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A211376
a(n) is the smallest 5-smooth number k such that both prime(n) - k and prime(n) + k are prime.
2
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
3,1
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
EXAMPLE
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.
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}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Lei Zhou, Feb 07 2013
STATUS
approved