A281113 -id:A281113 - cp's OEIS frontend

A302593 Numbers whose prime indices are powers of a common prime number.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 31, 32, 34, 36, 38, 40, 41, 42, 44, 46, 48, 49, 50, 53, 54, 56, 57, 59, 62, 63, 64, 67, 68, 72, 76, 80, 81, 82, 83, 84, 88, 92, 96, 97, 98, 100, 103, 106, 108, 109, 112

Offset: 1

Gus Wiseman, Apr 10 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

			Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of set systems.
01: {}
02: {{}}
03: {{1}}
04: {{},{}}
05: {{2}}
06: {{},{1}}
07: {{1,1}}
08: {{},{},{}}
09: {{1},{1}}
10: {{},{2}}
11: {{3}}
12: {{},{},{1}}
14: {{},{1,1}}
16: {{},{},{},{}}
17: {{4}}
18: {{},{1},{1}}
19: {{1,1,1}}
20: {{},{},{2}}
21: {{1},{1,1}}
22: {{},{3}}
23: {{2,2}}
24: {{},{},{},{1}}
25: {{2},{2}}
27: {{1},{1},{1}}
28: {{},{},{1,1}}
31: {{5}}
32: {{},{},{},{},{}}
34: {{},{4}}
36: {{},{},{1},{1}}
38: {{},{1,1,1}}
40: {{},{},{},{2}}

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000961, A001222, A003963, A005117, A007716, A056239, A275024, A047968, A281113, A295924, A301760, A302242, A302243.

filter:= proc(n) local F,q;
  uses numtheory;
  F:= map(pi, factorset(n));
  nops(`union`(op(map(factorset,F)))) <= 1
end proc:
select(filter, [$1..200]); # Robert Israel, Oct 22 2020

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],SameQ@@Join@@primeMS/@primeMS[#]&]

A277130 Number of planar branching factorizations of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 6, 2, 3, 1, 14, 1, 3, 3, 24, 1, 14, 1, 14, 3, 3, 1, 78, 2, 3, 6, 14, 1, 25, 1, 112, 3, 3, 3, 110, 1, 3, 3, 78, 1, 25, 1, 14, 14, 3, 1, 464, 2, 14, 3, 14, 1, 78, 3, 78, 3, 3, 1, 206, 1, 3, 14, 568, 3, 25, 1, 14, 3, 25, 1, 850, 1, 3, 14, 14

Offset: 1

Michel Marcus, Oct 01 2016

nonn

A planar branching factorization of n is either the number n itself or a sequence of at least two planar branching factorizations, one of each factor in an ordered factorization of n. - Gus Wiseman, Sep 11 2018

			From _Gus Wiseman_, Sep 11 2018: (Start)
The a(12) = 14 planar branching factorizations:
  12,
  (2*6), (3*4), (4*3), (6*2), (2*2*3), (2*3*2), (3*2*2),
  (2*(2*3)), (2*(3*2)), (3*(2*2)), ((2*2)*3), ((2*3)*2), ((3*2)*2).
(End)

Daniel Mondot, Table of n, a(n) for n = 1..9999
A. Knopfmacher, M. E. Mays, A survey of factorization counting functions, International Journal of Number Theory, 1(4):563-581,(2005). See page 15.

Cf. A277120.
Cf. A000108, A001055, A074206, A118376, A281113, A319122, A319123.
Cf. A277130, A281118, A281119, A292504, A292505, A295279, A295281, A319136, A319137, A319138.

#include 
#include 
#include 
#define MAX 10000
/* Number of planar branching factorizations of n. */
#define lu unsigned long
lu nbr[MAX]; /* number of branching */
lu a, b, d, e; /* temporary variables */
lu n; lu m, p; // factors of n
lu x; // square root of n
void main(unsigned argc, char *argv[])
{
  memset(nbr, 0, MAX*sizeof(lu));
  for (b=0, n=1; nDaniel Mondot, May 19 2017 */

ordfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@ordfacs[n/d],{d,Rest[Divisors[n]]}]]
otfs[n_]:=Prepend[Join@@Table[Tuples[otfs/@f],{f,Select[ordfacs[n],Length[#]>1&]}],n];
Table[Length[otfs[n]],{n,20}] (* Gus Wiseman, Sep 11 2018 *)

a(prime^n) = A118376(n). a(product of n distinct primes) = A319122(n). - Gus Wiseman, Sep 11 2018

