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A267000 a(n) is the smallest m such that A001414(m)=n and ((m=0) mod n) and m/n is both squarefree and prime to n, or 0 if no such m exists.

%I A267000 #15 Apr 06 2021 02:57:59
%S A267000 2,3,4,5,0,7,0,0,30,11,60,13,70,105,240,17,0,19,220,231,0,23,0,650,
%T A267000 286,1755,476,29,2730,31,1824,627,3570,805,4788,37,646,897,1160,41,
%U A267000 8778,43,1276,11385,8970,47,1776,36309,10850,1581,41860,53,2322,4070,2408,45885,16530,59
%N A267000 a(n) is the smallest m such that A001414(m)=n and ((m=0) mod n) and m/n is both squarefree and prime to n, or 0 if no such m exists.
%C A267000 The offset is 2 like A056240 since there is no number m with A001414(m) = 1
%C A267000 Alladi and Erdős state that there is only a finite number of zeros in this sequence.
%C A267000 When a(n) is not zero, A056240(n) <= a(n); a(n) <= A000792(n).
%H A267000 Amiram Eldar, <a href="/A267000/b267000.txt">Table of n, a(n) for n = 2..3000</a>
%H A267000 K. Alladi and P. Erdős, <a href="http://projecteuclid.org/euclid.pjm/1102811427">On an additive arithmetic function</a>, Pacific J. Math., Volume 71, Number 2 (1977), 275-294.
%F A267000 a(p) = p, for p prime.
%e A267000 a(10) = 30 since A001414(30)=10 and 30 is divisible by 10, and 30/10=3 is squarefree and prime to 10.
%o A267000 (PARI) sopfr(n) = {my(f=factor(n)); sum(k=1, #f~, f[k, 1]*f[k, 2]); }
%o A267000 first(n) = {my(k=1); while (sopfr(k) != n, k++); k;}
%o A267000 last(n) = polcoeff((1+x+2*x^2+x^4)/(1-3*x^3) + O(x^(n + 3)), n);
%o A267000 a(n) = {na = first(n); nb = last(n); for (m=na, nb, if ((sopfr(m) == n) && (! (m % n)) && issquarefree(m/n) && (gcd(m/n, n) == 1), return(m)););}
%Y A267000 Cf. A000792, A001414, A036844, A056240, A064364.
%K A267000 nonn
%O A267000 2,1
%A A267000 _Michel Marcus_, Jan 08 2016