A296133 -id:A296133 - cp's OEIS frontend

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

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

Gus Wiseman, Dec 05 2017

nonn

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

			The a(16) = 4 twice-factorizations: (2)*(2*2*2), (2*2*2*2), (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, A000009, A001055, A047966, A047968, A052409, A052410, A089723, A281113, A295923, A295924, A295931, A295935, A296133.

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

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

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)

More terms from Antti Karttunen, Jul 29 2018

A301765 Number of rooted twice-partitions of n where the first rooted partition is constant and the composite rooted partition is strict, i.e., of type (Q,R,Q).

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 5, 8, 7, 11, 11, 19, 16, 27, 23, 42, 33, 63, 47, 87, 71, 119, 90, 195

Offset: 1

Gus Wiseman, Mar 26 2018

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) = 8 rooted twice-partitions:
(7), (61), (52), (43), (421),
(3)(21), (21)(3),
()()()()()()()().

Cf. A002865, A032305, A063834, A093637, A127524, A279791, A296133, A301422, A301462, A301467, A301480, A301706.

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

A302591 One, powers of 2, and prime numbers of squarefree index.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 11, 13, 16, 17, 29, 31, 32, 41, 43, 47, 59, 64, 67, 73, 79, 83, 101, 109, 113, 127, 128, 137, 139, 149, 157, 163, 167, 179, 181, 191, 199, 211, 233, 241, 256, 257, 269, 271, 277, 283, 293, 313, 317, 331, 347, 349, 353, 367, 373, 389, 397, 401

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}}
08: {{},{},{}}
11: {{3}}
13: {{1,2}}
16: {{},{},{},{}}
17: {{4}}
29: {{1,3}}
31: {{5}}
32: {{},{},{},{},{}}
41: {{6}}
43: {{1,4}}
47: {{2,3}}
59: {{7}}
64: {{},{},{},{},{},{}}

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

Cf. A000079, A000961, A001222, A003963, A005117, A007716, A056239, A275024, A279791, A281113, A296133, A301765, A302242, A302243, A302491.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[400],Or[#===1,Union[primeMS[#]]==={1},PrimeQ[#]&&SquareFreeQ[PrimePi[#]]]&]

ok(n)={n>>valuation(n,2) == 1 || (isprime(n) && issquarefree(primepi(n)))} \\ Andrew Howroyd, Aug 26 2018

Union of A000079 and A302491. - Andrew Howroyd, Aug 26 2018

A302592 One, powers of 2, and prime numbers of prime index.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 11, 16, 17, 31, 32, 41, 59, 64, 67, 83, 109, 127, 128, 157, 179, 191, 211, 241, 256, 277, 283, 331, 353, 367, 401, 431, 461, 509, 512, 547, 563, 587, 599, 617, 709, 739, 773, 797, 859, 877, 919, 967, 991, 1024, 1031, 1063, 1087, 1153, 1171

Offset: 1

Gus Wiseman, Apr 10 2018

nonn

			Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of set multisystems.
001: {}
002: {{}}
003: {{1}}
004: {{},{}}
005: {{2}}
008: {{},{},{}}
011: {{3}}
016: {{},{},{},{}}
017: {{4}}
031: {{5}}
032: {{},{},{},{},{}}
041: {{6}}
059: {{7}}
064: {{},{},{},{},{},{}}
067: {{8}}
083: {{9}}
109: {{10}}
127: {{11}}
128: {{},{},{},{},{},{},{}}

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

Cf. A000961, A001222, A003963, A005117, A006450, A007716, A056239, A076610, A275024, A279791, A281113, A296133, A301765, A302242, A302243.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Or[#===1,Union[primeMS[#]]==={1},PrimeQ[#]&&PrimeQ[PrimePi[#]]]&]

ok(n)={n>>valuation(n,2) == 1 || (isprime(n) && isprime(primepi(n)))} \\ Andrew Howroyd, Aug 26 2018

Union of A000079 and A006450. - Andrew Howroyd, Aug 26 2018

cp's OEIS Frontend

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

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A301765 Number of rooted twice-partitions of n where the first rooted partition is constant and the composite rooted partition is strict, i.e., of type (Q,R,Q).

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A302591 One, powers of 2, and prime numbers of squarefree index.

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Mathematica

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A302592 One, powers of 2, and prime numbers of prime index.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula