A212172 -id:A212172 - cp's OEIS frontend

A212177 Number of exponents >= 2 in the canonical prime factorization of the n-th nonsquarefree number (A013929(n)).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1

Offset: 1

Matthew Vandermast, Jun 03 2012

Length of second signature of A013929(n) (cf. A212172).

			24 = 2^3*3 has 1 exponent of size 2 or greater in its prime factorization. Since 24 = A013929(8), a(8) = 1.

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.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
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, A212174.

Cf. A046028, A056170, A085548, A124010, A229099.

a212177 n = a212177_list !! (n-1)
a212177_list = filter (> 0) a056170_list
-- Reinhard Zumkeller, Dec 29 2012

f[n_] := Module[{c = Count[FactorInteger[n][[;; , 2]], ?(# > 1&)]}, If[n > 1 && c > 0, c, Nothing]]; f[1] = 0; Array[f, 300] (* _Amiram Eldar, Oct 01 2023 *)

from math import isqrt
from sympy import mobius, factorint
def A212177(n):
    def f(x): return n+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return sum(1 for e in factorint(m).values() if e>1) # Chai Wah Wu, Jul 22 2024

a(n) = A056170(A013929(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (Sum_{p prime} 1/p^2)/(1-1/zeta(2)) = A085548 / A229099 = 1.15347789194214704903... . - Amiram Eldar, Oct 01 2023

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)).

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)).

A375339 If n has exactly one non-unitary prime factor then a(n) is the exponent of the highest power of this prime that divides n, otherwise a(n) = 0.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 3, 2, 0, 3, 2, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 2, 2, 0, 0, 4, 2, 2, 0, 2, 0, 3, 0, 3, 0, 0, 0, 2, 0, 0, 2, 6, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 4, 4, 0, 0, 2, 0, 0, 0, 3, 0, 2, 0, 2, 0, 0, 0, 5, 0, 2, 2, 0, 0, 0, 0, 3, 0

Offset: 1

Amiram Eldar, Aug 12 2024

First differs from A212172, A275812 and A372603 at n = 36.
If n = m * p^e, such that m is squarefree, p is a prime that does not divide m and e >= 2, then a(n) = e, otherwise a(n) = 0.
By definition all the positive terms are larger than 1.
The asymptotic density of 0's in this sequence is 1 - Sum_{p prime} (1/(p^2-1)) / zeta(2) = 1 - A059956 * A154945 = 0.66461069244308962639... .
The asymptotic density of the occurrences of k >= 2 in this sequence is Sum_{p prime} (1/(p^(k-1)*(p+1))) / zeta(2). E.g., 0.200755... (A271971) for k = 2, 0.0741777... for k = 3, and 0.0320652... for k = 4.

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

Cf. A005361, A048109, A051903, A060687, A082293, A190641, A359466, A375341 (the positive terms).
Cf. A013661, A059956, A154945, A271971, A375340.
Cf. A212172, A275812, A372603.

a[n_] := Module[{e = Select[FactorInteger[n][[;; , 2]], # > 1 &]}, If[Length[e] == 1, e[[1]], 0]]; Array[a, 100]

a(n) = {my(e = select(x -> x > 1, factor(n)[,2])); if(#e == 1, e[1], 0);}

a(n) = A051903(n) * A359466(n).
a(n) = A005361(n) * A359466(n).
a(A190641(n)) >= 2.
a(n) = 2 if and only if n is in A060687.
a(n) = 3 if and only if n is in A048109.
a(n) <= 3 if and only if n is in A082293.
Asymptotic mean:  = Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} (2*p-1)/((p-1)^2*(p+1)) / zeta(2) = A375340 / A013661 = 0.92105359989459565838... .
Asymptotic second raw moment:  = Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)^2 = Sum_{p prime} (4*p^2-3*p+1)/((p-1)^3*(p+1)) / zeta(2) = 3.04027120804428071157... .
The asymptotic second central moment, or variance, is  - ^2 = 2.19193147416548680815... and the asymptotic standard deviation is sqrt( - ^2) = 1.48051729951577627898... .

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)).

A212178 Number of exponents >= 2 in canonical prime factorization of A025487(n) (first integer of each prime signature).

Original entry on oeis.org

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

Offset: 1

Matthew Vandermast, Jun 04 2012

nonn

Length of row n of A212175 equals a(n) if a(n) is positive, 1 otherwise.

			The canonical prime factorization of 24 (2^3*3) has 1 exponent that equals or exceeds 2. Since 24 = A025487(8), a(8) = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Will Nicholes, Prime Signatures
Index entries for sequences related to prime signature

Cf. A025487, A212172.

a(n) = A056170(A025487(n)).

A212183 Largest odd divisor of A002183(n) (number of divisors of n-th highly composite number).

Original entry on oeis.org

1, 1, 3, 1, 3, 1, 9, 5, 3, 1, 9, 5, 3, 15, 1, 9, 5, 3, 15, 1, 9, 5, 21, 45, 3, 25, 27, 15, 1, 9, 5, 21, 45, 3, 25, 27, 7, 15, 1, 9, 5, 21, 45, 3, 25, 27, 7, 15, 63, 1, 9, 75, 5, 21, 45, 3, 25, 27, 7, 15, 63, 1, 9, 75, 5, 21, 45, 3, 25, 105, 27, 7, 15, 63, 1, 9

Offset: 1

Matthew Vandermast, Jun 08 2012

nonn

The "odd part" (largest odd divisor) of the number of divisors of n is a function of the exponents >=2 in the prime factorization of n (cf. A212172, A212181).

