A045778 -id:A045778 - cp's OEIS frontend

A096111 If n = 2^k - 1, then a(n) = k+1, otherwise a(n) = (A000523(n)+1)*a(A053645(n)).

1, 2, 2, 3, 3, 6, 6, 4, 4, 8, 8, 12, 12, 24, 24, 5, 5, 10, 10, 15, 15, 30, 30, 20, 20, 40, 40, 60, 60, 120, 120, 6, 6, 12, 12, 18, 18, 36, 36, 24, 24, 48, 48, 72, 72, 144, 144, 30, 30, 60, 60, 90, 90, 180, 180, 120, 120, 240, 240, 360, 360, 720, 720, 7, 7, 14, 14, 21, 21

Offset: 0

Amarnath Murthy, Jun 29 2004

nonn

Each n > 1 occurs 2*A045778(n) times in the sequence.
f(n+2^k) = (k+1)*f(n) if 2^k > n+1. - Robert Israel, Apr 25 2016
If the binary indices of n (row n of A048793) are the positions 1's in its reversed binary expansion, then a(n) is the product of all binary indices of n + 1. The number of binary indices of n is A000120(n), their sum is A029931(n), and their average is A326699(n)/A326700(n). - Gus Wiseman, Jul 27 2019

Peter Kagey, Table of n, a(n) for n = 0..10000

Permutation of A096115, i.e. a(n) = A096115(A122198(n+1)) [Note the different starting offsets]. Bisection: A121663. Cf. A096113, A052330.
Cf. A029931.
Cf. A000120, A034797, A048793, A070939, A291166, A326031, A326672, A326673.

f:= proc(n) local L;
    L:= convert(2*n+2,base,2);
    convert(subs(0=NULL,zip(`*`,L, [$0..nops(L)-1])),`*`);
end proc:
map(f, [$0..100]); # Robert Israel, Apr 25 2016

CoefficientList[(Product[1 + k x^(2^(k - 1)), {k, 7}] - 1)/x, x] (* Michael De Vlieger, Apr 08 2016 *)
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];Table[Times@@bpe[n+1],{n,0,100}] (* Gus Wiseman, Jul 26 2019 *)

N=166; q='q+O('q^N);
gf= (prod(n=1,1+ceil(log(N)/log(2)), 1+n*q^(2^(n-1)) ) - 1) / q;
Vec(gf)
/* Joerg Arndt, Oct 06 2012 */

(define (A096111 n) (cond ((pow2? (+ n 1)) (+ 2 (A000523 n))) (else (* (+ 1 (A000523 n)) (A096111 (A053645 n))))))
(define (pow2? n) (and (> n 0) (zero? (A004198bi n (- n 1)))))

G.f.: ( prod(k>=1, 1+k*x^(2^(k-1)) )- 1 ) / x. - Vladeta Jovovic, Nov 08 2005

a(n) is the product of the exponents in the binary expansion of 2*n + 2. - Peter Kagey, Apr 24 2016

Edited, extended and Scheme code added by Antti Karttunen, Aug 25 2006

A050326 Number of factorizations of n into distinct squarefree numbers > 1.

Original entry on oeis.org

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

Offset: 1

Christian G. Bower, Oct 15 1999

nonn

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24=2^3*3 and 375=3*5^3 both have prime signature (3,1).
a(A212164(n)) = 0; a(A212166(n)) = 1; a(A006881(n)) = 2; a(A190107(n)) = 3; a(A085987(n)) = 4; a(A225228(n)) = 5; a(A179670(n)) = 7; a(A162143(n)) = 8; a(A190108(n)) = 11; a(A212167(n)) > 0; a(A212168(n)) > 1. - Reinhard Zumkeller, May 03 2013
The comment that a(A212164(n)) = 0 is incorrect. For example, 3600 belongs to A212164 but a(3600) = 1. The positions of zeros in this sequence are A293243. - Gus Wiseman, Oct 10 2017

			The a(30) = 5 factorizations are: 2*3*5, 2*15, 3*10, 5*6, 30. The a(180) = 5 factorizations are: 2*3*5*6, 2*3*30, 2*6*15, 3*6*10, 6*30. - _Gus Wiseman_, Oct 10 2017

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

Cf. A001055, A005117, A045778, A046523, A050320, A050327, a(p^k)=0 (p>1), a(A002110)=A000110, a(n!)=A103775(n), A206778, A293243.

