OFFSET
4,1
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
LINKS
David A. Corneth, Table of n, a(n) for n = 4..10003 (terms 4..1000 from Michel Marcus)
David A. Corneth, PARI program
EXAMPLE
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
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
David James Sycamore, Jun 16 2017
STATUS
approved