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A242950 Number of ordered ways to write n = k + m with k > 1 and m > 1 such that the least nonnegative residue of prime(k) modulo k is a square and the least nonnegative residue of prime(m) modulo m is a prime. 2
0, 0, 0, 0, 1, 1, 0, 1, 3, 2, 1, 1, 3, 4, 4, 1, 3, 5, 4, 4, 4, 3, 3, 3, 3, 4, 4, 4, 3, 5, 2, 5, 3, 5, 3, 6, 3, 7, 4, 6, 5, 7, 5, 9, 7, 6, 4, 6, 5, 9, 5, 6, 8, 7, 8, 5, 8, 5, 8, 4, 8, 6, 7, 4, 7, 4, 6, 4, 5, 4, 8, 2, 3, 4, 5, 4, 5, 6, 7, 7 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,9

COMMENTS

Conjecture: (i) a(n) > 0 for all n > 7.

(ii) Any integer n > 9 can be written as k + m with k > 1 and m > 1 such that the least nonnegative residue of prime(k) modulo k and the least nonnegative residue of prime(m) modulo m are both prime.

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

LINKS

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

EXAMPLE

a(11) = 1 since 11 = 2 + 9, prime(2) = 3 == 1^2 (mod 2), and prime(9) = 23 == 5 (mod 9) with 5 prime.

a(16) = 1 since 16 = 12 + 4, prime(12) = 37 == 1^2 (mod 12), and prime(4) = 7 == 3 (mod 4) with 3 prime.

MATHEMATICA

SQ[n_]:=IntegerQ[Sqrt[n]]

s[k_]:=SQ[Mod[Prime[k], k]]

p[k_]:=PrimeQ[Mod[Prime[k], k]]

a[n_]:=Sum[Boole[s[k]&&p[n-k]], {k, 2, n-2}]

Table[a[n], {n, 1, 80}]

CROSSREFS

Cf. A000290, A000040, A242425, A242748, A242753.

Sequence in context: A053542 A087284 A262311 * A304972 A152176 A152175

Adjacent sequences:  A242947 A242948 A242949 * A242951 A242952 A242953

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, May 27 2014

STATUS

approved

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Last modified October 29 18:53 EDT 2020. Contains 338067 sequences. (Running on oeis4.)