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A070003 Numbers divisible by the square of their largest prime factor. 35
4, 8, 9, 16, 18, 25, 27, 32, 36, 49, 50, 54, 64, 72, 75, 81, 98, 100, 108, 121, 125, 128, 144, 147, 150, 162, 169, 196, 200, 216, 225, 242, 243, 245, 250, 256, 288, 289, 294, 300, 324, 338, 343, 361, 363, 375, 392, 400, 432, 441, 450, 484, 486, 490, 500, 507 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Also numbers n such that P(phi(n)) - phi(P(n)) = 1, where P(x) is the largest prime factor of x. P(phi(n)) - phi(P(n)) = A006530(A000010(n)) - A000010(A006530(n)).

Also numbers n such that the value of commutator of phi and P functions at n is -1.

Equivalently, n such that n and phi(n) have the same largest prime factor since Phi(p)=p-1 if p is prime. - Benoit Cloitre, Jun 08 2002

Since n is divisible by P(n)^2, n cannot divide P(n)! and so A057109 is a supersequence. Hence all A002034(a(n)) are composite. - Jonathan Sondow, Dec 28 2004

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Paul Erdős and Ron L. Graham, On products of factorials, Bull. Inst. Math. Acad. Sinica 4:2 (1976), pp. 337-355. [alternate link]

Paul Erdős and Ilias Kastanas, Solution 6674:The smallest factorial that is a multiple of n, Amer. Math. Monthly 101 (1994) 179.

A. J. Kempner, Miscellanea, Amer. Math. Monthly, 25 (1918), 201-210. See Section II, "Concerning the smallest integer m! divisible by a given integer n."

Eric Weisstein's World of Mathematics, Greatest Prime Factor

Eric Weisstein's World of Mathematics, Totient Function

FORMULA

Erdős proved that there are x * exp(-(1 + o(1))sqrt(log x log log x)) members of this sequence up to x. - Charles R Greathouse IV, Mar 26 2012

MATHEMATICA

p[n_] := FactorInteger[n][[-1, 1]]; ep[n_] := EulerPhi[n]; fQ[n_] := p[ep[n]] == 1 + ep[p[n]]; Select[ Range[ 510], fQ] (* Robert G. Wilson v, Mar 26 2012 *)

Select[Range[500], FactorInteger[#][[-1, 2]] > 1 &] (* T. D. Noe, Dec 06 2012 *)

PROG

(PARI) for(n=3, 1000, if(component(component(factor(n), 1), omega(n))==component(component(factor(eulerphi(n)), 1), omega(eulerphi(n))), print1(n, ", ")))

(PARI) is(n)=my(f=factor(n)[, 2]); f[#f]>1 \\ Charles R Greathouse IV, Mar 21 2012

(PARI) sm(lim, mx)=if(mx==2, return(vector(log(lim+.5)\log(2)+1, i, 1<<(i-1)))); my(v=[1]); forprime(p=2, min(mx, lim), v=concat(v, p*sm(lim\p, p))); vecsort(v)

list(lim)=my(v=[]); forprime(p=2, sqrt(lim), v=concat(v, p^2*sm(lim\p^2, p))); vecsort(v) \\ Charles R Greathouse IV, Mar 27 2012

CROSSREFS

Subsequence of A122145.

Cf. A000010, A006530, A068211, A070777, A070812, A070002, A070004, A007283, A070813-A070816, A057109, A002034, A102067, A102068.

Sequence in context: A299117 A140269 A226385 * A073539 A090779 A166402

Adjacent sequences:  A070000 A070001 A070002 * A070004 A070005 A070006

KEYWORD

nonn

AUTHOR

Labos Elemer, May 07 2002

EXTENSIONS

New name from Jonathan Sondow and Charles R Greathouse IV, Mar 27 2012

STATUS

approved

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Last modified December 11 04:29 EST 2018. Contains 318049 sequences. (Running on oeis4.)