A065873 Numbers k such that usigma(k) +-1 are twin primes, where usigma(k) is the sum of the unitary divisors of k (A034448).

3, 5, 6, 10, 11, 17, 18, 20, 26, 29, 30, 38, 41, 44, 45, 46, 51, 55, 56, 59, 71, 80, 85, 88, 90, 98, 101, 105, 107, 114, 116, 118, 126, 132, 137, 140, 141, 145, 149, 150, 152, 153, 155, 158, 160, 161, 177, 178, 179, 185, 188, 191, 197, 203, 206, 207, 209, 212, 227

Offset

1,1

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Example

3 is a term since usigma(3) = 4 and (4-1, 4+1) = (3, 5) are twin primes.

Mathematica

f[n_] := Block[ {a = FactorInteger[n], k = l = s = 1}, l = Length[a]; While[k <= l, s = s * (a[[k, 1]]^a[[k, 2]] + 1); k++ ]; Return[s]]; Select[ Range[250], PrimeQ[ f[ # ] + 1] && PrimeQ[ f[ # ] - 1] & ]

Prog

(PARI) usigma(n)= { local(f, s=1); f=factor(n); for(i=1, matsize(f)[1], s*=1 + f[i, 1]^f[i, 2]); return(s) }
{ n=0; for (m=1, 10^9, u=usigma(m); if (isprime(u - 1) && isprime(u + 1), write("b065873.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Nov 02 2009
(Magma) usigma:=func<n|&+[d:d in Divisors(n)| Gcd(d, n div d) eq 1]>; [k:k in [1..230]| IsPrime(a) and (NextPrime(a)-2 eq a) where a is usigma(k)-1]; // Marius A. Burtea, Jan 16 2020

Crossrefs

Cf. A001097, A001359, A014574, A006512, A034448.

Sequence in context: A190243 A187127 A187126 * A072282 A028727 A257346

Adjacent sequences: A065870 A065871 A065872 * A065874 A065875 A065876

Keyword

nonn

Author

Robert G. Wilson v, Dec 07 2001

Status

approved