A050370 -id:A050370 - cp's OEIS frontend

A050371 Number of factorizations into composite numbers indexed by prime signatures. A050370(A025487).

1, 0, 1, 1, 1, 1, 2, 2, 1, 2, 3, 3, 3, 4, 4, 5, 5, 4, 8, 6, 7, 4, 8, 9, 7, 11, 12, 11, 8, 13, 14, 8, 19, 21, 15, 16, 22, 16, 23, 12, 23, 26, 20, 26, 36, 22, 29, 35, 29, 34, 14, 44, 41, 11, 40, 40, 58, 30, 50, 54, 57, 52, 21, 75, 56, 28, 77, 69, 61, 92, 42, 50

Offset: 1

Christian G. Bower, Nov 15 1999

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A025487, A050370.

More terms from Amiram Eldar, Jan 29 2019

A376602 Inflection and undulation points in the sequence of composite numbers (A002808).

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 14, 15, 16, 18, 20, 21, 22, 25, 27, 29, 32, 33, 34, 37, 38, 39, 41, 43, 44, 45, 48, 50, 52, 53, 54, 57, 60, 61, 62, 65, 66, 67, 68, 69, 72, 74, 76, 78, 80, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 96, 99, 100, 101, 103, 105, 106, 107, 108

Offset: 1

Gus Wiseman, Oct 05 2024

nonn

These are points at which the second differences (A073445) are zero.

			The composite numbers (A002808) are:
  4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, ...
with first differences (A073783):
  2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, ...
with first differences (A073445):
  0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, -1, 0, ...
with zeros at (A376602):
  1, 3, 5, 7, 9, 11, 14, 15, 16, 18, 20, 21, 22, 25, 27, 29, 32, 33, 34, 37, 38, ...

Wikipedia, Inflection point
Gus Wiseman, Inflection and undulation points in the sequence of composite numbers.

Partitions into composite numbers are counted by A023895, factorizations A050370.
For prime instead of composite we have A064113.
These are the positions of zeros in A073445.
For first differences we had A073783, ones A375929, complement A065890.
For concavity in primes we have A258025/A258026, weak A333230/A333231.
For upward concavity (instead of zero) we have A376651, downward A376652.
The complement is A376603.
For composite numbers: A002808 (terms), A073783 (first differences), A073445 (second differences), A376603 (nonzero curvature), A376651 (concave-up), A376652 (concave-down).
For inflection and undulation points: A064113 (prime), A376588 (non-perfect-power), A376591 (squarefree), A376594 (nonsquarefree), A376597 (prime-power), A376600 (non-prime-power).
Cf. A000040, A036263, A251092, A333214, A333216, A333254.

Join@@Position[Differences[Select[Range[100],CompositeQ],2],0]

A376603 Points of nonzero curvature in the sequence of composite numbers (A002808).

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 13, 17, 19, 23, 24, 26, 28, 30, 31, 35, 36, 40, 42, 46, 47, 49, 51, 55, 56, 58, 59, 63, 64, 70, 71, 73, 75, 77, 79, 81, 82, 94, 95, 97, 98, 102, 104, 112, 114, 118, 119, 123, 124, 126, 127, 131, 132, 136, 138, 146, 148, 150, 152, 162, 163

Offset: 1

Gus Wiseman, Oct 05 2024

nonn

These are points at which the second differences (A073445) are nonzero.

			The composite numbers (A002808) are:
  4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, ...
with first differences (A073783):
  2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, ...
with first differences (A073445):
  0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, -1, 0, ...
with nonzero terms at (A376603):
  2, 4, 6, 8, 10, 12, 13, 17, 19, 23, 24, 26, 28, 30, 31, 35, 36, 40, 42, 46, 47, ...

Dominic McCarty, Table of n, a(n) for n = 1..1000
Gus Wiseman, Points of nonzero curvature in the sequence of composite numbers.

Partitions into composite numbers are counted by A023895, factorizations A050370.
These are the positions of nonzero terms in A073445.
For first differences we had A073783, ones A375929, complement A065890.
For prime instead of composite we have A333214.
The complement is A376602.
For upward concavity (instead of nonzero) we have A376651, downward A376652.
For composite numbers: A002808 (terms), A073783 (first differences), A073445 (second differences), A376602 (zeros), A376651 (concave-up), A376652 (concave-down).
For nonzero curvature: A333214 (prime), A376589 (non-perfect-power), A376592 (squarefree), A376595 (nonsquarefree), A376598 (prime-power), A376601 (non-prime-power).
Cf. A000961, A064113, A246655, A251092, A258025, A333254.

