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a(n) is the smallest composite number > a(n-1) for which the sum of the prime factors (counted with multiplicity) is n.
1

%I #30 Jan 06 2019 19:30:53

%S 4,6,8,10,15,20,21,28,35,56,70,105,140,147,196,198,220,231,260,273,

%T 364,455,459,510,513,570,676,798,1064,1122,1242,1254,1288,1463,1976,

%U 2057,2277,2299,2320,2436,2480,2527,2552,2622,2728,2997,3224,3509,3885,4147,4216,4433,4592,4810,4816,5412,5661,5676,5875,6253

%N a(n) is the smallest composite number > a(n-1) for which the sum of the prime factors (counted with multiplicity) is n.

%C Conjecture: a(n) exists for every n >= 4.

%e For n=11, a(11)=28 because 28 is composite, the factors of 28=2*2*7 are (2,2,7 with multiplicity), and their sum is 11. The composite numbers 40,48,and 54 have the same sum of factors, but we take the smallest > a(10)=21, that is, 28.

%o (PARI) {

%o k=4;

%o for(n=1,10000,

%o if(!isprime(n),

%o f=factor(n);

%o s=sum(i=1, matsize(f)[1], f[i, 1]*f[i, 2]);

%o if(s==k,

%o print1(n", ")

%o ;k++

%o )

%o )

%o )

%o }

%Y Cf. A046343

%K nonn

%O 4,1

%A _Dimitris Valianatos_, Jan 12 2017