cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A295935 Number of twice-factorizations of n where the latter factorizations are constant, i.e., type (P,P,R).

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 1, 5, 3, 2, 1, 5, 1, 2, 2, 12, 1, 5, 1, 5, 2, 2, 1, 10, 3, 2, 5, 5, 1, 5, 1, 18, 2, 2, 2, 15, 1, 2, 2, 10, 1, 5, 1, 5, 5, 2, 1, 22, 3, 5, 2, 5, 1, 10, 2, 10, 2, 2, 1, 13, 1, 2, 5, 40, 2, 5, 1, 5, 2, 5, 1, 28, 1, 2, 5, 5, 2, 5, 1, 22, 12, 2, 1
Offset: 1

Views

Author

Gus Wiseman, Nov 29 2017

Keywords

Comments

a(n) is also the number of ways to choose a perfect divisor of each factor in a factorization of n.

Examples

			The a(24) = 10 twice-factorizations are:
(2)*(2)*(2)*(3), (2)*(3)*(2*2), (3)*(2*2*2)
(2)*(2)*(6), (2*2)*(6),
(2)*(3)*(4),
(2)*(12),
(3)*(8),
(4)*(6),
(24).

Crossrefs

Cf. A000005, A001055, A052409, A052410, A089723, A279784, A281113, A284639, A295923, A295924, A295931.

Programs

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

Formula

Dirichlet g.f.: 1/Product_{n > 1}(1 - A089723(n)/n^s).

A295924 Number of twice-factorizations of n of type (R,P,R).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 4, 3, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 4, 1, 1, 1, 1, 8, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Gus Wiseman, Nov 30 2017

Keywords

Comments

a(n) is the number of ways to choose an integer partition of a divisor of A052409(n).

Examples

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

Links

Crossrefs

Cf. A000005, A000041, A001055, A047968, A052409, A052410, A089723, A281113, A284639, A295923, A295931, A295935, A296134.

Programs

  • Mathematica
    Table[DivisorSum[GCD@@FactorInteger[n][[All,2]],PartitionsP],{n,100}]
  • PARI
    A052409(n) = { my(k=ispower(n)); if(k, k, n>1); }; \\ From A052409
A295924(n) = if(1==n,n,sumdiv(A052409(n),d,numbpart(d))); \\ Antti Karttunen, Jul 29 2018

Formula

a(1) = 1; for n > 1, a(n) = Sum_{d|A052409(n)} A000041(d). - Antti Karttunen, Jul 29 2018

Extensions

More terms from Antti Karttunen, Jul 29 2018

A296118 Number of ways to choose a factorization of each factor in a strict factorization of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 5, 2, 3, 1, 8, 1, 3, 3, 8, 1, 8, 1, 8, 3, 3, 1, 20, 2, 3, 5, 8, 1, 12, 1, 18, 3, 3, 3, 23, 1, 3, 3, 20, 1, 12, 1, 8, 8, 3, 1, 45, 2, 8, 3, 8, 1, 20, 3, 20, 3, 3, 1, 38, 1, 3, 8, 34, 3, 12, 1, 8, 3, 12, 1, 66, 1, 3, 8, 8, 3, 12, 1, 45, 8, 3
Offset: 1

Views

Author

Gus Wiseman, Dec 05 2017

Keywords

Examples

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

Links

Crossrefs

Cf. A000009, A005117, A045778, A261049, A271619, A281113, A294788, A295923, A296119, A296121.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Sum[Times@@(Length[facs[#]]&/@f),{f,Select[facs[n],UnsameQ@@#&]}],{n,100}]
  • PARI
    A001055(n, m=n) = if(1==n, 1, sumdiv(n, d, if((d>1)&&(d<=m), A001055(n/d, d))));
A296118(n, m=n) = ((n<=m)*A001055(n) + sumdiv(n, d, if((d>1)&&(d<=m)&&(dA001055(d)*A296118(n/d, d-1)))); \\ Antti Karttunen, Oct 08 2018

Formula

Dirichlet g.f.: Product_{n > 1}(1 + A001055(n)/n^s).

A295931 Number of ways to write n in the form n = (x^y)^z where x, y, and z are positive integers.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Gus Wiseman, Nov 29 2017

Keywords

Comments

By convention a(1) = 1.
Values can be 1, 3, 6, 9, 10, 15, 18, 21, 27, 28, 30, 36, 45, 54, 60, 63, 84, 90, etc. - Robert G. Wilson v, Dec 10 2017

Examples

			The a(256) = 10 ways are:
(2^1)^8    (2^2)^4   (2^4)^2  (2^8)^1
(4^1)^4    (4^2)^2   (4^4)^1
(16^1)^2   (16^2)^1
(256^1)^1

Links

Crossrefs

Cf. A000005, A007425, A052409, A052410, A089723, A093771, A175082, A277562, A281113, A284639, A294786, A294336, A294338, A295920, A295923, A295924, A295935.

Programs

  • Maple
    f:= proc(n) local m,d,t;
  m:= igcd(seq(t[2],t=ifactors(n)[2]));
  add(numtheory:-tau(d),d=numtheory:-divisors(m))
end proc:
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Dec 19 2017
  • Mathematica
    Table[Sum[DivisorSigma[0,d],{d,Divisors[GCD@@FactorInteger[n][[All,2]]]}],{n,100}]

Formula

a(A175082(k)) = 1, a(A093771(k)) = 3.
a(n) = Sum_{d|A052409(n)} A000005(d).

