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A010846 Number of numbers <= n whose set of prime factors is a subset of the set of prime factors of n. 35
1, 2, 2, 3, 2, 5, 2, 4, 3, 6, 2, 8, 2, 6, 5, 5, 2, 10, 2, 8, 5, 7, 2, 11, 3, 7, 4, 8, 2, 18, 2, 6, 6, 8, 5, 14, 2, 8, 6, 11, 2, 19, 2, 9, 8, 8, 2, 15, 3, 12, 6, 9, 2, 16, 5, 11, 6, 8, 2, 26, 2, 8, 8, 7, 5, 22, 2, 10, 6, 20, 2, 18, 2, 9, 9, 10, 5, 23, 2, 14, 5, 9, 2, 28, 5, 9, 7, 11, 2, 32, 5, 10 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

This function of n appears in an ABC-conjecture by Andrew Granville. See Goldfeld. - T. D. Noe, Jun 30 2009

LINKS

T. D. Noe and Michael De Vlieger, Table of n, a(n) for n = 1..10000 (first 5000 terms from T. D. Noe)

Dorian Goldfled, Modular forms, elliptic curves, and the ABC conjecture

FORMULA

a(n) = |{k<=n, k|n^(tau(k)-1)}|. - Vladeta Jovovic, Sep 13 2006

a(n) = Sum_{j = 1..n} Product_{primes p | j} delta(n mod p,0) where delta is the Kronecker delta. - Robert Israel, Feb 09 2015

a(n) = Sum_{1<=k<=n,(n,k)=1} mu(k)*floor(n/k). - Benoit Cloitre, May 07 2016

a(n) = Sum_{k=1..n} floor(n^k/k)-floor((n^k -1)/k). - Anthony Browne, May 28 2016

EXAMPLE

From Wolfdieter Lang, Jun 30 2014: (Start)

a(1) = 1 because the empty set is a subset of any set.

a(6) = 5 from the five numbers: 1 with the empty set, 2 with the set {2}, 3 with {3}, 4 with {2} and 6 with {2,3}, which are all subsets of {2,3}. 5 is out because {5} is not a subset of {2,3}. (End)

From David A. Corneth, Feb 10 2015: (Start)

Let p# be the product of primes up to p, A002110. Then,

a(13#) = 1161

a(17#) = 4843

a(19#) = 19985

a(23#) = 83074

a(29#) = 349670

a(31#) = 1456458

a(37#) = 6107257

a(41#) = 25547835

(End)

MAPLE

A:= proc(n) local F, S, s, j, p;

  F:= numtheory:-factorset(n);

  S:= {1};

  for p in F do

    S:= {seq(seq(s*p^j, j=0..floor(log[p](n/s))), s=S)}

  od;

  nops(S)

end proc;

seq(A(n), n=1..1000); # Robert Israel, Jun 27 2014

MATHEMATICA

pf[n_] := If[n==1, {}, Transpose[FactorInteger[n]][[1]]]; SubsetQ[lst1_, lst2_] := Intersection[lst1, lst2]==lst1; Table[pfn=pf[n]; Length[Select[Range[n], SubsetQ[pf[ # ], pfn] &]], {n, 100}] (* T. D. Noe, Jun 30 2009 *)

Table[Total[MoebiusMu[#] Floor[n/#] &@ Select[Range@ n, CoprimeQ[#, n] &]], {n, 92}] (* Michael De Vlieger, May 08 2016 *)

PROG

(PARI) a(n, f=factor(n)[, 1])=if(#f>1, my(v=f[1..#f-1], p=f[#f], s); while(n>0, s+=a(n, v); n\=p); s, if(#f&&n>0, log(n+.5)\log(f[1])+1, n>0)) \\ Charles R Greathouse IV, Jun 27 2013

(PARI) a(n) = sum(k=1, n, if(gcd(n, k)-1, 0, moebius(k)*(n\k))) \\ Benoit Cloitre, May 07 2016

CROSSREFS

Cf. A162306 (numbers for each n).

Sequence in context: A100565 A244098 A285573 * A073023 A173754 A180125

Adjacent sequences:  A010843 A010844 A010845 * A010847 A010848 A010849

KEYWORD

nonn,easy

AUTHOR

Olivier Gérard

EXTENSIONS

Definition made more precise at the suggestion of Wolfdieter Lang

STATUS

approved

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Last modified November 17 00:14 EST 2018. Contains 317275 sequences. (Running on oeis4.)