import Data.List (subsequences, genericIndex)
a050326 n = genericIndex a050326_list (n-1)
a050326_list = 1 : f 2 where
   f x = (if x /= s then a050326 s
                    else length $ filter (== x) $ map product $
                         subsequences $ tail $ a206778_row x) : f (x + 1)
         where s = a046523 x
-- Reinhard Zumkeller, May 03 2013

N:= 1000: # to get a(1)..a(N)
A:= Vector(N):
A[1]:= 1:
for n from 2 to N do
  if numtheory:-issqrfree(n) then
     S:= [$1..N/n]; T:= n*S; A[T]:= A[T]+A[S]
    fi;
od:
convert(A,list); # Robert Israel, Oct 10 2017

sqfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfacs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[sqfacs[n]],{n,100}] (* Gus Wiseman, Oct 10 2017 *)

Dirichlet g.f.: prod{n is squarefree and > 1}(1+1/n^s).

a(n) = A050327(A101296(n)). - R. J. Mathar, May 26 2017

A339890 Number of odd-length factorizations of n into factors > 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 2, 1, 4, 1, 1, 1, 4, 1, 1, 1, 3, 1, 2, 1, 2, 2, 1, 1, 6, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 5, 1, 1, 2, 5, 1, 2, 1, 2, 1, 2, 1, 8, 1, 1, 2, 2, 1, 2, 1, 6, 2, 1, 1, 5, 1, 1, 1

Offset: 1

Gus Wiseman, Dec 28 2020

nonn

			The a(n) factorizations for n = 24, 48, 60, 72, 96, 120:
  24      48          60       72          96          120
  2*2*6   2*3*8       2*5*6    2*4*9       2*6*8       3*5*8
  2*3*4   2*4*6       3*4*5    2*6*6       3*4*8       4*5*6
          3*4*4       2*2*15   3*3*8       4*4*6       2*2*30
          2*2*12      2*3*10   3*4*6       2*2*24      2*3*20
          2*2*2*2*3            2*2*18      2*3*16      2*4*15
                               2*3*12      2*4*12      2*5*12
                               2*2*2*3*3   2*2*2*2*6   2*6*10
                                           2*2*2*3*4   3*4*10
                                                       2*2*2*3*5

Alois P. Heinz, Table of n, a(n) for n = 1..20000

The case of set partitions (or n squarefree) is A024429.
The case of partitions (or prime powers) is A027193.
The ordered version is A174726 (even: A174725).
The remaining (even-length) factorizations are counted by A339846.
A000009 counts partitions into odd parts, ranked by A066208.
A001055 counts factorizations, with strict case A045778.
A027193 counts partitions of odd length, ranked by A026424.
A058695 counts partitions of odd numbers, ranked by A300063.
A160786 counts odd-length partitions of odd numbers, ranked by A300272.
A316439 counts factorizations by product and length.
A340101 counts factorizations into odd factors.
A340102 counts odd-length factorizations into odd factors.
Cf. A000700, A002033, A027187, A028260, A074206, A078408, A236914, A320732.

g:= proc(n, k, t) option remember; `if`(n>k, 0, t)+
      `if`(isprime(n), 0, add(`if`(d>k, 0, g(n/d, d, 1-t)),
          d=numtheory[divisors](n) minus {1, n}))
    end:
a:= n-> `if`(n<2, 0, g(n$2, 1)):
seq(a(n), n=1..100);  # Alois P. Heinz, Dec 30 2020

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],OddQ@Length[#]&]],{n,100}]

a(n) + A339846(n) = A001055(n).

A339846 Number of even-length factorizations of n into factors > 1.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 3, 0, 2, 0, 2, 1, 1, 0, 4, 1, 1, 1, 2, 0, 3, 0, 3, 1, 1, 1, 5, 0, 1, 1, 4, 0, 3, 0, 2, 2, 1, 0, 6, 1, 2, 1, 2, 0, 4, 1, 4, 1, 1, 0, 6, 0, 1, 2, 6, 1, 3, 0, 2, 1, 3, 0, 8, 0, 1, 2, 2, 1, 3, 0, 6, 3, 1, 0, 6, 1, 1, 1, 4, 0, 6, 1, 2, 1, 1, 1, 10, 0, 2, 2, 5, 0, 3, 0, 4, 3

