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A372406
a(n) is the size of the largest set of positive integers S from 1..prime(n)-1 such that for any subset R of S, Sum {R} + prime(n) is prime.
1
1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 3, 4, 4, 4, 4, 3, 4, 4, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4
OFFSET
1,4
COMMENTS
This sequence is not monotonically increasing.
EXAMPLE
Let n=5, so p=prime(5)=11. From A070046, there are 3 positive integers x such that 1 <= x < 11 and 11+x is prime, which are {2, 6, 8}, so a(5) <= 3. Next, we see that 11 + 2 + 6 + 8 = 27 which is not prime so a(5) < 3. Last, we see that 11 + 2 + 6 = 19 is prime, and we already checked that 11 + 2 and 11 + 6 were prime, so S = {2, 6} and a(5) = 2.
11 is the first n such that a(n) = 3. Here, prime(11) = 31, and there are multiple sets which work. One is S = {6, 22, 30}.
31 + {} = 31 (empty set subset of S),
31 + 6 = 37,
31 + 22 = 53,
31 + 30 = 61,
31 + 6 + 22 = 59,
31 + 6 + 30 = 67,
31 + 22 + 30 = 83,
31 + 6 + 22 + 30 = 89, all of which are prime.
28 is the first n such that a(n) = 4. Here, prime(28) = 107, and there are multiple sets which work. One is S = {2, 30, 42, 90}.
MAPLE
f:= proc(n)
local k, p, C, S, s, t, q;
p:= ithprime(n);
C:= select(isprime, [$p+1 .. 2*p-1]) -~ p;
S[1]:= map(t -> [{t}, {0, t}], C);
for k from 2 do
S[k]:= NULL;
for s in S[k-1] do
for t in select(`>`, C, max(s[1])) do
q:= s[2] +~ t;
if andmap(isprime, q +~ p) then
S[k]:= S[k], [s[1] union {t}, s[2] union q] ;
fi
od od;
S[k]:= {S[k]};
if S[k] = {} then return k-1 fi
od
end proc:
map(f, [$1..90]); # Robert Israel, May 06 2024
MATHEMATICA
nmax = 87; a372406 = {{1, 1}};
For[n = 2, n <= nmax, n++, d = {}; p = Prime[n];
For[a = 2, a < p, a += 2, If[PrimeQ[p + a], AppendTo[d, a]]]; q = 1; k = 0;
While[q == 1 && k <= Length[d], k++; su = Subsets[d, {k}];
For[i = 1, i <= Length[su], i++, s = su[[i]];
If[PrimeQ[Total[s] + p], y = Subsets[s]; t = 1;
For[z = 1, z <= Length[y], z++,
If[CompositeQ[Total[y[[z]]] + p], t = 0; q = 0; Break[]]];
If[t == 1, q = 1; Break[]], q = 0]]];
AppendTo[a372406, {n, k - 1}]]
Print[a372406]
CROSSREFS
Cf. A070046.
Sequence in context: A055980 A076080 A134914 * A329195 A204164 A257639
KEYWORD
nonn
AUTHOR
Samuel Harkness, Apr 29 2024
STATUS
approved