A288814 a(n) is the smallest composite number whose prime divisors (with multiplicity) sum to n.

4, 6, 8, 10, 15, 14, 21, 28, 35, 22, 33, 26, 39, 52, 65, 34, 51, 38, 57, 76, 95, 46, 69, 92, 115, 184, 161, 58, 87, 62, 93, 124, 155, 248, 217, 74, 111, 148, 185, 82, 123, 86, 129, 172, 215, 94, 141, 188, 235, 376, 329, 106, 159, 212, 265, 424, 371, 118, 177, 122, 183, 244, 305, 488, 427, 134, 201, 268, 335, 142

Offset: 4

Author

David James Sycamore, Jun 16 2017

Keywords

• nonn

Comments

Agrees with A056240(n) if n is composite (but not if n is prime).

For n prime, let P_n = greatest prime < n such that A056240(n-P_n) = A288313(m) for some m; then a(n) = Min{q*a(n-q): q prime, n-1 > q >= P_n}.

In most cases q is the greatest prime < p, but there are exceptions; e.g., p=211 is the smallest prime for which q (=197) is the second prime removed from 211, not the first. 541 is the next prime with this property (q=521). The same applies to p=16183, for which q=16139, the second prime removed from p. These examples all arise with q being the lesser of a prime pair.

For p prime, a(p) = q*a(p-q) for some prime q < p as described above. Then a(p-q) = 2,4,8 or 3*r for some prime r.

The subsequence of terms (4, 6, 8, 10, 14, 21, 22, 26, 34, ...), where for all m > n, a(m) > a(n) is the same as sequence A088686, and the sequence of its indices (4, 5, 6, 7, 9, 10, 13, 19, ...) is the same as A088685. - David James Sycamore, Jun 30 2017

Records are in A088685. - Robert G. Wilson v, Feb 26 2018

Number of terms less than 10^k, k=1,2,3,...: 3, 32, 246, 2046, 17053, 147488, ..., . - Robert G. Wilson v, Feb 26 2018

Examples

			a(5) = 6 = 2*3 is the smallest composite number whose prime divisors add to 5.
a(7) = 10 = 2*5 is the smallest composite number whose prime divisors add to 7.
12 = 2 * 2 * 3 is not in the sequence, since the sum of its prime divisors is 7, a value already obtained by the lesser 10. - _David A. Corneth_, Jun 22 2017
		

Links

• David A. Corneth, Table of n, a(n) for n = 4..10003 (terms 4..1000 from Michel Marcus)
• David A. Corneth, PARI program

Crossrefs

Cf. A046343, A056240, A088685, A288313.

Programs

Maple

N:= 100: # to get a(4)..a(N)
V:= Array(4..N): count:= 0:
for k from 4 while count < N-3 do
  if isprime(k) then next fi;
  s:= add(t[1]*t[2], t = ifactors(k)[2]);
if s <= N and V[s]=0 then
    V[s]:= k; count:= count+1;
fi
od:
convert(V,list); # Robert Israel, Feb 26 2018
# alternative
A288814 := proc(n)
    local k ;
    for k from 1 do
        if not isprime(k) and A001414(k) = n then
            return k ;
        end if;
    end do:
end proc:
seq(A288814(n),n=4..80) ; # R. J. Mathar, Apr 15 2024

Mathematica

Function[s, Table[FirstPosition[s, ?(# == n &)][[1]], {n, 4, 73}]]@ Table[Boole[CompositeQ@ n] Total@ Flatten@ Map[ConstantArray[#1, #2] & @@ # &, FactorInteger[n]], {n, 10^3}] (* _Michael De Vlieger, Jun 19 2017 *)
f[n_] := If[ PrimeQ@ n, 0, spf = Plus @@ Flatten[ Table[#1, {#2}] & @@@ FactorInteger@ n]]; t[] := 0; k = 1; While[k < 500, If[ t[f[k]] == 0, t[f[k]] = k]; k++]; t@# & /@ Range[4, 73] (* _Robert G. Wilson v, Feb 26 2018 *)

PARI

isok(k, n) = my(f=factor(k)); sum(j=1, #f~, f[j,1]*f[j,2]) == n;
a(n) = forcomposite(k=1,, if (isok(k, n), return(k))); \\ Michel Marcus, Jun 21 2017

PARI

lista(n) = {my(res = vector(n), s, todo); if(n < 4, return([]), todo = n-3); forcomposite(k=4, , f=factor(k); s = sum(j=1, #f~, f[j, 1]*f[j, 2]); if(s<=n, if(res[s]==0, res[s]=k; todo--; if(todo==0, return(vector(n-3, i, res[i+3]))))))} \\ David A. Corneth, Jun 21 2017

PARI

See PARI-link \\ David A. Corneth, Mar 23 2018