A062505 Numbers k such that if p is a prime that divides k, then either p + 2 or p - 2 is also prime.

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 25, 27, 29, 31, 33, 35, 39, 41, 43, 45, 49, 51, 55, 57, 59, 61, 63, 65, 71, 73, 75, 77, 81, 85, 87, 91, 93, 95, 99, 101, 103, 105, 107, 109, 117, 119, 121, 123, 125, 129, 133, 135, 137, 139, 143, 145, 147, 149, 151, 153, 155, 165

Offset

1,2

Comments

Multiplicative closure of twin primes (A001097).

References

Stephan Ramon Garcia and Steven J. Miller, 100 Years of Math Milestones: The Pi Mu Epsilon Centennial Collection, American Mathematical Society, 2019, pp. 35-37.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Example

35 is included because 35 = 5*7 and both (5+2) and (7-2) are primes.
65 = 5*13 where the factors are members of twin prime pairs: (3,5) and (11,13), therefore a(29) = 65 is a term; but 69 is not because 69 = 3*23 and 23 = A007510(2) is a single prime.

Mathematica

nmax = 15 (* corresponding to last twin prime pair (197, 199) *); tp[1] = 3; tp[n_] := tp[n] = (p = NextPrime[tp[n-1]]; While[ !PrimeQ[p+2], p = NextPrime[p]]; p); twins = Flatten[ Table[ {tp[n], tp[n]+2}, {n, 1, nmax}]]; max = Last[twins]; mult[twins_] := Select[ Union[ twins, Apply[ Times, Tuples[twins, {2}], {1}]], # <= max & ]; A062505 = Join[{1}, FixedPoint[mult, twins] ] (* Jean-François Alcover, Feb 23 2012 *)

Prog

(Magma) [k:k in [1..170] | forall{p:p in PrimeDivisors(k)| IsPrime(p+2) or IsPrime(p-2)}]; // Marius A. Burtea, Dec 30 2019

Crossrefs

Range of A072963.

Cf. A001097, A074480.

Sequence in context: A158333 A293167 A357377 * A230104 A093031 A378175

Adjacent sequences: A062502 A062503 A062504 * A062506 A062507 A062508

Keyword

nonn

Author

Leroy Quet, Jul 09 2001

Status

approved