A242879 Least positive integer k < n such that k*p == 1 (mod prime(k)) for some prime p < prime(k) and (n-k)*q == 1 (mod prime(n-k)) for some prime q < prime(n-k), or 0 if such a number k does not exist.

0, 0, 0, 2, 2, 2, 3, 2, 2, 3, 4, 2, 2, 3, 2, 3, 4, 7, 2, 2, 3, 4, 2, 3, 4, 13, 6, 7, 11, 13, 10, 11, 2, 3, 4, 18, 6, 7, 2, 3, 4, 2, 2, 3, 4, 6, 6, 2, 3, 2, 2, 3, 4, 2, 2, 3, 4, 6, 6, 2, 3, 2, 3, 4, 7, 2, 3, 2, 3, 4, 7, 2, 2, 2, 2, 3, 2, 3, 4, 7

Offset

1,4

Comments

According to the conjecture in A242753, a(n) should be positive for all n > 3.

We have verified that a(n) > 0 for all n = 4, ..., 10^8.

Links

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

Example

a(4) = 2 since 4 = 2 + 2 and 2*2 == 1 (mod prime(2)=3).
a(7) = 3 since 7 = 3 + 4, 3*2 == 1 (mod prime(3)=5) with 2 prime, and also 4*2 == 1 (mod prime(4)=7) with 2 prime, but 5*9 == 1 (mod prime(5)=11) with 9 not prime.

Mathematica

p[n_]:=PrimeQ[PowerMod[n, -1, Prime[n]]]
Do[Do[If[p[k]&&p[n-k], Print[n, " ", k]; Goto[aa]]; Continue, {k, 1, n/2}]; Print[n, " ", 0]; Label[aa]; Continue, {n, 1, 80}]

Crossrefs

Cf. A000040, A242425, A242748, A242753, A242754, A242755.

Sequence in context: A186233 A226056 A104011 * A176775 A175778 A357039

Adjacent sequences: A242876 A242877 A242878 * A242880 A242881 A242882

Keyword

nonn

Author

Zhi-Wei Sun, May 25 2014

Status

approved