Join@@Position[Sign[Differences[Select[Range[100],CompositeQ],2]],1|-1]

A376651 Points of upward concavity in the sequence of composite numbers (A002808).

Original entry on oeis.org

4, 8, 12, 17, 23, 26, 30, 35, 40, 46, 49, 55, 58, 63, 70, 73, 77, 81, 94, 97, 102, 112, 118, 123, 126, 131, 136, 146, 150, 162, 173, 176, 180, 185, 195, 200, 205, 210, 216, 219, 229, 242, 245, 249, 262, 267, 276, 280, 285, 292, 297, 302, 305, 310, 317, 320

Offset: 1

Gus Wiseman, Oct 06 2024

nonn

These are points at which the second differences (A073445) are positive.

Also positions of strict ascents in the first differences (A073783) of composite numbers (A002808).

			The composite numbers are (A002808):
  4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, ...
with first differences (A073783):
  2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, ...
with first differences (A073445):
  0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, -1, 0, ...
with positive terms at (A376651):
  4, 8, 12, 17, 23, 26, 30, 35, 40, 46, 49, 55, 58, 63, 70, 73, 77, 81, 94, 97, ...

Dominic McCarty, Table of n, a(n) for n = 1..1000
Gus Wiseman, Points of upward concavity in the sequence of composite numbers.

The version for A000002 is A022297, negative A156242.
Partitions into composite numbers are counted by A023895, factorizations A050370.
For first differences we had A065310 or A073783, ones A375929.
These are the positions of positive terms in A073445, negative A376652.
For prime instead of composite we have A258025, negative A258026.
For zero second differences (instead of positive) we have A376602.
For composite numbers: A002808 (terms), A073783 (first differences), A073445 (second differences), A376602 (inflections and undulations), A376603 (nonzero curvature), A376652 (concave-down).
Cf. A000961, A057820, A064113, A065890, A251092, A333254, A376604.

Join@@Position[Sign[Differences[Select[Range[1000],CompositeQ],2]],1]

A376652 Points of downward concavity in the sequence of composite numbers (A002808).

Original entry on oeis.org

2, 6, 10, 13, 19, 24, 28, 31, 36, 42, 47, 51, 56, 59, 64, 71, 75, 79, 82, 95, 98, 104, 114, 119, 124, 127, 132, 138, 148, 152, 163, 174, 178, 181, 187, 196, 201, 206, 212, 217, 221, 230, 243, 247, 250, 263, 268, 278, 281, 286, 293, 298, 303, 306, 311, 318, 321

Offset: 1

Gus Wiseman, Oct 06 2024

nonn

These are points at which the second differences (A073445) are negative.

Also positions of strict descents in the first differences (A073783) of composite numbers (A002808).

			The composite numbers are (A002808):
  4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, ...
with first differences (A073783):
  2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, ...
with second differences (A073445):
  0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, -1, 0, ...
with negative terms at (A376651):
  2, 6, 10, 13, 19, 24, 28, 31, 36, 42, 47, 51, 56, 59, 64, 71, 75, 79, 82, 95, 98, ...

Robert Israel, Table of n, a(n) for n = 1..10000
Gus Wiseman, Points of downward concavity in the sequence of composite numbers.

The version for A000002 is A156242, positive A022297.
Partitions into composite numbers are counted by A023895, factorizations A050370.
For first differences we had A065310 or A073783, ones A375929.
These are the positions of negative terms in A073445, positive A376651.
For prime instead of composite we have A258026, positive A258025.
For zero second differences instead of negative we have A376602.
For composite numbers: A002808 (terms), A073783 (first differences), A073445 (second differences), A376602 (inflections and undulations), A376603 (nonzero curvature), A376651 (concave-up).
Cf. A000961, A007916, A057820, A064113, A065890, A251092, A333254, A376604.

Comps:= remove(isprime, [seq(i,i=4..1000)]):
D1:= Comps[2..-1]-Comps[1..-2]:
D2:= D1[2..-1]-D1[1..-2]:
select(t -> D2[t] < 0, [$1..nops(D2)]); # Robert Israel, Nov 06 2024

Join@@Position[Sign[Differences[Select[Range[1000],CompositeQ],2]],-1]

A320633 Composite numbers whose prime indices are also composite.

