A316441 -id:A316441 - cp's OEIS frontend

A319240 Positions of zeros in A316441, the list of coefficients in the expansion of Product_{n > 1} 1/(1 + 1/n^s).

4, 6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 25, 26, 28, 33, 34, 35, 38, 39, 44, 45, 46, 48, 49, 50, 51, 52, 55, 57, 58, 62, 63, 65, 68, 69, 72, 74, 75, 76, 77, 80, 82, 85, 86, 87, 91, 92, 93, 94, 95, 98, 99, 106, 108, 111, 112, 115, 116, 117, 118, 119, 121, 122

Offset: 1

Gus Wiseman, Sep 15 2018

From Tian Vlasic, Dec 31 2021: (Start)
Numbers that have an equal number of even and odd-length unordered factorizations.
There are infinitely many terms since p^2 is a term for prime p.
Out of all numbers of the form p^k with p prime (listed in A000961), only the numbers of the form p^2 (A001248) are terms.
Out of all numbers of the form p*q^k, p and q prime, only the numbers of the form p*q (A006881), p*q^2 (A054753), p*q^4 (A178739) and p*q^6 (A189987) are terms.
Similar methods can be applied to all prime signatures. (End)

			12 = 2*6 = 3*4 = 2*2*3 has an equal number of even-length factorizations and odd-length factorizations (2). - _Tian Vlasic_, Dec 09 2021

David A. Corneth, Table of n, a(n) for n = 1..10000

Complement of A319239.

Cf. A001055, A025487, A045778, A114592, A162247, A190938, A281116, A281118, A303386, A316441, A319238.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Join@@Position[Table[Sum[(-1)^Length[f],{f,facs[n]}],{n,100}],0]

A319239 Positions of nonzero terms in A316441, the list of coefficients in the expansion of Product_{n > 1} 1/(1 + 1/n^s).

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 11, 13, 16, 17, 19, 23, 24, 27, 29, 30, 31, 32, 36, 37, 40, 41, 42, 43, 47, 53, 54, 56, 59, 60, 61, 64, 66, 67, 70, 71, 73, 78, 79, 81, 83, 84, 88, 89, 90, 96, 97, 100, 101, 102, 103, 104, 105, 107, 109, 110, 113, 114, 120, 125, 126, 127, 128

Offset: 1

Gus Wiseman, Sep 15 2018

nonn

Complement of A319240.

Cf. A001055, A045778, A114592, A162247, A190938, A281116, A281118, A303386, A316441, A319237.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Join@@Position[Table[Sum[(-1)^Length[f],{f,facs[n]}],{n,100}],_Integer?(Abs[#]>0&)]

A045778 Number of factorizations of n into distinct factors greater than 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 5, 1, 2, 2, 3, 1, 5, 1, 3, 2, 2, 2, 5, 1, 2, 2, 5, 1, 5, 1, 3, 3, 2, 1, 7, 1, 3, 2, 3, 1, 5, 2, 5, 2, 2, 1, 9, 1, 2, 3, 4, 2, 5, 1, 3, 2, 5, 1, 9, 1, 2, 3, 3, 2, 5, 1, 7, 2, 2, 1, 9, 2, 2, 2, 5, 1, 9, 2, 3, 2, 2, 2, 10, 1, 3, 3, 5, 1, 5, 1, 5

Offset: 1

David W. Wilson

This sequence depends only on the prime signature of n and not on the actual value of n.
Also the number of strict multiset partitions (sets of multisets) of the prime factors of n. - Gus Wiseman, Dec 03 2016
Number of sets of integers greater than 1 whose product is n. - Antti Karttunen, Feb 20 2024

			24 can be factored as 24, 2*12, 3*8, 4*6, or 2*3*4, so a(24) = 5. The factorization 2*2*6 is not permitted because the factor 2 is present twice. a(1) = 1 represents the empty factorization.

David W. Wilson, Table of n, a(n) for n = 1..10000
Philippe A. J. G. Chevalier, On the discrete geometry of physical quantities, Preprint, 2012.
P. A. J. G. Chevalier, On a Mathematical Method for Discovering Relations Between Physical Quantities: a Photonics Case Study, Slides from a talk presented at ICOL2014.
P. A. J. G. Chevalier, A "table of Mendeleev" for physical quantities?, Slides from a talk, May 14 2014, Leuven, Belgium.
Arnold Knopfmacher and Michael Mays, Ordered and Unordered Factorizations of Integers, The Mathematica Journal, Vol 10 (1), 2006.
R. J. Mathar, Factorizations of n = 1..1500
Eric Weisstein's World of Mathematics, Unordered Factorization
Index entries for sequences computed from exponents in factorization of n

Cf. A001055, A045779, A045780, A050323. a(p^k) = A000009. a(A002110) = A000110.
Cf. A036469, A114591, A114592, A316441 (Dirichlet inverse).
Cf. A156648 (2*Dgf at s=2), A073017 (2*Dgf at s=3), A258870 (2*Dgf at s=4).
Cf. also A069626 (Number of sets of integers > 1 whose least common multiple is n).
Cf. A287549 (partial sums).