The number 1 appears a total of 18 times (see Graeme link for proof). Ramanujan proved that no number appears an infinite number of times (see Ramanujan link). It would be interesting to know more about a) which odd numbers appear in the sequence and b) how many times a number of a given size can appear in the sequence. See also A160233.

			The highly composite number 48 has a total of 10 divisors. Since 48 = A002182(8), A002183(8) = 10. Since the largest odd divisor of 10 is 5, a(8) = 5.

Amiram Eldar, Table of n, a(n) for n = 1..10000
A. Flammenkamp, List of the first 1200 highly composite numbers
Graeme McRae, Highly Composite Numbers
S. Ramanujan, Highly Composite Numbers (p. 34).
S. Ramanujan, Highly composite numbers, Proc. Lond. Math. Soc. 14 (1915), 347-409; reprinted in Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962.

Cf. A000265, A002182, A002183, A212181.

A160233 gives the n-th integer that is the largest member of A002183 with its particular odd part.

a(n) = A000265(A002183(n)) = A212181(A002182(n)).

A212184 Row n of table gives exponents >= 2 in canonical prime factorization of n-th highly composite number (A002182(n)), in nonincreasing order, or 0 if no such exponent exists.

Original entry on oeis.org

0, 0, 2, 0, 2, 3, 2, 2, 4, 2, 3, 2, 2, 4, 3, 2, 4, 2, 3, 2, 2, 4, 3, 2, 4, 2, 3, 3, 5, 2, 4, 3, 6, 2, 4, 2, 2, 3, 2, 4, 4, 5, 2, 2, 4, 2, 3, 3, 5, 2, 4, 3, 6, 2, 4, 2, 2, 5, 3, 4, 4, 5, 2, 2, 6, 3, 4, 2, 3, 3, 5, 2, 4, 3, 6, 2, 4, 2, 2, 5, 3, 4, 4, 5, 2, 2, 6

Offset: 1

Matthew Vandermast, Jul 01 2012

Length of row n equals A212185(n) if A212185(n) is positive, or 1 if A212185(n) = 0.
Row n of table represents second signature of A002182(n) (cf. A212172). The use of 0 in the table to represent squarefree highly composite numbers 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 { }.
No row is repeated an infinite number of times in the table.  The contrary to this would imply that at least one integer appeared in A212183 an infinite number of times - something that Ramanujan proved to be false (cf. Ramanujan link). It would be interesting to know if there is an upper bound on the number of times a row can appear.

			First rows read: 0; 0; 2; 0; 2; 3; 2,2; 4; 2; 3; 2,2; 4;...
12 = 2^2*3 has positive exponents 2 and 1 in its canonical prime factorization (1s are often left implicit as exponents). Only exponents that are 2 or greater appear in a number's second signature; therefore, 12's second signature is {2}.  Since 12 = A002182(5), row 5 represents the second signature {2}.

Srinivasa Ramanujan, Highly composite numbers, Proc. Lond. Math. Soc. 14 (1915), 347-409; reprinted in Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962.

Amiram Eldar, Table of n, a(n) for n = 1..5594 (first 1000 rows)
Achim Flammenkamp, List of the first 1200 highly composite numbers.
Srinivasa Ramanujan, Highly Composite Numbers (p. 34).

Cf. A002182, A212172, A212182, A212183, A212185.

With[{v = Import["https://oeis.org/A002182/b002182.txt", "Table"][[;; , 2]]}, exp[n_] := Select[FactorInteger[n][[;; , 2]], # > 1 &]; exp /@ v[[1 ;; 100]] /. {} -> {0} // Flatten] (* Amiram Eldar, Jan 20 2025 *)

Row n is identical to row A002182(n) of table A212172.

A212185 Number of exponents >= 2 in canonical prime factorization of n-th highly composite number (A002182(n)).

Original entry on oeis.org

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

Offset: 1

Matthew Vandermast, Jul 16 2012

nonn

Length of row n of A212184 equals a(n) if a(n) is positive, 1 otherwise.

			The canonical prime factorization of 720 (2^4*3^2*5) has 2 exponents that equal or exceed 2. Since 720 = A002182(14), a(14) = 2.

S. Ramanujan, Highly composite numbers, Proc. Lond. Math. Soc. 14 (1915), 347-409; reprinted in Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962.

Amiram Eldar, Table of n, a(n) for n = 1..10000
A. Flammenkamp, List of the first 1200 highly composite numbers
S. Ramanujan, Highly Composite Numbers

Cf. A002182, A212172, A212182, A212184.

s={}; dm=0; Do[d = DivisorSigma[0, n]; If[d > dm, dm = d; e = FactorInteger[n][[;;,2]]; AppendTo[s, Count[e, ?(# > 1 &)]]], {n, 1, 10^6}]; s (* _Amiram Eldar, Jun 30 2019 *)

cp's OEIS Frontend

A212177 Number of exponents >= 2 in the canonical prime factorization of the n-th nonsquarefree number (A013929(n)).

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

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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)).

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A375339 If n has exactly one non-unitary prime factor then a(n) is the exponent of the highest power of this prime that divides n, otherwise a(n) = 0.

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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.

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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.

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A212178 Number of exponents >= 2 in canonical prime factorization of A025487(n) (first integer of each prime signature).

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A212183 Largest odd divisor of A002183(n) (number of divisors of n-th highly composite number).

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