Offset: 1

Gus Wiseman, Dec 28 2020

nonn

			The a(n) factorizations for n = 12, 16, 24, 36, 48, 72, 96, 120:
  2*6  2*8      3*8      4*9      6*8      8*9      2*48         2*60
  3*4  4*4      4*6      6*6      2*24     2*36     3*32         3*40
       2*2*2*2  2*12     2*18     3*16     3*24     4*24         4*30
                2*2*2*3  3*12     4*12     4*18     6*16         5*24
                         2*2*3*3  2*2*2*6  6*12     8*12         6*20
                                  2*2*3*4  2*2*2*9  2*2*3*8      8*15
                                           2*2*3*6  2*2*4*6      10*12
                                           2*3*3*4  2*3*4*4      2*2*5*6
                                                    2*2*2*12     2*3*4*5
                                                    2*2*2*2*2*3  2*2*2*15
                                                                 2*2*3*10

The case of set partitions (or n squarefree) is A024430.
The case of partitions (or prime powers) is A027187.
The ordered version is A174725, odd: A174726.
The odd-length factorizations are counted by A339890.
A001055 counts factorizations, with strict case A045778.
A001358 lists semiprimes, with squarefree case A006881.
A027187 counts partitions of even length, ranked by A028260.
A058696 counts partitions of even numbers, ranked by A300061.
A316439 counts factorizations by product and length.
A340102 counts odd-length factorizations into odd factors.
Cf. A002033, A007716, A027193, A050320, A058695, A074206, A236913, A320655, A320656, A320732.

g:= proc(n, k, t) option remember; `if`(n>k, 0, t)+
      `if`(isprime(n), 0, add(`if`(d>k, 0, g(n/d, d, 1-t)),
          d=numtheory[divisors](n) minus {1, n}))
    end:
a:= n-> `if`(n=1, 1, g(n$2, 0)):
seq(a(n), n=1..100);  # Alois P. Heinz, Dec 30 2020

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],EvenQ@Length[#]&]],{n,100}]

A339846(n, m=n, e=1) = if(1==n, e, sumdiv(n, d, if((d>1)&&(d<=m), A339846(n/d, d, 1-e)))); \\ Antti Karttunen, Oct 22 2023

a(n) + A339890(n) = A001055(n).

Data section extended up to a(105) by Antti Karttunen, Oct 22 2023

A181796 a(n) = number of divisors of n whose canonical prime factorizations contain no repeated positive exponents (cf. A130091).

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 5, 2, 3, 3, 5, 2, 5, 2, 5, 3, 3, 2, 7, 3, 3, 4, 5, 2, 4, 2, 6, 3, 3, 3, 7, 2, 3, 3, 7, 2, 4, 2, 5, 5, 3, 2, 9, 3, 5, 3, 5, 2, 7, 3, 7, 3, 3, 2, 7, 2, 3, 5, 7, 3, 4, 2, 5, 3, 4, 2, 10, 2, 3, 5, 5, 3, 4, 2, 9, 5, 3, 2, 7, 3, 3, 3, 7, 2, 7, 3, 5, 3, 3, 3, 11, 2, 5, 5, 7, 2, 4, 2, 7, 4

Offset: 1

Matthew Vandermast, Nov 22 2010

nonn

The canonical factorization of n into prime powers can be written as Product p(i)^e(i), for example. A host of equivalent notations can also be used (for another example, see Weisstein link). a(n) depends only on prime signature of n (cf. A025487).
a(n) >= A085082(n). (A085082(n) equals the number of members of A025487 that divide A046523(n), and each member of A025487 is divisible by at least one member of A130091 that divides no smaller member of A025487.) a(n) > A085082(n) iff n has in its canonical prime factorization at least two exponents greater than 1.
a(n) = number of such divisors of n that in their prime factorization all exponents are unique. - Antti Karttunen, May 27 2017
First differs from A335549 at a(90) = 7, A335549(90) = 8. First differs from A335516 at a(180) = 9, A335516(180) = 10. - Gus Wiseman, Jun 28 2020