Terms a(65) and beyond from Daniel Mondot, May 19 2017

A295281 Number of complete strict tree-factorizations of n > 1.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 4, 1, 0, 1, 1, 1, 3, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 0, 1, 4, 1, 1, 1, 4, 1, 6, 1, 1, 1, 1, 1, 4, 1, 1, 0, 1, 1, 9, 1, 1, 1, 1, 1, 9, 1

Offset: 2

Gus Wiseman, Nov 19 2017

nonn

A strict tree-factorization (see A295279 for definition) is complete if its leaves are all prime numbers.
From Andrew Howroyd, Nov 18 2018: (Start)
a(n) depends only on the prime signature of n.
This sequence is very similar but not identical to the number of complete orderless identity tree-factorizations of n. The first difference is at n=900 (square of three primes). Here a(n) = 191 whereas the other sequence would have 197. (End)

			The a(72) = 6 complete strict tree-factorizations are: 2*3*(2*(2*3)), 2*(2*3*(2*3)), 2*(2*(3*(2*3))), 2*(3*(2*(2*3))), 3*(2*(2*(2*3))), (2*3)*(2*(2*3)).

Andrew Howroyd, Table of n, a(n) for n = 2..10000

Cf. A000311, A025475, A085971, A273873, A281113 A281118, A281119, A292505, A295279.

postfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[postfacs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
sftc[n_]:=Prepend[Join@@Function[fac,Tuples[sftc/@fac]]/@Select[postfacs[n],And[Length[#]>1,UnsameQ@@#]&],n];
Table[Length[Select[sftc[n],FreeQ[#,_Integer?(!PrimeQ[#]&)]&]],{n,2,100}]

seq(n)={my(v=vector(n), w=vector(n)); v[1]=1; for(k=2, n, w[k]=v[k]+isprime(k); forstep(j=n\k*k, k, -k, v[j]+=w[k]*v[j/k])); w[2..n]} \\ Andrew Howroyd, Nov 18 2018

a(product of n distinct primes) = A000311(n).

Positions of zeros are proper prime powers A025475. Positions of nonzero entries are A085971.

A295923 Number of twice-factorizations of n where the first factorization is constant, i.e., type (P,R,P).

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 1, 4, 3, 2, 1, 4, 1, 2, 2, 10, 1, 4, 1, 4, 2, 2, 1, 7, 3, 2, 4, 4, 1, 5, 1, 8, 2, 2, 2, 13, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 12, 3, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 29, 2, 5, 1, 4, 2, 5, 1, 16, 1, 2, 4, 4, 2, 5, 1, 12, 10, 2, 1, 11

Offset: 1

Gus Wiseman, Nov 30 2017

nonn

a(n) is also the number of ways to choose a perfect divisor d|n and then a sequence of log_d(n) factorizations of d.

			The a(16) = 10 twice-factorizations are (2*2*2*2), (2*2*4), (2*8), (4*4), (16), (2*2)*(2*2), (2*2)*(4), (4)*(2*2), (4)*(4), (2)*(2)*(2)*(2).

Cf. A000005, A001055, A052409, A052410, A279787, A281113, A284639, A295920, A295924, A295931, A295935.

postfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[postfacs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
a[n_]:=Sum[Length[postfacs[n^(1/g)]]^g,{g,Divisors[GCD@@FactorInteger[n][[All,2]]]}];
Array[a,50]

A296121 Number of twice-factorizations of n with no repeated factorizations.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 5, 2, 3, 1, 8, 1, 3, 3, 10, 1, 8, 1, 8, 3, 3, 1, 20, 2, 3, 5, 8, 1, 12, 1, 20, 3, 3, 3, 25, 1, 3, 3, 20, 1, 12, 1, 8, 8, 3, 1, 47, 2, 8, 3, 8, 1, 20, 3, 20, 3, 3, 1, 38, 1, 3, 8, 40, 3, 12, 1, 8, 3, 12, 1, 68, 1, 3, 8, 8, 3, 12, 1, 47, 10

Offset: 1

Gus Wiseman, Dec 05 2017

nonn

From Robert G. Wilson v, Dec 05 2017: (Start)
a(n) = 1 iff n equals 1 or is a prime;
a(n) = 2 iff n is a prime squared;
a(n) = 3 iff n is a squarefree semiprime;
a(n) = 5 iff n is a prime cube;
a(n) = 8 iff n is of the form p^2*q, etc.
(End)

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

Robert G. Wilson v, Table of n, a(n) for n = 1..1000

Cf. A001055, A045778, A050345, A063834, A089723, A281113, A296118, A296119, A296120, A296122.

