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
Keywords
Examples
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.
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
Programs
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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 -
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 *)
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