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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: A244098 A285573 A325339 * A073023 A173754 A180125 Adjacent sequences:  A010843 A010844 A010845 * A010847 A010848 A010849 KEYWORD nonn,easy AUTHOR EXTENSIONS Definition made more precise at the suggestion of Wolfdieter Lang STATUS approved

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Last modified October 18 01:04 EDT 2019. Contains 328135 sequences. (Running on oeis4.)