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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. 1
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 (list; graph; refs; listen; history; text; internal format)
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 A226182

Adjacent sequences:  A242876 A242877 A242878 * A242880 A242881 A242882

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, May 25 2014

STATUS

approved

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Last modified May 28 17:37 EDT 2020. Contains 334684 sequences. (Running on oeis4.)