A212176 -id:A212176 - cp's OEIS frontend

A212644 If an integer's second signature (cf. A212172) is the n-th to appear among positive integers, a(n) = number of distinct second signatures represented among its divisors.

1, 2, 3, 4, 5, 3, 6, 5, 7, 7, 6, 8, 9, 9, 9, 11, 12, 4, 10, 13, 10, 15, 7, 11, 15, 14, 18, 10, 12, 17, 18, 9, 21, 13, 15, 13, 19, 22, 14, 24, 16, 20, 14, 21, 26, 19, 10, 27, 19, 25, 16, 15, 23, 30, 24, 5, 21, 16, 30, 22, 30, 23, 16, 25, 34, 29, 9, 27, 22, 33

Offset: 1

Matthew Vandermast, Jun 07 2012

nonn

Also, number of divisors of A181800(n) that are members of A181800.

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 (cf. A212642). Let m be any integer with second signature {S}. Then A212180(m) = k and A085082(m) is congruent to j modulo k. If {S} is the second signature of A181800(n), then A085082(m) is congruent to A212643(n) modulo a(n).

			The divisors of 72 represent 5 distinct second signatures (cf. A212172), as can be seen from the exponents >=2, if any, in the canonical prime factorization of each divisor:
{ }: 1, 2 (prime), 3 (prime), 6 (2*3)
{2}: 4 (2^2), 9 (3^2), 12 (2^2*3), 18 (2*3^2)
{3}: 8 (2^3), 24 (2^3*3)
{2,2}: 36 (2^2*3^2)
{3,2}: 72 (2^3*3^2)
Since 72 = A181800(8), a(8) = 5.

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

Cf. A181800, A085082, A212172, A212176, A212642, A212643.

a(n) = A212180(A181800(n)).

Data corrected by Amiram Eldar, Jul 14 2019

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)).

A212174 Row n of table represents second signature of A013929(n): list of exponents >= 2 in canonical prime factorization of A013929(n), in nonincreasing order.

Original entry on oeis.org

2, 3, 2, 2, 4, 2, 2, 3, 2, 3, 2, 5, 2, 2, 3, 2, 2, 4, 2, 2, 2, 3, 3, 2, 2, 6, 2, 3, 2, 2, 2, 4, 4, 2, 3, 2, 2, 5, 2, 2, 2, 2, 3, 3, 2, 4, 2, 2, 3, 2, 2, 3, 2, 7, 2, 3, 3, 2, 4, 2, 2, 2, 2, 3, 2, 2, 5, 4, 2, 3, 2, 2, 2, 2, 4, 2, 2, 3, 2, 3, 6, 2, 2, 2, 3, 2, 2, 2

Offset: 1

Matthew Vandermast, Jun 03 2012

Length of row n equals A212177(n).

			First rows of table read: 2; 3; 2; 2; 4; 2; 2; 3;...
12 = 2^2*3 has positive exponents 2 and 1 in its prime factorization, but only exponents that are 2 or greater appear in a number's second signature. Hence, 12's second signature is {2}. Since 12 = A013929(4), row 4 of the table represents the second signature {2}.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.

Jason Kimberley, Table of i, a(i) for i = 1..4492 (n = 1..3917)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Primefan, The First 2500 Integers Factored (1st of 5 pages)

Cf. A013929, A212172, A212175, A212176, A212177.

&cat[Reverse(Sort([pe[2]:pe in Factorisation(n)|pe[2]gt 1])):n in[1..247]]; // Jason Kimberley, Jun 13 2012

a(n) = A212172(A013929(n)).

This sequence is both the subsequence of A212171 formed by omitting all 1s and the subsequence of A212172 formed by omitting all 0's. - Jason Kimberley, Jun 13 2012

A212175 List of exponents >= 2 in canonical prime factorization of A025487(n) (first integer of each prime signature), in nonincreasing order, or 0 if no such exponent exists.

Original entry on oeis.org

0, 0, 2, 0, 3, 2, 4, 3, 0, 5, 2, 2, 4, 2, 6, 3, 2, 5, 3, 7, 4, 2, 2, 2, 6, 0, 3, 3, 4, 8, 5, 2, 3, 2, 7, 2, 4, 3, 5, 9, 6, 2, 4, 2, 8, 3, 5, 3, 2, 2, 2, 6, 10, 3, 3, 7, 2, 2, 2, 4, 4, 5, 2, 9, 4, 6, 3, 3, 2, 2, 7, 11, 4, 3, 8, 2, 0, 3, 2, 5, 4, 6, 2, 10, 5, 7, 3

Offset: 1

Matthew Vandermast, Jun 03 2012

Length of row n equals A212178(n) if A212178(n) is positive, or 1 if A212178(n) = 0.

Row n of table represents second signature of A025487(n) (cf. A212172). The use of 0 in the table to represent numbers with no exponents >=2 in their prime factorization accords with the usual OEIS practice of using 0 to represent nonexistent elements when possible. In comments, the second signature of squarefree numbers is represented as { }.

			240 = 2^4*3*5 has 1 exponent in its canonical prime factorization that equals or exceeds 2 (namely, 4). Hence, 240's second signature is {4}. Since 240 = A025487(24), row 24 of the table represents the second signature {4}.

Will Nicholes, Prime Signatures

Cf. A025487, A212172, A212176, A212178.

A124832 gives all positive exponents in prime factorization of A025487(n) for n > 1.

a(n) = A212172(A025487(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)).

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

A212644 If an integer's second signature (cf. A212172) is the n-th to appear among positive integers, a(n) = number of distinct second signatures represented among its divisors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

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

A212174 Row n of table represents second signature of A013929(n): list of exponents >= 2 in canonical prime factorization of A013929(n), in nonincreasing order.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Formula

A212175 List of exponents >= 2 in canonical prime factorization of A025487(n) (first integer of each prime signature), in nonincreasing order, or 0 if no such exponent exists.

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

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