A336016 -id:A336016 - cp's OEIS frontend

A336015 Irregular triangle where row n lists primes q below the n-th primorial such that the multiplicative order of q mod the n-th primorial is 2. I.e., such primes q having the least k such that q^k (mod primorial(n)) == 1 is 2.

5, 11, 19, 29, 29, 41, 71, 139, 181, 419, 461, 659, 769, 881, 1231, 1429, 2309, 1429, 2729, 4159, 5279, 5851, 8009, 8581, 10009, 12011, 12739, 13441, 13859, 14741, 15289, 17291, 20021, 23869, 24179, 30029, 1429, 23869, 77351, 95369, 102101, 116689, 120121, 188189

Offset: 2

Michael De Vlieger, David James Sycamore, David A. Corneth, Jul 08 2020

			Table begins:
5;
11, 19, 29;
29, 41, 71, 139, 181;
419, 461, 659, 769, 881, 1231, 1429, 2309;
...
For row 2 we look for primes q such that q^2 == 1 (mod primorial(2)) == 1 (mod 6) where q is coprime to 6. It turns out the only prime with this property is 5 as 5^2 == 1 (mod 6). - _David A. Corneth_, Aug 15 2020

David A. Corneth, Table of n, a(n) for n = 2..10001

Cf. A000010, A002110, A005867, A336016.

Table[Function[P, Select[Prime@ Range[n, PrimePi[P - 1]], MultiplicativeOrder[#, P] == 2 &]][Product[Prime@ i, {i, n}]], {n, 8}] // Flatten

row(n) = my(pp = vecprod(primes(n)), res=List()); forstep(i=pp/prime(n)+1, pp-1, 2, if(gcd(i,pp) == 1 && znorder(Mod(i,pp)) == 2 && isprime(i), listput(res,i))); res \\ David A. Corneth, Jul 08 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

A336015 Irregular triangle where row n lists primes q below the n-th primorial such that the multiplicative order of q mod the n-th primorial is 2. I.e., such primes q having the least k such that q^k (mod primorial(n)) == 1 is 2.

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