A295920 -id:A295920 - cp's OEIS frontend

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

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

Gus Wiseman, Nov 29 2017

nonn

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

			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

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

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

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

Table[Sum[DivisorSigma[0,d],{d,Divisors[GCD@@FactorInteger[n][[All,2]]]}],{n,100}]

a(A175082(k)) = 1, a(A093771(k)) = 3.

a(n) = Sum_{d|A052409(n)} A000005(d).

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]

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}]

A302498 Numbers that are a power of a prime number whose prime index is itself a power of a prime number.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 16, 17, 19, 23, 25, 27, 31, 32, 41, 49, 53, 59, 64, 67, 81, 83, 97, 103, 109, 121, 125, 127, 128, 131, 157, 179, 191, 211, 227, 241, 243, 256, 277, 283, 289, 311, 331, 343, 353, 361, 367, 401, 419, 431, 461, 509, 512, 529, 547, 563

Offset: 1

Gus Wiseman, Apr 09 2018

nonn

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

			49 is in the sequence because 49 = prime(prime(1)^2)^2.
Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of constant constant-multiset multisystems.
01: {}
02: {{}}
03: {{1}}
04: {{},{}}
05: {{2}}
07: {{1,1}}
08: {{},{},{}}
09: {{1},{1}}
11: {{3}}
16: {{},{},{},{}}
17: {{4}}
19: {{1,1,1}}
23: {{2,2}}
25: {{2},{2}}
27: {{1},{1},{1}}
31: {{5}}
32: {{},{},{},{},{}}
41: {{6}}
49: {{1,1},{1,1}}
53: {{1,1,1,1}}
59: {{7}}
64: {{},{},{},{},{},{}}

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

Cf. A000961, A001222, A003963, A005117, A007716, A056239, A076610, A275024, A279789, A295920, A301763, A302242, A302243, A302492, A302493, A302496, A302497.

Select[Range[100],Or[#===1,PrimePowerQ[#]&&And@@(Or[#===1,PrimePowerQ[#]]&/@PrimePi/@FactorInteger[#][[All,1]])]&]

ok(n)={my(p); n == 1 || (isprimepower(n, &p) && (p == 2 || isprimepower(primepi(p))))} \\ Andrew Howroyd, Aug 26 2018

cp's OEIS Frontend

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

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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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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).

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A302498 Numbers that are a power of a prime number whose prime index is itself a power of a prime number.

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