Original entry on oeis.org

49, 91, 133, 161, 169, 203, 247, 259, 299, 301, 329, 343, 361, 371, 377, 427, 437, 481, 497, 511, 529, 551, 553, 559, 611, 623, 637, 667, 679, 689, 703, 707, 721, 749, 791, 793, 817, 841, 851, 893, 917, 923, 931, 949, 959, 973, 989, 1007, 1027, 1043, 1057

Offset: 1

Gus Wiseman, Oct 18 2018

nonn

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

			The sequence of terms begins:
   49 = prime(4)^2
   91 = prime(4)*prime(6)
  133 = prime(4)*prime(8)
  161 = prime(4)*prime(9)
  169 = prime(6)^2
  203 = prime(4)*prime(10)
  247 = prime(6)*prime(8)
  259 = prime(4)*prime(12)
  299 = prime(6)*prime(9)
  301 = prime(4)*prime(14)
  329 = prime(4)*prime(15)
  343 = prime(4)^3
  361 = prime(8)^2
  371 = prime(4)*prime(16)
  377 = prime(6)*prime(10)
  427 = prime(4)*prime(18)
  437 = prime(8)*prime(9)
  481 = prime(6)*prime(12)
  497 = prime(4)*prime(20)
  511 = prime(4)*prime(21)
  529 = prime(9)^2
  551 = prime(8)*prime(10)
  553 = prime(4)*prime(22)
  559 = prime(6)*prime(14)
  611 = prime(6)*prime(15)
  623 = prime(4)*prime(24)
  637 = prime(4)^2*prime(6)

Cf. A000040, A006450, A007821, A018252, A050370, A056239, A076610, A112798, A302242, A302478, A320533, A320628, A320629.

Select[Range[2,1000],And[OddQ[#],!PrimeQ[#],And@@Not/@PrimeQ/@PrimePi/@First/@FactorInteger[#]]&]

A317748 Irregular triangle where T(n,k) is the number of factorizations of n into factors > 1 with GCD d = A027750(n, k).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 06 2018

			Triangle begins:
   1:  0
   2:  0  1
   3:  0  1
   4:  0  1  1
   5:  0  1
   6:  1  0  0  1
   7:  0  1
   8:  0  2  0  1
   9:  0  1  1
  10:  1  0  0  1
  11:  0  1
  12:  2  1  0  0  0  1
  13:  0  1
  14:  1  0  0  1
  15:  1  0  0  1
  16:  0  3  1  0  1
  17:  0  1
  18:  2  0  1  0  0  1
  19:  0  1
  20:  2  1  0  0  0  1

Row lengths are A000005. Row sums are A001055. First column is A281116. Number of nonzero terms in each row is A317751.
Cf. A007562, A007716, A014963, A027750, A045778, A050370, A162247, A289509, A303386, A303546.
Cf. A317752, A317755, A317757.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
goc[n_,m_]:=Length[Select[facs[n],And[And@@(Divisible[#,m]&/@#),GCD@@(#/m)==1]&]];
Table[goc[n,d],{n,30},{d,Divisors[n]}]

Name edited by Peter Munn, Mar 05 2025

A317751 Number of divisors d of n such that there exists a factorization of n into factors > 1 with GCD d.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 3, 2, 2, 2, 3, 1, 2, 1, 3, 2, 2, 2, 5, 1, 2, 2, 3, 1, 2, 1, 3, 3, 2, 1, 4, 2, 3, 2, 3, 1, 3, 2, 3, 2, 2, 1, 3, 1, 2, 3, 4, 2, 2, 1, 3, 2, 2, 1, 5, 1, 2, 3, 3, 2, 2, 1, 4, 3, 2, 1, 3, 2, 2, 2, 3, 1, 3, 2, 3, 2, 2, 2, 4, 1, 3, 3, 5, 1, 2, 1, 3, 2

Offset: 1

Gus Wiseman, Aug 06 2018

nonn

Also the number of distinct possible GCDs of factorizations of n into factors > 1.
Also the number of nonzero terms in row n of A317748.
a(prime^n) = A008619(n).
If n is squarefree and composite, a(n) = 2.

