cp's OEIS Frontend

A051904 Minimal exponent in prime factorization of n.

%I A051904 #67 Mar 24 2025 08:55:09
%S A051904 0,1,1,2,1,1,1,3,2,1,1,1,1,1,1,4,1,1,1,1,1,1,1,1,2,1,3,1,1,1,1,5,1,1,
%T A051904 1,2,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,6,1,1,1,1,
%U A051904 1,1,1,2,1,1,1,1,1,1,1,1,4,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1
%N A051904 Minimal exponent in prime factorization of n.
%C A051904 The asymptotic mean of this sequence is 1 (Niven, 1969). - _Amiram Eldar_, Jul 10 2020
%C A051904 Let k = A007947(n), then for n > 1 k^a(n) is the greatest power of k which divides n; see example. - _David James Sycamore_, Sep 07 2023
%H A051904 Daniel Forgues, <a href="/A051904/b051904.txt">Table of n, a(n) for n = 1..100000</a>
%H A051904 Cao Hui-Zhong, <a href="http://www.math.bas.bg/infres/MathBalk/MB-05/MB-05-105-108.pdf">The Asymptotic Formulas Related to Exponents in Factoring Integers</a>, Math. Balkanica, Vol. 5 (1991), Fasc. 2.
%H A051904 Ivan Niven, <a href="https://doi.org/10.1090/S0002-9939-1969-0241373-5">Averages of Exponents in Factoring Integers</a>, Proc. Amer. Math. Soc., Vol. 22, No. 2 (1969), pp. 356-360.
%H A051904 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/NivensConstant.html">Niven's Constant</a>
%H A051904 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A051904 a(n) = min_{k=1..A001221(n)} A124010(n,k). - _Reinhard Zumkeller_, Aug 27 2011
%F A051904 a(1) = 0, for n > 1, if A001221(n) = 1 (when n is in A000961), a(n) = A001222(n), otherwise a(n) = min(A067029(n), a(A028234(n))). - _Antti Karttunen_, Jul 12 2017
%F A051904 Sum_{k=1..n} a(k) ~ n + zeta(3/2)*n^(1/2)/zeta(3) + (zeta(2/3)/zeta(2) + c0)*n^(1/3), where c0 = A362974 = Product_{p prime} (1 + 1/p^(4/3) + 1/p^(5/3)) [Cao Hui-Zhong, 1991]. - _Vaclav Kotesovec_, Mar 24 2025
%e A051904 For n = 72 = 2^3*3^2, a(72) = min(exponents) = min(3,2) = 2.
%e A051904 For n = 72, using alternative definition: rad(72) = 6; and 6^2 = 36 divides 72 but no higher power of 6 divides 72, so a(72) = 2.
%e A051904 For n = 432, rad(432) = 6 and 6^3 = 216 divides 432 but no higher power of 6 divides 432, therefore a(432) = 3. - _David James Sycamore_, Sep 08 2023
%p A051904 a := proc (n) if n = 1 then 0 else min(seq(op(2, op(j, op(2, ifactors(n)))), j = 1 .. nops(op(2, ifactors(n))))) end if end proc: seq(a(n), n = 1 .. 100); # _Emeric Deutsch_, May 20 2015
%t A051904 Table[If[n == 1, 0, Min @@ Last /@ FactorInteger[n]], {n, 100}] (* _Ray Chandler_, Jan 24 2006 *)
%o A051904 (Haskell)
%o A051904 a051904 1 = 0
%o A051904 a051904 n = minimum $ a124010_row n  -- _Reinhard Zumkeller_, Jul 15 2012
%o A051904 (PARI) a(n)=vecmin(factor(n)[,2]) \\ _Charles R Greathouse IV_, Nov 19 2012
%o A051904 (Scheme) (define (A051904 n) (cond ((= 1 n) 0) ((= 1 (A001221 n)) (A001222 n)) (else (min (A067029 n) (A051904 (A028234 n)))))) ;; _Antti Karttunen_, Jul 12 2017
%o A051904 (Python)
%o A051904 from sympy import factorint
%o A051904 def a(n):
%o A051904     f = factorint(n)
%o A051904     l = [f[p] for p in f]
%o A051904     return 0 if n == 1 else min(l)
%o A051904 print([a(n) for n in range(1, 51)]) # _Indranil Ghosh_, Jul 13 2017
%Y A051904 Cf. A000961, A001221, A005361, A008479, A051903, A052409, A076558, A124010, A362974.
%Y A051904 Cf. A007947.
%K A051904 nonn,easy
%O A051904 1,4
%A A051904 _Labos Elemer_, Dec 16 1999