			12 has a total of six divisors (1, 2, 3, 4, 6 and 12). Of those divisors, the number 1 has no prime factors, hence, no positive exponents at all (and no repeated positive exponents) in its canonical prime factorization. The lists of positive exponents for 2, 3, 4, 6 and 12 are (1), (1), (2), (1,1) and (2,1) respectively (cf. A124010). Of all six divisors, only the number 6 (2^1*3^1) has at least one positive exponent repeated (namely, 1). The other five do not; hence, a(12) = 5.
For n = 90 = 2 * 3^2 * 5, the divisors that satisfy the condition are: 1, 2, 3, 3^2, 5, 2 * 3^2, 3^2 * 5, altogether 7, (but for example 90 itself is not included), thus a(90) = 7.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Factorization.
Gus Wiseman, Sequences counting and encoding certain classes of multisets

Diverges from A088873 at n=24 and from A085082 at n=36. a(36) = 7, while A085082(36) = 6.
Partitions with distinct multiplicities are A098859.
Sorted prime signature is A118914.
Unsorted prime signature is A124010.
a(n) is the number of divisors of n in A130091.
Factorizations with distinct multiplicities are A255231.
The largest of the counted divisors is A327498.
Factorizations using the counted divisors are A327523.
Cf. A000005, A007916, A045778, A056239, A212168, A327498, A335519, A335549.

Table[DivisorSum[n, 1 &, Length@ Union@ # == Length@ # &@ FactorInteger[#][[All, -1]] &], {n, 105}] (* Michael De Vlieger, May 28 2017 *)

no_repeated_exponents(n) = { my(es = factor(n)[, 2]); if(length(Set(es)) == length(es),1,0); }
A181796(n) = sumdiv(n,d,no_repeated_exponents(d)); \\ Antti Karttunen, May 27 2017

from sympy import factorint, divisors
def ok(n):
    f=factorint(n)
    ex=[f[i] for i in f]
    for i in ex:
        if ex.count(i)>1: return 0
    return 1
def a(n): return sum([1 for i in divisors(n) if ok(i)]) # Indranil Ghosh, May 27 2017

a(A000079(n)) = a(A002110(n)) = n+1.
a(A006939(n)) = A000110(n+1).
a(A181555(n)) = A002720(n).

A320656 Number of factorizations of n into squarefree semiprimes. Number of multiset partitions of the multiset of prime factors of n, into strict pairs.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0

Offset: 1

Gus Wiseman, Oct 18 2018

nonn

			The a(4620) = 6 factorizations into squarefree semiprimes:
  4620 = (6*10*77)
  4620 = (6*14*55)
  4620 = (6*22*35)
  4620 = (10*14*33)
  4620 = (10*21*22)
  4620 = (14*15*22)
The a(4620) = 6 multiset partitions into strict pairs:
  {{1,2},{1,3},{4,5}}
  {{1,2},{1,4},{3,5}}
  {{1,2},{1,5},{3,4}}
  {{1,3},{1,4},{2,5}}
  {{1,3},{2,4},{1,5}}
  {{1,4},{2,3},{1,5}}
The a(69300) = 10 factorizations into squarefree semiprimes:
  69300 = (6*6*35*55)
  69300 = (6*10*15*77)
  69300 = (6*10*21*55)
  69300 = (6*10*33*35)
  69300 = (6*14*15*55)
  69300 = (6*15*22*35)
  69300 = (10*10*21*33)
  69300 = (10*14*15*33)
  69300 = (10*15*21*22)
  69300 = (14*15*15*22)
The a(69300) = 10 multiset partitions into strict pairs:
  {{1,2},{1,2},{3,4},{3,5}}
  {{1,2},{1,3},{2,3},{4,5}}
  {{1,2},{1,3},{2,4},{3,5}}
  {{1,2},{1,3},{2,5},{3,4}}
  {{1,2},{1,4},{2,3},{3,5}}
  {{1,2},{2,3},{1,5},{3,4}}
  {{1,3},{1,3},{2,4},{2,5}}
  {{1,3},{1,4},{2,3},{2,5}}
  {{1,3},{2,3},{2,4},{1,5}}
  {{1,4},{2,3},{2,3},{1,5}}.
The a(210) = 3 factorizations into squarefree semiprimes: 210 = (6*35) = (10*21) = (14*15). - _Antti Karttunen_, Nov 02 2022

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Cf. A001055, A002110, A006881, A079275, A007716, A007717, A045778, A123023, A318871, A318953, A320462, A320655, A320658, A320659.

bepfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[bepfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ[#]&&PrimeOmega[#]==2&]}]];
Table[Length[bepfacs[n]],{n,100}]

