A211678 Twin primes p, p+2 with unique values of sigma(p) and sigma(p+2); sigma(n) = A000203(n) = sum of divisors of n.

3, 5, 7, 197, 199, 281, 283, 347, 349, 461, 463, 641, 643, 821, 823, 857, 859, 1289, 1291, 1697, 1699, 1721, 1723, 1787, 1789, 1877, 1879, 2081, 2083, 2141, 2143, 2381, 2383, 2549, 2551, 2801, 2803, 3257, 3259, 3539, 3541, 3557, 3559, 3929, 3931, 4019, 4021

Offset

1,1

Links

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

Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems: Inversion of Multiplicative Functions (invphi.gp).

Example

Twin primes 197 and 199 are in sequence because sigma(197) = 198, sigma(199) = 200 and there are no other numbers m, n with sigma(m) = 198 or sigma(n) = 200.

Mathematica

d = DivisorSigma[1, Range[4100]]; t = Transpose[Select[Tally[Sort[d]], #[[2]] == 1 && #[[1]] <= Length[d] &]][[1]]; t2 = Sort[Flatten[Table[Position[d, i], {i, t}]]]; t3 = Select[t2, PrimeQ]; tp = {}; Do[If[t3[[i + 1]] - t3[[i]] == 2 && DivisorSigma[1, t3[[i]]] != DivisorSigma[1, t3[[i + 1]]], AppendTo[tp, t3[[i]]]; AppendTo[tp, t3[[i]] + 2]], {i, Length[t3] - 1}]; Union[tp] (* T. D. Noe, Apr 26 2012 *)

Prog

(PARI) is(k) = isprime(k) && invsigmaNum(sigma(k)) == 1 && ((isprime(k+2) && invsigmaNum(sigma(k+2)) == 1) || (isprime(k-2) && invsigmaNum(sigma(k-2)) == 1)); \\ Amiram Eldar, Aug 08 2024, using Max Alekseyev's invphi.gp

Crossrefs

Subsequence of A211656 and A211660.

Cf. A211767 (lesser of twin primes p, p+2 with unique values of sigma(p) and sigma(p+2)), A211769 (greater of twin primes p, p+2 with unique values of sigma(p) and sigma(p+2)).

Cf. A000203.

Sequence in context: A114366 A088092 A174271 * A082756 A268693 A068832

Adjacent sequences: A211675 A211676 A211677 * A211679 A211680 A211681

Keyword

nonn

Author

Jaroslav Krizek, Apr 20 2012

Status

approved