A233547 a(n) = |{0 < k < n/2: phi(k)*phi(n-k) - 1 and phi(k)*phi(n-k) + 1 are both prime}|, where phi(.) is Euler's totient function.

0, 0, 0, 0, 0, 1, 2, 1, 3, 4, 3, 2, 3, 2, 3, 1, 1, 2, 1, 5, 2, 3, 1, 2, 1, 1, 3, 4, 5, 4, 3, 2, 3, 2, 5, 2, 5, 5, 3, 5, 3, 1, 5, 3, 7, 6, 3, 2, 4, 7, 5, 1, 4, 6, 6, 5, 2, 4, 6, 9, 9, 6, 8, 5, 8, 8, 6, 6, 9, 4, 8, 6, 8, 5, 7, 9, 7, 9, 5, 7, 3, 9, 5, 6, 7, 7, 10, 5, 12, 7, 5, 7, 5, 7, 5, 7, 8, 4, 7, 13

Offset

1,7

Comments

Conjecture: (i) a(n) > 0 for all n > 5.

(ii) For any n > 3, sigma(k)*phi(n-k) - 1 and sigma(k)*phi(n-k) + 1 are both prime for some 0 < k < n, where sigma(k) is the sum of all (positive) divisors of k.

(iii) For any n > 5 not equal to 35, there is a positive integer k < n such that phi(k)*phi(n-k) - 1 is a Sophie Germain prime.

Note that part (i) implies the twin prime conjecture. We have verified it for n up to 10^7.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Example

a(6) = 1 since phi(1)*phi(5) = 1*4 = 4 with 4 - 1 and 4 + 1 twin primes.
a(8) = 1 since phi(1)*phi(7) = 1*6 = 6 with 6 - 1 and 6 + 1 twin primes.
a(16) = 1 since phi(2)*phi(14) = 1*6 = 6 with 6 - 1 and 6 + 1 twin primes.
a(17) = 1 since phi(3)*phi(14) = 2*6 = 12 with 12 - 1 and 12 + 1 twin primes.
a(19) = 1 since phi(1)*phi(18) = 1*6 = 6 with 6 - 1 and 6 + 1 twin primes.
a(23) = 1 since phi(2)*phi(21) = 1*12 = 12 with 12 - 1 and 12 + 1 twin primes.
a(25) = 1 since phi(11)*phi(14) = 10*6 = 60 with 60 - 1 and 60 + 1 twin primes.
a(26) = 1 since phi(7)*phi(19) = 6*18 = 108 with 108 - 1 and 108 + 1 twin primes.
a(42) = 1 since phi(14)*phi(28) = 6*12 = 72 with 72 - 1 and 72 +1 twin primes.
a(52) = 1 since phi(14)*phi(38) = 6*18 = 108 with 108 - 1 and 108 + 1 twin primes.

Mathematica

TQ[n_]:=PrimeQ[n-1]&&PrimeQ[n+1]
a[n_]:=Sum[If[TQ[EulerPhi[k]*EulerPhi[n-k]], 1, 0], {k, 1, (n-1)/2}]
Table[a[n], {n, 1, 100}]

Crossrefs

Cf. A000010, A000040, A001359, A005384, A006512, A014574, A232270, A233544.

Sequence in context: A193020 A301471 A237124 * A358193 A122530 A301453

Adjacent sequences: A233544 A233545 A233546 * A233548 A233549 A233550

Keyword

nonn

Author

Zhi-Wei Sun, Dec 12 2013

Status

approved