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A349093 a(n) is the smallest nonprime number m (m = A018252(t)) such that n divides the product P(t) of all nonprime numbers up to and including m (P(t) = A036691(t-1)).

%I A349093 #10 Feb 16 2025 08:34:02
%S A349093 1,4,6,4,10,6,14,6,9,10,22,6,26,14,10,8,34,9,38,10,14,22,46,6,15,26,9,
%T A349093 14,58,10,62,8,22,34,14,9,74,38,26,10,82,14,86,22,10,46,94,8,21,15,34,
%U A349093 26,106,9,22,14,38,58
%N A349093 a(n) is the smallest nonprime number m (m = A018252(t)) such that n divides the product P(t) of all nonprime numbers up to and including m (P(t) = A036691(t-1)).
%C A349093 a(n) >= 2*gpf(n) for n > 1, where gpf(n) denotes the greatest prime factor of n (A006530(n)).
%C A349093 Conjecture: the equation a(n) = a(n+1) has no solutions. This holds up to at least n = 10^7.
%C A349093 Consecutive solutions of the equation a(n) = 2*K(n) (where K(n) is the Kempner number A002034(n)) are consecutive terms of A048839.
%H A349093 J. Sondow and E. W. Weisstein, <a href="https://mathworld.wolfram.com/SmarandacheFunction.html">MathWorld: Smarandache Function</a>
%F A349093 a(p) = 2*p for prime p.
%F A349093 a(p_1*p_2*...*p_u) = 2*p_u, where p_i's are distinct primes and p_1 < p_2 < ... < p_u.
%F A349093 a(n) where n is factored as n = p_1^k_1*p_2^k_2*...*p_u^k_u is given by a(n) = max( a(p_1^k_1), a(p_2^k_2), ..., a(p_u^k_u) ), where a(p_i^k_i) = w*p_i and w is the smallest m >= 2 satisfying the inequality:
%F A349093 -1 + Sum_{t=1..m} floor((m*p_i)/(p_i)^t) >= k_i.
%e A349093 a(15) = 10 because:
%e A349093 15 does not divide 1=A036691(0)=1, 1*4=A036691(1)=4, 1*4*6=A036691(2)=24, 1*4*6*8=A036691(3)=192, 1*4*6*8*9=A036691(4)=1728 and does divide 1*4*6*8*9*10=A036691(5)=17280.
%o A349093 (Maxima)
%o A349093 f(p,k):=(z:2, for m:2 while -1+sum(floor((p*m)/(p^t)),t,1,m)<k do (z:z+1), z*p);
%o A349093 a(n):=(for d:1 thru length(ifactors(n)) do (A[d]:f(ifactors(n)[d][1],ifactors(n)[d][2])), lmax(makelist(A[d],d,1,length(ifactors(n)))));
%o A349093 makelist(a(n),n,2,200);
%Y A349093 Cf. A002034, A006530, A018252, A036691.
%K A349093 nonn
%O A349093 1,2
%A A349093 _Lechoslaw Ratajczak_, Mar 25 2022