8, 15, 1, 13, 10, 22, 0, 7, 14, 12, 2, 9, 20, 1, 14, 11, 26, 7, 8, 18, 13, 2, 10, 25, 1, 15, 15, 12, 30, 9

Offset: 1

Matthew Vandermast, Jun 05 2012

Significance of the sequence: Consider a member of A181800 with second signature {S} whose divisors represent a total of k distinct second signatures and a total of (j+k) distinct prime signatures. For all integers n with second signature {S}, A212180(n) = k and A085082(n) is congruent to j modulo k; see examples.

Note: b(n) = A212642(n); c(n) = A212644(n).

			4 is the smallest integer with second signature {2}, and its divisors represent 3 distinct prime signatures and 2 distinct second signatures. 1 = 3 mod 2. Since 4 = A181800(2), a(2) = 1. For all integers m with second signature {2}, A085082(m) is congruent to 1 modulo 2.
10800 is the smallest integer with second signature {4,3,2}, and its divisors represent 28 distinct prime signatures and 14 distinct second signatures. 0 = 28 mod 14.  Since 10800 = A181800(39), a(39) = 0. For all integers m with second signature {4,3,2}, A085082(m) is congruent to 0 modulo 14.

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

Cf. A085082, A212171, A212172, A212180, A212642, A212644.

a(n) = A212642(n)-A212644(n), reduced modulo A212644(n).

A212638 a(n) = n-th powerful number that is the first integer of its prime signature, divided by its largest squarefree divisor: A003557(A181800(n)).

Original entry on oeis.org

1, 2, 4, 8, 16, 6, 32, 12, 64, 24, 36, 128, 48, 72, 256, 96, 144, 30, 512, 192, 216, 288, 60, 1024, 384, 432, 576, 120, 2048, 768, 864, 180, 1152, 240, 1296, 4096, 1536, 1728, 360, 2304, 480, 2592, 8192, 3072, 3456, 720, 900, 4608, 960, 5184, 1080, 16384

Offset: 1

Matthew Vandermast, Jun 05 2012

nonn

The number of second signatures represented by the divisors of A181800(n) equals the number of prime signatures represented among the divisors of a(n). Cf. A212172, A212644.

A permutation of A025487.

			6 (whose prime factorization is 2*3) is the largest squarefree divisor of 144 (whose prime factorization is 2^4*3^2). Since 144 = A181800(10), and 144/6 = 24, a(10) = 24.

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

Cf. A003557, A007947, A085082, A181800, A212172, A212642, A212644.

Other permutations of A025487 include A036035, A059901, A063008, A077569, A085988, A086141, A087443, A108951, A181821, A181822.

a(n) = A003557(A181800(n)).

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)).

A212640 Number of divisors of A181800(n) (n-th powerful number that is the first integer of its prime signature).

Original entry on oeis.org

1, 3, 4, 5, 6, 9, 7, 12, 8, 15, 16, 9, 18, 20, 10, 21, 24, 27, 11, 24, 25, 28, 36, 12, 27, 30, 32, 45, 13, 30, 35, 48, 36, 54, 36, 14, 33, 40, 60, 40, 63, 42, 15, 36, 45, 72, 64, 44, 72, 48, 75, 16, 39, 50, 84, 81, 49, 80, 48, 81, 54, 90, 17, 42, 55, 96, 108

Offset: 1

Matthew Vandermast, Jun 09 2012

nonn

			A181800(6) = 36 has 9 divisors (1, 2, 3, 4, 6, 9, 12, 18 and 36).  Hence, a(6) = 9.

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

Cf. A181800, A000005, A212641.

a(n) = A000005(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

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A212642 a(n) = number of distinct prime signatures represented among divisors of A181800(n) (n-th powerful number that is the first integer of its prime signature).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A212179 Number of distinct prime factors of A181800(n) (n-th powerful number that is the first integer of its prime signature).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A212643 Let b(n) and c(n) be the total numbers of distinct prime signatures and second signatures, respectively, represented among divisors of A181800(n) (first integers of each second signature; cf. A212172). b(n) mod c(n) = a(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A212638 a(n) = n-th powerful number that is the first integer of its prime signature, divided by its largest squarefree divisor: A003557(A181800(n)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A212640 Number of divisors of A181800(n) (n-th powerful number that is the first integer of its prime signature).

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs