A211376 a(n) is the smallest 5-smooth number k such that both prime(n) - k and prime(n) + k are prime.

2, 4, 6, 6, 6, 12, 6, 12, 12, 6, 12, 24, 6, 6, 12, 18, 6, 12, 6, 18, 24, 18, 30, 12, 6, 6, 30, 24, 24, 18, 30, 12, 18, 12, 6, 36, 30, 6, 12, 18, 60, 30, 30, 72, 12, 60, 30, 48, 6, 12, 30, 12, 6, 6, 12, 60, 6, 12, 54, 24, 24, 48, 36, 36, 18, 30, 36, 18, 6, 90

Offset: 3

Author

Lei Zhou, Feb 07 2013

Keywords

• nonn
• easy

Comments

The three numbers prime(n) - k, prime(n), prime(n) + k are an arithmetic progression of primes.

Conjecture: a(n) is defined for all integers n > 2.

Conjecture confirmed true up to n = 300000, no exceptions.

Note that if (p1, n, p2) is an arithmetic progression where p1 and p2 are prime, then 2n = p1 + p2 is a Goldbach pair. There are numbers n such that no such sequence (p1, n, p2) exists for which the common difference n - p1 = p2 - n is 5-smooth. The first such number is 90. The first such odd number is 1845.

a(n) is defined for 3 <= n <= 10^7. - David A. Corneth, Jul 10 2021

Examples

			Let n = 43. The 43rd prime is 191, and 191-42 = 149 and 191+42 = 233 are both prime. However, 42 = 2*3*7 is not a 5-smooth number, so a(43) != 42. But 191-60 = 31 and 191+60 = 251 are both prime numbers, and 60 = 2^2*3*5 is the smallest such 5-smooth number. So a(43) = 60.
		

Links

• Lei Zhou, Table of n, a(n) for n = 3..10000

Crossrefs

Cf. A078611, A051037, A001031.

Programs

Mathematica

Table[p=Prime[i];j=0;While[j=j+2;If[(PrimeQ[p-j])&&(PrimeQ[p+j]), f=Last[FactorInteger[j]][[1]],f=p];f>5];j,{i,3,72}]