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A322453 Number of factorizations of n into factors > 1 using only primes and perfect powers. 3
1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 7, 1, 1, 2, 2, 1, 1, 1, 5, 5, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 7, 1, 2, 2, 5, 1, 1, 1, 3, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

First differs from A000688 at a(36) = 5, A000688(36) = 4.

Terms in this sequence only depend on the prime signature of n. - David A. Corneth, Dec 26 2018

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16384

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Index entries for sequences computed from exponents in factorization of n

EXAMPLE

The a(144) = 13 factorizations:

  (144),

  (4*36), (9*16),

  (2*2*36), (2*8*9), (3*3*16), (4*4*9),

  (2*2*4*9), (2*3*3*8), (3*3*4*4),

  (2*2*2*2*9), (2*2*3*3*4),

  (2*2*2*2*3*3).

MATHEMATICA

perpowQ[n_]:=GCD@@FactorInteger[n][[All, 2]]>1;

pfacs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[pfacs[n/d], Min@@#>=d&]], {d, Select[Rest[Divisors[n]], Or[PrimeQ[#], perpowQ[#]]&]}]];

Table[Length[pfacs[n]], {n, 100}]

PROG

(PARI) A322453(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&(ispower(d)||isprime(d)), s += A322453(n/d, d))); (s)); \\ Antti Karttunen, Dec 26 2018

CROSSREFS

Cf. A000688, A000961, A001055, A001597, A025487, A050336, A284696, A294068, A320322, A322452.

Sequence in context: A336736 A000688 A295879 * A327012 A328855 A327658

Adjacent sequences:  A322450 A322451 A322452 * A322454 A322455 A322456

KEYWORD

nonn

AUTHOR

Gus Wiseman, Dec 09 2018

EXTENSIONS

More terms from Antti Karttunen, Dec 24 2018

STATUS

approved

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Last modified September 21 02:29 EDT 2020. Contains 337266 sequences. (Running on oeis4.)