			The divisors of 36 that are possible GCDs of factorizations of 36 are {1, 2, 3, 6, 36}, so a(36) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A000837, A001055, A014963, A045778, A050370, A162247, A281116, A289509.

Cf. A317748, A317752, A317755, A317757.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
goc[n_,m_]:=Length[Select[facs[n],And[And@@(Divisible[#,m]&/@#),GCD@@(#/m)==1]&]];
Table[Length[Select[Divisors[n],goc[n,#]!=0&]],{n,100}]

A317751aux(n, m, facs, gcds) = if(1==n, setunion([gcd(Vec(facs))],gcds), my(newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); gcds = setunion(gcds,A317751aux(n/d, d, newfacs, gcds)))); (gcds));
A317751(n) = if(1==n,0,length(A317751aux(n, n, List([]), Set([])))); \\ Antti Karttunen, Sep 08 2018

More terms from Antti Karttunen, Sep 08 2018

A050372 Number of ways to factor n into distinct composite factors.

Original entry on oeis.org

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

Offset: 1

Christian G. Bower, Nov 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).

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A002808, A045778, A050370-A050375. a(p^k)=A025147. a(A002110)=A000296.

with(numtheory):
b:= proc(n, k) option remember;
     `if`(isprime(n), 0, `if`(n>k, 0, 1)+
      add(`if`(d>k or isprime(d), 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

b[n_, k_] := b[n, k] = If[PrimeQ[n], 0, If[n>k, 0, 1] + Sum[If[d>k || PrimeQ[d], 0, b[n/d, d-1]], {d, Divisors[n] ~Complement~ {1, n}}]];
a[n_] := b[n, n];
Array[a, 120] (* Jean-François Alcover, Mar 21 2017, after Alois P. Heinz *)

Dirichlet g.f.: Product_{n is composite}(1+1/n^s).

A050374 Number of ordered factorizations of n into composite factors.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 1, 0, 3, 1, 1, 1, 1, 0, 1, 0, 3, 1, 1, 1, 4, 0, 1, 1, 3, 0, 1, 0, 1, 1, 1, 0, 5, 1, 1, 1, 1, 0, 3, 1, 3, 1, 1, 0, 5, 0, 1, 1, 5, 1, 1, 0, 1, 1, 1, 0, 7, 0, 1, 1, 1, 1, 1, 0, 5, 2, 1, 0, 5, 1, 1, 1, 3, 0, 5, 1, 1, 1, 1, 1, 10, 0, 1, 1, 4, 0, 1

Offset: 1

Christian G. Bower, Nov 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).

The Dirichlet inverse is given by A005171, all but the first term in A005171 turned negative. - R. J. Mathar, Jul 15 2010

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n

Cf. A002033, A002808, A005171, A050370-A050375, A000045, A002110, A032032.

read(transforms):
[1, seq(-A005171(n), n=2..100)] ;
a050374 := DIRICHLETi(%) ; # R. J. Mathar, May 26 2017

A050374(n) = if(1==n,n,sumdiv(n,d,if(dA050374(d),0))); \\ Antti Karttunen, Oct 20 2017

Dirichlet g.f.: 1/(1-B(s)) where B(s) is D.g.f. of characteristic function of composite numbers.
a(n) = A050375(A101296(n)). - R. J. Mathar, May 26 2017
For n >= 1, a(p^n) = A000045(n-1), for any prime p.
For n >= 0, a(A002110(n)) = A032032(n).

cp's OEIS Frontend

A050371 Number of factorizations into composite numbers indexed by prime signatures. A050370(A025487).

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A376602 Inflection and undulation points in the sequence of composite numbers (A002808).

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Mathematica

A376603 Points of nonzero curvature in the sequence of composite numbers (A002808).

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A376651 Points of upward concavity in the sequence of composite numbers (A002808).

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A376652 Points of downward concavity in the sequence of composite numbers (A002808).

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Maple

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A320633 Composite numbers whose prime indices are also composite.

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A317748 Irregular triangle where T(n,k) is the number of factorizations of n into factors > 1 with GCD d = A027750(n, k).

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Extensions

A317751 Number of divisors d of n such that there exists a factorization of n into factors > 1 with GCD d.

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PARI

Extensions

A050372 Number of ways to factor n into distinct composite factors.