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

A301829 Number of ways to choose a nonempty submultiset of a factorization of n into factors greater than one.

Original entry on oeis.org

0, 1, 1, 3, 1, 4, 1, 7, 3, 4, 1, 12, 1, 4, 4, 15, 1, 12, 1, 12, 4, 4, 1, 29, 3, 4, 7, 12, 1, 17, 1, 29, 4, 4, 4, 37, 1, 4, 4, 29, 1, 17, 1, 12, 12, 4, 1, 64, 3, 12, 4, 12, 1, 29, 4, 29, 4, 4, 1, 53, 1, 4, 12, 54, 4, 17, 1, 12, 4, 17, 1, 92, 1, 4, 12, 12, 4, 17

Offset: 1

Gus Wiseman, Mar 27 2018

nonn

			The a(12) = 12 submultisets ("<" means subset or equal):
(2)<(2*2*3), (3)<(2*2*3), (2*2)<(2*2*3), (2*3)<(2*2*3), (2*2*3)<(2*2*3),
(2)<(2*6), (6)<(2*6), (2*6)<(2*6),
(3)<(3*4), (4)<(3*4), (3*4)<(3*4),
(12)<(12).

Cf. A000712, A001055, A001222, A001405, A122768, A276024, A281113, A284640, A295632, A299701, A299702, A299729, A301830.

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

a(n) = Sum_{d|n, d>1} f(d) * f(n/d) where f(n) = A001055(n) is the number of factorizations of n into factors greater than 1.

A302521 Odd numbers whose prime indices are squarefree and have disjoint prime indices. Numbers n such that the n-th multiset multisystem is a set partition.

Original entry on oeis.org

1, 3, 5, 11, 13, 15, 17, 29, 31, 33, 41, 43, 47, 51, 55, 59, 67, 73, 79, 83, 85, 93, 101, 109, 113, 123, 127, 137, 139, 141, 143, 145, 149, 155, 157, 163, 165, 167, 177, 179, 181, 187, 191, 199, 201, 205, 211, 215, 219, 221, 233, 241, 249, 255, 257, 269, 271

Offset: 1

Gus Wiseman, Apr 09 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

			Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of set partitions.
01: {}
03: {{1}}
05: {{2}}
11: {{3}}
13: {{1,2}}
15: {{1},{2}}
17: {{4}}
29: {{1,3}}
31: {{5}}
33: {{1},{3}}
41: {{6}}
43: {{1,4}}
47: {{2,3}}
51: {{1},{4}}
55: {{2},{3}}
59: {{7}}
67: {{8}}
73: {{2,4}}
79: {{1,5}}
83: {{9}}
85: {{2},{4}}
93: {{1},{5}}

Cf. A000110, A000961, A001222, A003963, A005117, A007716, A056239, A275024, A279790, A281113, A294788, A301750, A302242, A302243, A302505.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1,100,2],UnsameQ@@Join@@primeMS/@primeMS[#]&]

A316220 Number of triangles whose weight is the n-th Fermi-Dirac prime in the multiorder of integer partitions of Fermi-Dirac primes into Fermi-Dirac primes.

Original entry on oeis.org

1, 1, 3, 3, 9, 21, 46, 95, 273, 363, 731, 3088, 6247, 24152, 46012, 319511, 1141923, 2138064, 7346404, 13530107, 45297804, 271446312

Offset: 1

Gus Wiseman, Jun 26 2018

A Fermi-Dirac prime (A050376) is a number of the form p^(2^k) where p is prime and k >= 0. An FD-partition is an integer partition of a Fermi-Dirac prime into Fermi-Dirac primes. a(n) is the number of sequences of FD-partitions whose sums are weakly decreasing and sum to the n-th Fermi-Dirac prime.

Gus Wiseman, Comcategories and Multiorders
Gus Wiseman, Illustration of the first six terms of A316220.