⍝ Dyalog dialect
divisors ← {ð←⍵{(0=⍵|⍺)/⍵}⍳⌊⍵*÷2 ⋄ 1=⍵:ð ⋄ ð, (⍵∘÷)¨(⍵=(⌊⍵*÷2)*2)↓⌽ð}
A045778 ← { D←1↓divisors(⍵) ⋄ T←(⍴D)⍴2 ⋄ +/⍵⍷{×/D/⍨T⊤⍵}¨(-∘1)⍳2*⍴D } ⍝ (simple, but a memory hog)
A045778 ← { ⍺←⌽divisors(⍵) ⋄ 1=⍵:1 ⋄ 0=≢⍺:0 ⋄ R←⍺↓⍨⍺⍳⍵∘÷ ⋄ Ð←{⍺/⍨0=⍺|⍵} ⋄ +/(((R)Ð⊢)∇⊢)¨(⍵∘÷)¨⍺ } ⍝ (more efficient) - Antti Karttunen, Feb 20 2024

with(numtheory):
b:= proc(n, k) option remember;
      `if`(n>k, 0, 1) +`if`(isprime(n), 0,
      add(`if`(d>k, 0, b(n/d, d-1)), d=divisors(n) minus {1, n}))
    end:
a:= n-> b(n$2):
seq(a(n), n=1..120);  # Alois P. Heinz, May 26 2013

gd[m_, 1] := 1; gd[1, n_] := 0; gd[1, 1] := 1; gd[0, n_] := 0; gd[m_, n_] := gd[m, n] = Total[gd[# - 1, n/#] & /@ Select[Divisors[n], # <= m &]]; Array[ gd[#, #] &, 100]  (* Alexander Adam, Dec 28 2012 *)

v=vector(100,k,k==1); for(n=2,#v, v+=dirmul(v,vector(#v,k,k==n)) ); v /* Max Alekseyev, Jul 16 2014 */

A045778(n, k=n) = ((n<=k) + sumdiv(n, d, if(d > 1 && d <= k && d < n, A045778(n/d, d-1)))); \\ After Alois P. Heinz's Maple-code by Antti Karttunen, Jul 23 2017, edited Feb 20 2024

A045778(n, m=n) = if(1==n, 1, sumdiv(n,d,if((d>1)&&(d<=m),A045778(n/d,d-1)))); \\ Antti Karttunen, Feb 20 2024

from sympy.core.cache import cacheit
from sympy import divisors, isprime
@cacheit
def b(n, k): return (0 if n>k else 1) + (0 if isprime(n) else sum(0 if d>k else b(n//d, d - 1) for d in divisors(n)[1:-1]))
def a(n): return b(n, n)
print([a(n) for n in range(1, 121)]) # Indranil Ghosh, Aug 19 2017, after Maple code

Dirichlet g.f.: Product_{n>=2} (1 + 1/n^s).
Let p and q be two distinct prime numbers and k a natural number. Then a(p^k) = A000009(k) and a(p^k*q) = A036469(k). - Alexander Adam, Dec 28 2012
Let p_i with 1<=i<=k k distinct prime numbers. Then a(Product_{i=1..k} p_i) = A000110(k). - Alexander Adam, Dec 28 2012

Edited by Franklin T. Adams-Watters, Jun 04 2009

A114592 Sum_{n>=1} a(n)/n^s = Product_{k>=2} (1 - 1/k^s).

Original entry on oeis.org

1, -1, -1, -1, -1, 0, -1, 0, -1, 0, -1, 1, -1, 0, 0, 0, -1, 1, -1, 1, 0, 0, -1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, 0, 0, 1, -1, 0, 0, 1, -1, 1, -1, 1, 1, 0, -1, 1, -1, 1, 0, 1, -1, 1, 0, 1, 0, 0, -1, 1, -1, 0, 1, 0, 0, 1, -1, 1, 0, 1, -1, 1, -1, 0, 1, 1, 0, 1

Offset: 1

Leroy Quet, Dec 11 2005

sign

For n >= 2, Sum_{k|n} A001055(n/k) * a(k) = 0. A114591(n) = Sum_{k|n} a(k).

