A296134 -id:A296134 - cp's OEIS frontend

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

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

Gus Wiseman, Nov 30 2017

nonn

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

			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).

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

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

Table[DivisorSum[GCD@@FactorInteger[n][[All,2]],PartitionsP],{n,100}]

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

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

More terms from Antti Karttunen, Jul 29 2018

A301766 Number of rooted twice-partitions of n where the first rooted partition is strict and the composite rooted partition is constant, i.e., of type (R,Q,R).

Original entry on oeis.org

1, 1, 1, 3, 4, 6, 7, 9, 11, 13, 16, 19, 22, 26, 32, 36, 42, 52, 59, 66, 79, 93, 108, 125, 141, 162, 192, 222, 248, 285, 331, 375, 430, 492, 555, 632, 719, 816, 929, 1051, 1177, 1327, 1510, 1701, 1908, 2146, 2408, 2705, 3035, 3388, 3792, 4257, 4751, 5284, 5894

Offset: 1

Gus Wiseman, Mar 26 2018

nonn

A rooted partition of n is an integer partition of n - 1. A rooted twice-partition of n is a choice of a rooted partition of each part in a rooted partition of n.

			The a(9) = 11 rooted twice-partitions:
(7), (1111111),
(6)(), (33)(), (222)(), (111111)(), (11111)(1), (22)(2), (1111)(11),
(1111)(1)(), (111)(11)().

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A002865, A032305, A047966, A063834, A093637, A296134, A300383, A301422, A301462, A301467, A301480, A301706.

twirtns[n_]:=Join@@Table[Tuples[IntegerPartitions[#-1]&/@ptn],{ptn,IntegerPartitions[n-1]}];
Table[Select[twirtns[n],UnsameQ@@Total/@#&&SameQ@@Join@@#&]//Length,{n,20}]

a(n)=if(n<3, 1, sum(k=1, n-2, polcoef(prod(j=0, (n-2)\k, 1 + x^(j*k + 1) + O(x^n)), n-1))) \\ Andrew Howroyd, Aug 26 2018

Terms a(26) and beyond from Andrew Howroyd, Aug 26 2018

A302597 Squarefree numbers whose prime indices are powers of a common prime number.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 14, 17, 19, 21, 22, 23, 31, 34, 38, 41, 42, 46, 53, 57, 59, 62, 67, 82, 83, 97, 103, 106, 109, 114, 115, 118, 127, 131, 133, 134, 157, 159, 166, 179, 191, 194, 206, 211, 218, 227, 230, 241, 254, 262, 266, 277, 283, 311, 314, 318, 331

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}}
05: {{2}}
06: {{},{1}}
07: {{1,1}}
10: {{},{2}}
11: {{3}}
14: {{},{1,1}}
17: {{4}}
19: {{1,1,1}}
21: {{1},{1,1}}
22: {{},{3}}
23: {{2,2}}
31: {{5}}
34: {{},{4}}
38: {{},{1,1,1}}

Cf. A000961, A001222, A003963, A005117, A007716, A056239, A275024, A047966, A281113, A296134, A301766, A302242, A302243.

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

A302600 1, 2, prime numbers of prime index, and prime numbers of prime index times 2.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 11, 17, 22, 31, 34, 41, 59, 62, 67, 82, 83, 109, 118, 127, 134, 157, 166, 179, 191, 211, 218, 241, 254, 277, 283, 314, 331, 353, 358, 367, 382, 401, 422, 431, 461, 482, 509, 547, 554, 563, 566, 587, 599, 617, 662, 706, 709, 734, 739, 773

Offset: 1

Gus Wiseman, Apr 10 2018

nonn

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

Also squarefree numbers whose prime indices other than 1 are equal prime numbers.

			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}}
05: {{2}}
06: {{},{1}}
10: {{},{2}}
11: {{3}}
17: {{4}}
22: {{},{3}}
31: {{5}}
34: {{},{4}}
41: {{6}}
59: {{7}}
62: {{},{5}}
67: {{8}}
82: {{},{6}}
83: {{9}}

Cf. A000961, A001222, A003963, A005117, A007716, A056239, A275024, A047966, A281113, A296134, A301766, A302242, A302243.

Select[Range[1000],SquareFreeQ[#]&&SameQ@@DeleteCases[primeMS[#],1]&&And@@PrimeQ/@DeleteCases[primeMS[#],1]&]

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A301766 Number of rooted twice-partitions of n where the first rooted partition is strict and the composite rooted partition is constant, i.e., of type (R,Q,R).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A302597 Squarefree numbers whose prime indices are powers of a common prime number.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A302600 1, 2, prime numbers of prime index, and prime numbers of prime index times 2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica