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A309158
The smallest prime, a(n), larger than prime(n) for which every even difference from 2 to prime(n) - 1 occurs at least once for some pair of primes from prime(n) to a(n) inclusive.
0
5, 11, 13, 23, 31, 47, 47, 53, 67, 67, 73, 101, 101, 107, 113, 131, 139, 151, 151, 151, 173, 179, 193, 193, 227, 227, 233, 241, 241, 283, 283, 293, 293, 313, 313, 353, 353, 353, 353, 397, 397, 397, 421, 421, 421, 461, 461, 467, 467, 503, 503, 503, 521, 563, 569, 599, 599
OFFSET
2,1
COMMENTS
The "prime differences prime" a(n) is the smallest prime greater than prime(n), n > 1, for which every even difference from 2 to prime(n)-1 occurs for some pair of primes from prime(n) to a(n) inclusive.
a(n) is at least prime(n) + (prime(n) - 1) = 2 * prime(n) - 1.
If the sequence of prime differences primes is infinite, there are infinitely many pairs of primes for each even difference. If there are only finitely many pairs of primes for some even difference, the sequence ends.
Ratios a(n)/prime(n), n = 2 to 15 are 1.67, 2.20, 1.86, 2.09, 2.38, 2.76, 2.47, 2.30, 2.31, 2.16, 1.97, 2.46, 2.35, 2.28.
Conjecture: The sequence is infinite.
Conjecture: There are finitely many values of n with a(n) = 2 * prime(n) - 1.
Conjecture: There are infinitely many values of n with a(n) = a(n-1).
Conjecture: For all n, a(n) <= 3 * prime(n). (This is true for n <= 101.)
EXAMPLE
For n = 4, prime(4) = 7 and 7 - 1 = 6. Check differences for 7 and 11: 11 - 7 = 4. For 7, 11, and 13: 11 - 7 = 4, 13 - 7 = 6, 13 - 11 = 2, so a(4) = 13.
Also prime(6) = 13, 13 - 1 = 12. For 13, 17, 19, 23, 29 and 31, 29 - 17 = 12, 23 - 13 = 10, 31 - 23 = 8, 19 - 13 = 6, 17 - 13 = 4, 19 - 17 = 2, and a(6) = 31.
MAPLE
for n from 2 to 58 do
a := ithprime(n):
for d from 2 by 2 to a - 1 do
p := ithprime(n);
while not isprime(p + d) do
p := nextprime(p)
od;
if p + d > a then a := p + d fi
od;
print(n, a)
od: # Peter Luschny, Jul 17 2019
MATHEMATICA
For [n=2, n <= 101, n++,
Clear[d]; d=0;
Clear[a]; a=Prime[n];
While[d < Prime[n]-1,
d=d+2;
Clear[m]; m=n;
While[CompositeQ[d+Prime[m]], m++];
If[d+Prime[m] > a, a=d+Prime[m]]];
Print[{n, Prime[n], a, N[a/Prime[n]]}]
]
CROSSREFS
KEYWORD
nonn
AUTHOR
Sally Myers Moite, Jul 14 2019
STATUS
approved