OFFSET
1,1
COMMENTS
Conjecture: all these primes are isolated primes (A007510). - Davide Rotondo, Dec 31 2024
Stronger conjecture: all p are 7 or 23 mod 30. - Charles R Greathouse IV, Jan 21 2025
Above conjectures are true. Proof sketch: If n + n+1 + n+2 + n+3 + n+4 = 5n+10, so there must be at least one prime sandwiched between the five composite numbers. If p and p+4 are prime, then p-1 + p+1 + p+2 + p+3 + p+5 = 5p + 10 is composite. If neither p-2 nor p+2 are prime, the sums p-4 + p-3 + p-2 + p-1 + p+1, etc., are 5p-9, 5p-3, 5p+3, and 5p+9 which are even for p > 2 (and p = 2 does not work). So we must have p and p+2 prime, which yield p-3 + p-2 + p-1 + p+1 + p+3 = 5p-2, p-2 + p-1 + p+1 + p+3 + p+4 = 5p+5, and p-1 + p+1 + p+3 + p+4 + p+5 = 5p+12. 5p+5 is composite, but the others can work. Now note that the first form yields only 5p-2 = 23 mod 30 and the second 5p+12 = 7 mod 30. - Charles R Greathouse IV, Jan 23 2025
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
FORMULA
a(n) >> n log^3 n. - Charles R Greathouse IV, Jan 23 2025
MATHEMATICA
composite[ n_Integer ] := (k = n + PrimePi[ n ] + 1; While[ k - PrimePi[ k ] - 1 != n, k++ ]; k); a = {}; Do[ p = composite[ n ] + composite[ n + 1 ] + composite[ n + 2 ] + composite[ n + 3 ] + composite[ n + 4 ]; If[ PrimeQ[ p ], a = Append[ a, p ] ], {n, 1, 1500} ]; a
PROG
(PARI) list(lim)=my(v=List(), u=[4, 6, 8, 9, 0], i=5); forcomposite(n=10, lim\1, u[i]=n; if(i++>5, i=1); my(p=vecsum(u)); if(p>lim, break); if(isprime(p), listput(v, p))); Vec(v) \\ Charles R Greathouse IV, Dec 27 2024
(PARI) ok(p)=p=p%30; p==11 || p==17 || p==29
list(lim)=my(v=List([37]), p=11); forprime(q=13, (lim+12)\5, if(q-p>2 || !ok(p), p=q; next); if(isprime(5*p-2), listput(v, 5*p-2)); if(isprime(5*p+12), listput(v, 5*p+12)); p=q); if(v[#v]>lim, listpop(v)); Vec(v) \\ Charles R Greathouse IV, Jan 23 2025
CROSSREFS
KEYWORD
nonn
AUTHOR
Robert G. Wilson v, Mar 30 2001
STATUS
approved