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

4, 8, 6, 10, 28, 22, 52, 34, 76, 184, 58, 213, 148, 82, 172, 309, 424, 118, 393, 268, 142, 584, 316, 664, 573, 388, 202, 412, 214, 436, 753, 508, 813, 274, 1465, 298, 933, 974, 652, 1336, 1384, 358, 1137, 382, 772, 394, 1257, 1329, 892, 454, 916, 1864, 478, 1497, 1538, 1569

Offset: 1

Author

David James Sycamore, Jul 01 2017

Keywords

• nonn

Comments

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).

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.

Examples

			a(5)=6 because 6 = 2*3 is the smallest number whose sum of prime divisors (2+3 = 5) is divisible by 5.
a(37) = 213 = A288814(74) = A288814(2*37).
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A001414, A006512, A056240, A108605, A288814.

Programs

Mathematica

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 *)

PARI

sopfr(k) = my(f=factor(k)); sum(j=1, #f~, f[j, 1]*f[j, 2]);
a(n) = my(pn=prime(n)); forcomposite(c=pn, , if (sopfr(c) % pn == 0, return(c))); \\ Michel Marcus, Jul 03 2017