First entry greater than 1 in absolute value is a(360) = -2. - Gus Wiseman, Sep 15 2018

			24 can be factored into distinct integers (each >= 2) as 24; as 4*6, 3*8 and 2*12; and as 2*3*4. (A045778(24) = 5).
So a(24) = (-1)^1 + 3*(-1)^2 + (-1)^3 = 1, where the 1 exponent is due to the 1 factor of the 24 = 24 factorization and the 2 exponent is due to the 3 cases of 2 factors each of the 24 = 4*6 = 3*8 = 2*12 factorizations and the 3 exponent is due to the 24 = 2*3*4 factorization.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A001055, A045778, A114591.

Cf. A001222, A162247, A190938, A259936, A281116, A303386, A316441, A319237, A319238.

strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
Table[Sum[(-1)^Length[f],{f,strfacs[n]}],{n,100}] (* Gus Wiseman, Sep 15 2018 *)

A114592aux(n, k) = if(1==n, 1, sumdiv(n, d, if(d > 1 && d <= k && d < n, (-1)*A114592aux(n/d, d-1))) - (n<=k)); \\ After code in A045778.
A114592(n) = A114592aux(n,n); \\ Antti Karttunen, Jul 23 2017

a(1) = 1; for n>= 2, a(n) = sum, over ways to factor n into any number of distinct integers >= 2, of (-1)^(number of integers in a factorization). (See example.)

More terms from Antti Karttunen, Jul 23 2017

A319238 Positions of zeros in A114592, the list of coefficients in the expansion of Product_{n > 1} (1 - 1/n^s).

Original entry on oeis.org

6, 8, 10, 14, 15, 16, 21, 22, 26, 27, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 64, 65, 69, 74, 77, 81, 82, 85, 86, 87, 91, 93, 94, 95, 96, 106, 111, 115, 118, 119, 120, 122, 123, 125, 129, 133, 134, 141, 142, 143, 144, 145, 146, 155, 158, 159, 160, 161, 166

Offset: 1

Gus Wiseman, Sep 15 2018

nonn

From Tian Vlasic, Jan 01 2022: (Start)
Numbers that have an equal number of even- and odd-length unordered factorizations into distinct factors.
For prime p, by the pentagonal number theorem, p^k is a term if and only if k is in A090864.
For primes p and q, p*q^k is a term if and only if k = A000326(m)+N with 0 <= N < m. (End)

			16 = 2*8 = 4*4 = 2*2*4 = 2*2*2*2 has an equal number of even-length factorizations and odd-length factorizations into distinct factors (1). - _Tian Vlasic_, Dec 31 2021

Leonhard Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005.
Leonhard Euler, De mirabilibus proprietatibus numerorum pentagonalium, par. 2.
Eric Weisstein's World of Mathematics, Pentagonal Number Theorem
Wikipedia, Pentagonal number theorem

Complement of A319237.

Cf. A001055, A045778, A114592, A162247, A190938, A281116, A281118, A303386, A316441, A319240.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Join@@Position[Table[Sum[(-1)^Length[f],{f,Select[facs[n],UnsameQ@@#&]}],{n,100}],0]

A339717 Dirichlet g.f.: Product_{k>=2} 1 / (1 + k^(-s))^2.

Original entry on oeis.org

1, -2, -2, 1, -2, 2, -2, -2, 1, 2, -2, 0, -2, 2, 2, 4, -2, 0, -2, 0, 2, 2, -2, 4, 1, 2, -2, 0, -2, 2, -2, -4, 2, 2, 2, 2, -2, 2, 2, 4, -2, 2, -2, 0, 0, 2, -2, -4, 1, 0, 2, 0, -2, 4, 2, 4, 2, 2, -2, 0, -2, 2, 0, 5, 2, 2, -2, 0, 2, 2, -2, -4, -2, 2, 0, 0, 2, 2, -2, -4

Offset: 1

Ilya Gutkovskiy, Dec 14 2020

sign

Cf. A022597, A316441, A328706, A339718, A339719, A339720, A339721, A339722.

a(1) = 1; a(n) = -Sum_{d|n, d < n} A328706(n/d) * a(d).
a(n) = Sum_{d|n} A316441(n/d) * A316441(d).
a(p^k) = A022597(k) for prime p.

A339718 Dirichlet g.f.: Product_{k>=2} 1 / (1 + k^(-s))^3.

