A060330 Primes that are the sum of five consecutive composite numbers.

37, 53, 67, 83, 97, 157, 293, 307, 353, 367, 503, 547, 683, 743, 757, 907, 953, 967, 983, 997, 1193, 1553, 1567, 1733, 1747, 2153, 2617, 2843, 2857, 3083, 3203, 3217, 3307, 4057, 4133, 4283, 4297, 5107, 5153, 5167, 5303, 6143, 6397, 6607, 7253, 7417

Offset: 1

Author

Robert G. Wilson v, Mar 30 2001

Keywords

• nonn

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: 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

Crossrefs

Subsequence of A007510.

Programs

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

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

Formula

a(n) >> n log^3 n. - Charles R Greathouse IV, Jan 23 2025