A320656(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&issquarefree(d)&&2==bigomega(d), s += A320656(n/d, d))); (s)); \\ Antti Karttunen, Nov 02 2022

a(A002110(n)) = A123023(n). - Antti Karttunen, Nov 02 2022

Data section extended up to a(120) and the secondary offset added by Antti Karttunen, Nov 02 2022

A303386 Number of aperiodic factorizations of n > 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 7, 1, 2, 2, 4, 1, 5, 1, 6, 2, 2, 2, 7, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 12, 1, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 7, 2, 5, 1, 4, 2, 5, 1, 16, 1, 2, 4, 4, 2, 5, 1, 12, 3, 2, 1, 11, 2, 2, 2, 7, 1, 11, 2, 4, 2, 2, 2, 19, 1, 4, 4, 7, 1, 5, 1, 7, 5

Offset: 2

Gus Wiseman, Apr 23 2018

nonn

An aperiodic factorization of n is a finite multiset of positive integers greater than 1 whose product is n and whose multiplicities are relatively prime.

			The a(36) = 7 aperiodic factorizations are (2*2*9), (2*3*6), (2*18), (3*3*4), (3*12), (4*9), and (36). Missing from this list are (2*2*3*3) and (6*6).

Antti Karttunen, Table of n, a(n) for n = 2..65537

Cf. A000740, a(2^n) = A000837(n), A001055, A007716, A007916, A045778, A052409, A052410, A100953, A162247, A275024, A275870, A281113, A281116, A301700, A302242, A303431.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],GCD@@Length/@Split[#]===1&]],{n,2,100}]

A001055(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A001055(n/d, d))); (s));
A052409(n) = { my(k=ispower(n)); if(k, k, n>1); }; \\ From A052409
A303386(n) = if(1==n,n,my(r); sumdiv(A052409(n),d, ispower(n,d,&r); moebius(d)*A001055(r))); \\ Antti Karttunen, Sep 25 2018

a(n) = Sum_{d|A052409(n)} mu(d) * A001055(n^(1/d)), where mu = A008683.

More terms from Antti Karttunen, Sep 25 2018

A296119 Number of ways to choose a strict factorization of each factor in a factorization of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 7, 1, 3, 3, 7, 1, 7, 1, 7, 3, 3, 1, 16, 2, 3, 4, 7, 1, 12, 1, 12, 3, 3, 3, 21, 1, 3, 3, 16, 1, 12, 1, 7, 7, 3, 1, 33, 2, 7, 3, 7, 1, 16, 3, 16, 3, 3, 1, 34, 1, 3, 7, 23, 3, 12, 1, 7, 3, 12, 1, 50, 1, 3, 7, 7, 3, 12, 1, 33, 7, 3

Offset: 1

Gus Wiseman, Dec 05 2017

nonn

			The a(24) = 16 twice-factorizations:
(2)*(2)*(2)*(3),
(2)*(2)*(2*3), (2)*(2)*(6), (2)*(3)*(4),
(2)*(2*6), (2)*(3*4), (2)*(12), (3)*(2*4), (3)*(8), (4)*(2*3), (4)*(6),
(2*3*4), (2*12), (3*8), (4*6), (24).

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A000009, A005117, A045778, A270995, A281113, A294788, A296118, A296120, A296121.

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

A045778(n, m=n) = ((n<=m) + sumdiv(n, d, if((d>1)&&(d<=m)&&(dA045778(n/d, d-1))));
A296119(n, m=n) = if(1==n, 1, sumdiv(n, d, if((d>1)&&(d<=m), A045778(d)*A296119(n/d, d)))); \\ Antti Karttunen, Oct 08 2018

Dirichlet g.f.: 1/Product_{n > 1}(1 - A045778(n)/n^s).

