%I #18 May 09 2024 03:11:31

%S 6,4,30,1530,40530,37838430,900569670,781767956970

%N The smallest composite number k that shares exactly n distinct prime factors with sopfr(k), the sum of the primes dividing k, with repetition.

%C Conjecture: A001221(a(n)) = n+1, for n >= 2. - _Daniel Suteu_, May 08 2024

%e a(0) = 6 as 6 = 2 * 3 while sopfr(6) = 5, which shares 0 distinct prime factors with 6.

%e a(1) = 4 as 4 = 2 * 2 while sopfr(4) = 4 = 2 * 2, which shares 1 distinct prime factor, 2, with 4.

%e a(2) = 30 as 30 = 2 * 3 * 5 while sopfr(30) = 10 = 2 * 5, which shares 2 distinct prime factors, 2 and 5, with 30.

%e a(3) = 1530 as 1530 = 2 * 3^2 * 5 * 17 while sopfr(1530) = 30 = 2 * 3 * 5, which shares 3 distinct primes factors, 2, 3 and 5, with 1530.

%e a(4) = 40530 as 40530 = 2 * 3 * 5 * 7 * 193 while sopfr(40530) = 210 = 2 * 3 * 5 * 7, which shares 4 distinct prime factors, 2, 3, 5 and 7, with 40530.

%e a(5) = 37838430 as 37838430 = 2 * 3^2 * 5 * 7 * 17 * 3533 while sopfr(37838430) = 3570 = 2 * 3 * 5 * 7 * 17, which shares 5 distinct prime factors, 2, 3, 5, 7 and 17, with 37838430.

%e a(6) = 900569670 as 900569670 = 2 * 3 * 5 * 7 * 11 * 13 * 29989 while sopfr(900569670) = 30030 = 2 * 3 * 5 * 7 * 11 * 13, which shares 6 distinct prime factors, 2, 3, 5, 7, 11 and 13, with 900569670.

%Y Cf. A001414, A370354, A002808, A027746.

%K nonn,more,new

%O 0,1

%A _Scott R. Shannon_, May 05 2024

%E a(7) from _Daniel Suteu_, May 08 2024