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

1,1

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.

Links

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

Example

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).

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A011515 A005531 A328278 * A335159 A064494 A119800

Adjacent sequences: A288186 A288187 A288188 * A288190 A288191 A288192

Keyword

nonn

Author

David James Sycamore, Jul 01 2017

Status

approved