Cf. A050376, A063834, A064547, A213925, A269134, A281113, A299757, A305829, A316202, A316210, A316211, A316219, A316228.

nn=60;
FDpQ[n_]:=With[{f=FactorInteger[n]},n>1&&Length[f]==1&&MatchQ[FactorInteger[2f[[1,2]]],{{2,_}}]];
FDpl=Select[Range[nn],FDpQ];
fen[n_]:=fen[n]=SeriesCoefficient[Product[1/(1-x^p),{p,Select[Range[n],FDpQ]}],{x,0,n}];
Table[Sum[Times@@fen/@p,{p,Select[IntegerPartitions[FDpl[[n]]],And@@FDpQ/@#&]}],{n,Length[FDpl]}]

A321514 Number of ways to choose a factorization of each integer from 2 to n into factors > 1.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 4, 12, 24, 48, 48, 192, 192, 384, 768, 3840, 3840, 15360, 15360, 61440, 122880, 245760, 245760, 1720320, 3440640, 6881280, 20643840, 82575360, 82575360, 412876800, 412876800, 2890137600, 5780275200, 11560550400, 23121100800, 208089907200

Offset: 1

Gus Wiseman, Nov 11 2018

nonn

			The a(8) = 12 ways to choose a factorization of each integer from 2 to 8:
  (2)*(3)*(4)*(5)*(6)*(7)*(8)
  (2)*(3)*(4)*(5)*(6)*(7)*(2*4)
  (2)*(3)*(4)*(5)*(2*3)*(7)*(8)
  (2)*(3)*(2*2)*(5)*(6)*(7)*(8)
  (2)*(3)*(4)*(5)*(6)*(7)*(2*2*2)
  (2)*(3)*(4)*(5)*(2*3)*(7)*(2*4)
  (2)*(3)*(2*2)*(5)*(6)*(7)*(2*4)
  (2)*(3)*(2*2)*(5)*(2*3)*(7)*(8)
  (2)*(3)*(4)*(5)*(2*3)*(7)*(2*2*2)
  (2)*(3)*(2*2)*(5)*(6)*(7)*(2*2*2)
  (2)*(3)*(2*2)*(5)*(2*3)*(7)*(2*4)
  (2)*(3)*(2*2)*(5)*(2*3)*(7)*(2*2*2)

Dominates A321468.

Cf. A001055, A050336, A066723, A076716, A157612, A281113, A300383, A317144, A317145, A321467, A321470, A321471, A321472.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Array[Length[facs[#]]&,n,1,Times],{n,30}]

a(n) = Product_{k = 1..n} A001055(k).

A296132 Number of twice-factorizations of n where the first factorization is constant and the latter factorizations are strict, i.e., type (P,R,Q).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 05 2017

nonn

a(n) is also the number of ways to choose a perfect divisor d|n and then a sequence of log_d(n) strict factorizations of d.

			The a(36) = 9 twice-factorizations are (2*3)*(2*3), (2*3)*(6), (6)*(2*3), (6)*(6), (2*3*6), (2*18), (3*12), (4*9), (36).

Cf. A000005, A001055, A005117, A045778, A052409, A052410, A089723, A279788, A281113, A284639, A295920, A295931.

sfs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sfs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
Table[Sum[Length[sfs[n^(1/g)]]^g,{g,Divisors[GCD@@FactorInteger[n][[All,2]]]}],{n,100}]

cp's OEIS Frontend

A302593 Numbers whose prime indices are powers of a common prime number.

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Maple

Mathematica

A277130 Number of planar branching factorizations of n.

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C

Mathematica

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Extensions

A295281 Number of complete strict tree-factorizations of n > 1.

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PARI

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A295923 Number of twice-factorizations of n where the first factorization is constant, i.e., type (P,R,P).

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Mathematica

A296121 Number of twice-factorizations of n with no repeated factorizations.

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Mathematica

A301829 Number of ways to choose a nonempty submultiset of a factorization of n into factors greater than one.

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Mathematica

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A302521 Odd numbers whose prime indices are squarefree and have disjoint prime indices. Numbers n such that the n-th multiset multisystem is a set partition.

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Mathematica

A316220 Number of triangles whose weight is the n-th Fermi-Dirac prime in the multiorder of integer partitions of Fermi-Dirac primes into Fermi-Dirac primes.

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