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A288189 a(n) is the smallest composite number whose sum of prime divisors (with multiplicity) is divisible by prime(n).

%I A288189 #42 Apr 15 2018 03:38:44
%S A288189 4,8,6,10,28,22,52,34,76,184,58,213,148,82,172,309,424,118,393,268,
%T A288189 142,584,316,664,573,388,202,412,214,436,753,508,813,274,1465,298,933,
%U A288189 974,652,1336,1384,358,1137,382,772,394,1257,1329,892,454,916,1864,478,1497,1538,1569
%N A288189 a(n) is the smallest composite number whose sum of prime divisors (with multiplicity) is divisible by prime(n).
%C A288189 In most cases a(n) = A288814(prime(n)) but there are exceptions, e.g., a(37)=213, whereas A288814(37)=248. Other exceptions include a(53), a(67), a(127), a(137), etc. These examples occur when there is a number r such that A001414(r*p) is less than A288814(p).
%C A288189 The strictly increasing subsequence of terms (10, 22, 34, 58, 82, 118, 142, 202, 214, 274, 298, ...) where for all m>n, a(m)>a(n) gives the semiprimes with prime sum of prime factors, A108605. The sequence of the indices of this subsequence (5, 7, 13, 19, 31, 43, 61, 73, 103, 109, 139, 151, ...) gives the greater of twin primes, A006512.
%H A288189 Charles R Greathouse IV, <a href="/A288189/b288189.txt">Table of n, a(n) for n = 1..10000</a>
%e A288189 a(5)=6 because 6 = 2*3 is the smallest number whose sum of prime divisors (2+3 = 5) is divisible by 5.
%e A288189 a(37) = 213 = A288814(74) = A288814(2*37).
%t A288189 With[{s = Array[Boole[CompositeQ@ #] Total@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger@ #] &, 10^4] /. 0 -> ""}, Table[FirstPosition[s, _?(Mod[#, p] == 0 &)][[1]], {p, Prime@ Range@ 56}]] (* _Michael De Vlieger_, Apr 14 2018 *)
%o A288189 (PARI) sopfr(k) = my(f=factor(k)); sum(j=1, #f~, f[j, 1]*f[j, 2]);
%o A288189 a(n) = my(pn=prime(n)); forcomposite(c=pn, , if (sopfr(c) % pn == 0, return(c))); \\ _Michel Marcus_, Jul 03 2017
%Y A288189 Cf. A000040, A001414, A006512, A056240, A108605, A288814.
%K A288189 nonn
%O A288189 1,1
%A A288189 _David James Sycamore_, Jul 01 2017