A295920 Number of twice-factorizations of n of type (P,R,R).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Gus Wiseman, Nov 30 2017

Keywords

Comments

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

Examples

			The a(64) = 17 twice-factorizations are:
(2)*(2)*(2)*(2)*(2)*(2)  (2*2)*(2*2)*(2*2)  (2*2*2)*(2*2*2)  (2*2*2*2*2*2)
(2*2)*(2*2)*(4)          (2*2)*(4)*(2*2)    (4)*(2*2)*(2*2)
(2*2)*(4)*(4)            (4)*(2*2)*(4)      (4)*(4)*(2*2)
(2*2*2)*(8)              (8)*(2*2*2)
(4)*(4)*(4)              (4*4*4)
(8)*(8)                  (8*8)
(64)

Links

Crossrefs

Cf. A000005, A001055, A052409, A052410, A089723, A279789, A281113, A295923, A295924, A295931, A295935.

Programs

  • Mathematica
    Table[Sum[Length[Divisors[GCD@@FactorInteger[n^(1/d)][[All,2]]]]^d,{d,Divisors[GCD@@FactorInteger[n][[All,2]]]}],{n,100}]
  • PARI
    A052409(n) = { my(k=ispower(n)); if(k, k, n>1); }; \\ From A052409
A295920(n) = if(1==n,n,my(r); sumdiv(A052409(n), d, if(!ispower(n,d,&r),(1/0),numdiv(A052409(r))^d))); \\ Antti Karttunen, Dec 06 2018, after Mathematica-code

Formula

a(n) = Sum_{d|A052409(n)} A000005(A052409(n^(1/d)))^d. - Antti Karttunen, Dec 06 2018, after Mathematica-code

Extensions

More terms from Antti Karttunen, Dec 06 2018

A296134 Number of twice-factorizations of n of type (R,Q,R).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 4, 1, 1, 1, 2, 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, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Gus Wiseman, Dec 05 2017

Keywords

Comments

a(n) is the number of ways to choose a strict integer partition of a divisor of A052409(n).

Examples

			The a(16) = 4 twice-factorizations: (2)*(2*2*2), (2*2*2*2), (4*4), (16).

Links

Crossrefs

Cf. A000005, A000009, A001055, A047966, A047968, A052409, A052410, A089723, A281113, A295923, A295924, A295931, A295935, A296133.

Programs

  • Mathematica
    Table[DivisorSum[GCD@@FactorInteger[n][[All,2]],PartitionsQ],{n,100}]
  • PARI
    A000009(n,k=(n-!(n%2))) = if(!n,1,my(s=0); while(k >= 1, if(k<=n, s += A000009(n-k,k)); k -= 2); (s));
A052409(n) = { my(k=ispower(n)); if(k, k, n>1); }; \\ From A052409
A296134(n) = if(1==n,n,sumdiv(A052409(n),d,A000009(d))); \\ Antti Karttunen, Jul 29 2018

Formula

From Antti Karttunen, Jul 31 2018: (Start)
a(1) = 1; for n > 1, a(n) = Sum_{d|A052409(n)} A000009(d).
a(n) = A047966(A052409(n)). (End)

Extensions

More terms from Antti Karttunen, Jul 29 2018

A320887 Number of multiset partitions of factorizations of n into factors > 1 such that all the parts have the same product.

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 1, 4, 3, 2, 1, 4, 1, 2, 2, 9, 1, 4, 1, 4, 2, 2, 1, 7, 3, 2, 4, 4, 1, 5, 1, 8, 2, 2, 2, 12, 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, 22, 2, 5, 1, 4, 2, 5, 1, 16, 1, 2, 4, 4, 2, 5, 1, 12, 9, 2, 1, 11, 2, 2, 2, 7, 1, 11, 2, 4, 2, 2, 2, 19, 1, 4, 4, 12, 1, 5, 1, 7, 5
Offset: 1

Views

Author

Gus Wiseman, Oct 23 2018

Keywords

Examples

			The a(36) = 12 multiset partitions:
  (2*2*3*3)    (6)*(2*3)  (6)*(6)  (36)
  (2*3)*(2*3)  (2*2*9)    (2*18)
               (2*3*6)    (3*12)
               (3*3*4)    (4*9)
                          (6*6)

Links

Crossrefs

Cf. A001055, A001970, A046523, A050336, A052409, A279375, A294786, A295923, A320886, A320888, A320889.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[With[{g=GCD@@FactorInteger[n][[All,2]]},Sum[Binomial[Length[facs[n^(1/d)]]+d-1,d],{d,Divisors[g]}]],{n,100}]
  • PARI
    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
A320887(n) = if(1==n,n,my(r); sumdiv(A052409(n), d, binomial(A001055(sqrtnint(n,d)) + d - 1, d))); \\ Antti Karttunen, Nov 17 2019

Formula

a(n) = Sum_{d|A052409(n)} binomial(A001055(n^(1/d)) + d - 1, d).
a(n) = a(A046523(n)). - Antti Karttunen, Nov 17 2019

Extensions

Data section extended up to term a(105) by Antti Karttunen, Nov 17 2019
Showing 1-7 of 7 results.

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