A127925 Primes p such that 2p < prime(k-i) + prime(k+i) for i=1..k-1, where p=prime(k).

3, 7, 19, 23, 43, 47, 73, 109, 113, 199, 283, 293, 313, 317, 463, 467, 503, 509, 523, 619, 661, 683, 691, 887, 1063, 1069, 1109, 1129, 1303, 1307, 1321, 1327, 1613, 1621, 1627, 1637, 1669, 1789, 2143, 2161, 2383, 2393, 2399, 2477, 2731, 2753, 2803, 2861, 2971

Offset

1,1

Comments

One of several sets of "good primes" in section A14 of Guy.

McNew calls these numbers "midpoint convex primes". - Peter Munn, Jul 04 2025

From Thomas Ordowski, Jun 01 2026: (Start)

Equivalently, numbers k > 1 such that 2*pi(k) > pi(k-m) + pi(k+m) for every 0 < m < k, where the pi function is A000720.

Conjecture: these are the numbers k > 1 for which 2*pi(k) = pi(k-m) + pi(k+m) has no solution 0 < m < k.

The conjecture is true if and only if for every composite number k this equation has such a solution m. (End)

References

R. K. Guy, Unsolved Problems in Number Theory, 3rd ed. Springer, 2004.

Links

T. D. Noe, Table of n, a(n) for n=1..10001

Nathan McNew, Popular values of the largest prime divisor function (corrected version), page 16, November 2015.

Mathematica

t={}; n=1; While[Length[t]<100, n++; p=Prime[n]; i=1; While[i<n && 2p<Prime[n-i]+Prime[n+i], i++ ]; If[i==n, AppendTo[t, p]]]; t

Crossrefs

Cf. A028388.

A246033 is a subset.

Subset of A124661, A178954.

Sequence in context: A018432 A381590 A205302 * A032388 A050866 A226923

Adjacent sequences: A127922 A127923 A127924 * A127926 A127927 A127928

Keyword

nonn

Author

T. D. Noe, Feb 06 2007

Status

approved