Original entry on oeis.org

1, -3, -3, 3, -3, 6, -3, -4, 3, 6, -3, -3, -3, 6, 6, 9, -3, -3, -3, -3, 6, 6, -3, 9, 3, 6, -4, -3, -3, -3, -3, -12, 6, 6, 6, 3, -3, 6, 6, 9, -3, -3, -3, -3, -3, 6, -3, -18, 3, -3, 6, -3, -3, 9, 6, 9, 6, 6, -3, -3, -3, 6, -3, 15, 6, -3, -3, -3, 6, -3, -3, -15, -3, 6, -3, -3, 6, -3, -3, -18

Offset: 1

Ilya Gutkovskiy, Dec 14 2020

sign

Cf. A022598, A316441, A339335, A339717, A339719, A339720, A339721, A339722.

a(1) = 1; a(n) = -Sum_{d|n, d < n} A339335(n/d) * a(d).

a(p^k) = A022598(k) for prime p.

A319237 Positions of nonzero terms in A114592, the list of coefficients in the expansion of Product_{n > 1} (1 - 1/n^s).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 12, 13, 17, 18, 19, 20, 23, 24, 25, 28, 29, 30, 31, 32, 36, 37, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 60, 61, 63, 66, 67, 68, 70, 71, 72, 73, 75, 76, 78, 79, 80, 83, 84, 88, 89, 90, 92, 97, 98, 99, 100, 101, 102

Offset: 1

Gus Wiseman, Sep 15 2018

nonn

Complement of A319238.

Cf. A001055, A045778, A114592, A162247, A190938, A281116, A281118, A303386, A316441, A319239.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Join@@Position[Table[Sum[(-1)^Length[f],{f,Select[facs[n],UnsameQ@@#&]}],{n,100}],_Integer?(Abs[#]>0&)]

A339719 Dirichlet g.f.: Product_{k>=2} 1 / (1 + k^(-s))^4.

Original entry on oeis.org

1, -4, -4, 6, -4, 12, -4, -8, 6, 12, -4, -12, -4, 12, 12, 17, -4, -12, -4, -12, 12, 12, -4, 20, 6, 12, -8, -12, -4, -20, -4, -28, 12, 12, 12, 10, -4, 12, 12, 20, -4, -20, -4, -12, -12, 12, -4, -48, 6, -12, 12, -12, -4, 20, 12, 20, 12, 12, -4, 4, -4, 12, -12, 38, 12, -20, -4, -12, 12, -20

Offset: 1

Ilya Gutkovskiy, Dec 14 2020

sign

Cf. A022599, A316441, A339336, A339717, A339718, A339720, A339721, A339722.

a(1) = 1; a(n) = -Sum_{d|n, d < n} A339336(n/d) * a(d).

a(p^k) = A022599(k) for prime p.

A339720 Dirichlet g.f.: Product_{k>=2} 1 / (1 + k^(-s))^5.

Original entry on oeis.org

1, -5, -5, 10, -5, 20, -5, -15, 10, 20, -5, -30, -5, 20, 20, 30, -5, -30, -5, -30, 20, 20, -5, 45, 10, 20, -15, -30, -5, -55, -5, -56, 20, 20, 20, 35, -5, 20, 20, 45, -5, -55, -5, -30, -30, 20, -5, -105, 10, -30, 20, -30, -5, 45, 20, 45, 20, 20, -5, 45, -5, 20, -30, 85, 20

Offset: 1

Ilya Gutkovskiy, Dec 14 2020

sign

Cf. A022600, A316441, A339337, A339717, A339718, A339719, A339721, A339722.

a(1) = 1; a(n) = -Sum_{d|n, d < n} A339337(n/d) * a(d).

a(p^k) = A022600(k) for prime p.

cp's OEIS Frontend

A319240 Positions of zeros in A316441, the list of coefficients in the expansion of Product_{n > 1} 1/(1 + 1/n^s).

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A319239 Positions of nonzero terms in A316441, the list of coefficients in the expansion of Product_{n > 1} 1/(1 + 1/n^s).

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A045778 Number of factorizations of n into distinct factors greater than 1.

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A114592 Sum_{n>=1} a(n)/n^s = Product_{k>=2} (1 - 1/k^s).

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A319238 Positions of zeros in A114592, the list of coefficients in the expansion of Product_{n > 1} (1 - 1/n^s).

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A339717 Dirichlet g.f.: Product_{k>=2} 1 / (1 + k^(-s))^2.

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A339718 Dirichlet g.f.: Product_{k>=2} 1 / (1 + k^(-s))^3.

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A319237 Positions of nonzero terms in A114592, the list of coefficients in the expansion of Product_{n > 1} (1 - 1/n^s).

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A339719 Dirichlet g.f.: Product_{k>=2} 1 / (1 + k^(-s))^4.

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