A336015 -id:A336015 - cp's OEIS frontend

A336016 a(n) is the number of primes q less than primorial(n) having k = 2 as the least exponent such that q^k == 1 (mod primorial(n)).

0, 1, 3, 5, 8, 19, 22, 51, 89, 145, 263, 453, 851, 1575, 2880, 5469, 10338, 19115, 35782, 67569, 128601, 243600, 463840, 883589

Offset: 1

Michael De Vlieger, David James Sycamore, David A. Corneth, Jul 10 2020

a(n) = length of row n of A336015 for n > 1.

			a(4) = 5 as there are 5 primes q coprime to primorial(4) = 210 such that 2 is the least positive integer exponent k where q^k == 1 (mod 210). Those primes are 29, 41, 71, 139, 181 and indeed we have 29^2 == 1 (mod 210), 41^2 == 1 (mod 210), 71^2 == 1 (mod 210), 139^2 == 1 (mod 210) and 181^2 == 1 (mod 210) and no more below 210. So as these are five such primes in row 4, a(4) = 5. - _David A. Corneth_, Aug 15 2020

Cf. A000010, A002110, A005867, A131577, A336015.

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

a(n) = {if(n <= 2, return(n-1)); my(pp = vecprod(primes(n))/2, d = divisors(pp), res = 0); for(i = 1, #d, c = lift(chinese(Mod(-1, d[i]), Mod(1, pp/d[i]))); forstep(i = c, pp*2, pp, if(isprime(i), res++ ) ) ); res } \\ David A. Corneth, Aug 16 2020

New name from David A. Corneth, Aug 15 2020

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)).

Original entry on oeis.org

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

David James Sycamore, Michael De Vlieger Jul 09 2020

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.

			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.

Cf. A000010, A336015, A336016, A116979.

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);

{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}]

cp's OEIS Frontend

A336016 a(n) is the number of primes q less than primorial(n) having k = 2 as the least exponent such that q^k == 1 (mod primorial(n)).

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