A381454 Number of multisets that can be obtained by choosing a strict integer partition of each prime index of n and taking the multiset union.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 4, 2, 2, 1, 5, 1, 6, 2, 2, 3, 8, 1, 3, 4, 1, 2, 10, 2, 12, 1, 3, 5, 4, 1, 15, 6, 4, 2, 18, 2, 22, 3, 2, 8, 27, 1, 3, 3, 5, 4, 32, 1, 6, 2, 6, 10, 38, 2, 46, 12, 2, 1, 8, 3, 54, 5, 8, 4, 64, 1, 76, 15, 3, 6, 6, 4, 89, 2, 1

Offset: 1

Gus Wiseman, Mar 08 2025

nonn

First differs from A357982 at a(25) = 3, A357982(25) = 4.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
A multiset partition can be regarded as an arrow in the ranked poset of integer partitions. For example, we have {{1},{1,2},{1,3},{1,2,3}}: {1,1,1,1,2,2,3,3} -> {1,3,4,6}, or (33221111) -> (6431) (depending on notation).
Set multipartitions are generally not transitive. For example, we have arrows: {{1},{1,2}}: {1,1,2} -> {1,3} and {{1,3}}: {1,3} -> {4}, but there is no set multipartition {1,1,2} -> {4}.

			The a(25) = 3 multisets are: {3,3}, {1,2,3}, {1,1,2,2}.

For constant instead of strict partitions see A381453, A355733, A381455, A000688.
Positions of 1 are A003586.
The upper version is A381078, before sums A050320.
For distinct block-sums see A381634, A381633, A381806.
Multiset partitions of prime indices:
- For multiset partitions (A001055) see A317141 (upper), A300383 (lower).
- For strict multiset partitions (A045778) see A381452.
- For set systems (A050326, zeros A293243) see A381441 (upper).
- For sets of constant multisets (A050361) see A381715.
- For strict multiset partitions with distinct sums (A321469) see A381637.
- For sets of constant multisets with distinct sums (A381635, zeros A381636) see A381716.
More on set systems: A050342, A116539, A296120, A318361.
More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
More on set multipartitions with distinct sums: A279785, A381717, A381718.
A000041 counts integer partitions, strict A000009.
A000040 lists the primes.
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
A265947 counts refinement-ordered pairs of integer partitions.
A358914 counts twice-partitions into distinct strict partitions.
Cf. A000720, A001222, A002846, A005117, A066328, A213242, A213385, A213427, A293511, A299200, A299201, A299202, A300385, A317142.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Union[Sort/@Join@@@Tuples[Select[IntegerPartitions[#],UnsameQ@@#&]&/@prix[n]]]],{n,100}]

a(A002110(n)) = A381808(n).

A316439 Irregular triangle where T(n,k) is the number of factorizations of n into k factors > 1, with k ranging from 1 to Omega(n).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 03 2018

			The factorizations of 24 are (2*2*2*3), (2*2*6), (2*3*4), (2*12), (3*8), (4*6), (24) so the 24th row is {1, 3, 2, 1}.
Triangle begins:
  {}
  1
  1
  1  1
  1
  1  1
  1
  1  1  1
  1  1
  1  1
  1
  1  2  1
  1
  1  1
  1  1
  1  2  1  1
  1
  1  2  1
  1
  1  2  1
  1  1
  1  1
  1
  1  3  2  1
  1  1
  1  1
  1  1  1
  1  2  1
  1
  1  3  1

Alois P. Heinz, Rows n = 1..20000, flattened

Cf. A001222 (row lengths), A001055 (row sums), A001970, A007716, A045778, A162247, A259936, A281116, A303386.

g:= proc(n, k) option remember; `if`(n>k, 0, x)+
      `if`(isprime(n), 0, expand(x*add(`if`(d>k, 0,
      g(n/d, d)), d=numtheory[divisors](n) minus {1, n})))
    end:
T:= n-> `if`(n=1, [][], (p-> seq(coeff(p, x, i)
        , i=1..degree(p)))(g(n$2))):
seq(T(n), n=1..50);  # Alois P. Heinz, Aug 11 2019

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],Length[#]==k&]],{n,100},{k,PrimeOmega[n]}]

cp's OEIS Frontend

A096111 If n = 2^k - 1, then a(n) = k+1, otherwise a(n) = (A000523(n)+1)*a(A053645(n)).

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A050326 Number of factorizations of n into distinct squarefree numbers > 1.

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A339890 Number of odd-length factorizations of n into factors > 1.

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A339846 Number of even-length factorizations of n into factors > 1.

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A181796 a(n) = number of divisors of n whose canonical prime factorizations contain no repeated positive exponents (cf. A130091).

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A320656 Number of factorizations of n into squarefree semiprimes. Number of multiset partitions of the multiset of prime factors of n, into strict pairs.

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A303386 Number of aperiodic factorizations of n > 1.

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