A336093 Let P(n) = primorial(n) = A002110(n); a(n) is the number of primes q < P(n) such that P(n) - q is also prime and q^2==1 (mod P(n)).

0, 0, 2, 4, 2, 8, 6, 28, 36, 40, 56, 106, 192, 304, 526, 926, 1644, 2756, 4944, 8840, 15958, 28402, 51102, 92372

Offset: 1

Author

David James Sycamore, Michael De Vlieger Jul 09 2020

Keywords

• nonn
• more

Comments

A totative of k is a number <= k and relatively prime to k.

A336016(n) gives the number of prime totatives p of P(n) for which p^2==1 (mod P(n)). Whereas P(n) - p also has this property, it is not always prime. This sequence gives the number of prime totatives q of P(n) such that q^2==1 mod P(n) and P(n) - q is also prime. All terms are even; a(n) <= A336016(n) for all n. The number of distinct representations of P(n) as the sum of two primes each having multiplicative order 2 (mod P(n)) is given by a(n)/2, for which A116979(n) is an upper bound.

Examples

			P(4)=210; all totatives 29,41,71,139,181 are prime. However 210 - 41 = 169 is not prime, whereas 210-29 = 181, 210-71 = 139. Therefore the totatives we count in this case are 29,71,139,181, so a(4) = 4.
		

Crossrefs

Cf. A000010, A336015, A336016, A116979.

Programs

Maple

with(NumberTheory):
P := proc (k)
local n, v, W, H;
n := 1; v := 0;
W := product(ithprime(j), j = 1 .. k);
H := PrimeCounting(W);
for n from 1 to H do
if mod(ithprime(n)^2, W) = 1 and isprime(W-ithprime(n)) then v := v+1 else v := v end if:
end do:
v;
end proc:
seq(P(k), k = 1 .. 8);

Mathematica

{0, 0}~Join~Table[Block[{P = #, k = 0}, Do[If[AllTrue[{#, P - #}, And[PrimeQ@ #, MultiplicativeOrder[#, P] == 2] &], k++] &@ Prime[i], {i, PrimePi[n + 1], PrimePi[P/2]}]; 2 k ] &@ Product[Prime@ j, {j, n}], {n, 3, 8}]