A369111 -id:A369111 - cp's OEIS frontend

A369105 Primes p such that p+2 has only prime factors congruent to -1 modulo 4.

5, 7, 17, 19, 29, 31, 41, 47, 61, 67, 79, 97, 101, 127, 131, 137, 139, 149, 197, 199, 211, 229, 241, 251, 269, 277, 281, 307, 359, 379, 397, 421, 439, 461, 467, 487, 499, 521, 569, 571, 587, 601, 617, 619, 631, 641, 647, 691, 709, 719, 727, 751, 757, 787, 809, 811

Offset: 1

Stefano Spezia, Jan 13 2024

nonn

Jones and Zvonkin call these primes BCC primes, where BCC stands for Bujalance, Cirre, and Conder.

Amiram Eldar, Table of n, a(n) for n = 1..10000
E. Bujalance, F. J. Cirre, and M. D. E. Conder, Bounds on the orders of groups of automorphisms of a pseudo-real surface of given genus, Journal of the London Mathematical Society, Volume 101, Issue 2, p. 877-906, (2019).
Gareth A. Jones and Alexander K. Zvonkin, A number-theoretic problem concerning pseudo-real Riemann surfaces, arXiv:2401.00270 [math.NT], 2023. See page 1.

Cf. A001221, A004614, A004767, A027748, A369107, A369108, A369109, A369111.

Select[Prime[Range[150]], PrimeQ[f=First/@FactorInteger[#+2]] == Table[True,{j,PrimeNu[#+2]}] && Mod[f,4] == Table[3, {m,PrimeNu[#+2]}] &]

is1(n) = {my(p = factor(n)[, 1]); for(i = 1, #p, if(p[i] % 4 == 1, return(0))); 1;};
lista(pmax) = forprime(p = 3, pmax, if(is1(p+2), print1(p, ", "))); \\ Amiram Eldar, Jun 03 2024

A369107 a(n) is the number of numbers less than or equal to 10^n that are divisible only by primes congruent to 3 mod 4.

Original entry on oeis.org

4, 26, 201, 1680, 14902, 135124, 1243370, 11587149, 108941388, 1031330156, 9816605847

Offset: 1

Stefano Spezia, Jan 13 2024

Gareth A. Jones and Alexander K. Zvonkin, A number-theoretic problem concerning pseudo-real Riemann surfaces, arXiv:2401.00270 [math.NT], 2023. See Table 1 at page 6 and Table 2 at page 7.

Cf. A004614, A004767, A011557, A369105, A369108, A369109, A369111.

a[n_] := Length[Join[{1}, Select[Range[10^n], PrimeQ[f = First/@FactorInteger[#]] == Table[True, {j,PrimeNu[#]}] && Mod[f,4] == Table[3, {m,PrimeNu[#]}] && #<=10^n &]]]; Array[a, 10]

is1(n) = {my(p = factor(n)[, 1]); for(i = 1, #p, if(p[i] % 4 == 1, return(0))); 1;};
lista(nmax) = {my(c = 0, pow = 10, n = 1, nm = nmax + 1); forstep(k = 1, 10^nmax + 1, 2, if(k > pow, print1(c, ", "); pow *= 10; n++; if(n == nm, break)); if(is1(k), c++));} \\ Amiram Eldar, Jun 03 2024

a(11) from Amiram Eldar, Jun 03 2024

A369108 a(n) is the number of numbers less than or equal to 10^n that are divisible only by primes congruent to 1 mod 4.

Original entry on oeis.org

2, 15, 123, 1074, 9623, 87882, 814183, 7618317, 71838469, 681591775, 6499182987

Offset: 1

Stefano Spezia, Jan 13 2024

Gareth A. Jones and Alexander K. Zvonkin, A number-theoretic problem concerning pseudo-real Riemann surfaces, arXiv:2401.00270 [math.NT], 2023. See Table 2 at page 7.

Cf. A004613, A369105, A369107, A369109, A369111.

a[n_] := Length[Join[{1}, Select[Range[10^n], PrimeQ[f = First/@FactorInteger[#]] == Table[True, {j,PrimeNu[#]}] && Mod[f,4] == Table[1, {m,PrimeNu[#]}] && #<=10^n &]]]; Array[a, 9]

is1(n) = n % 4 == 1 && factorback(factor(n)[, 1] % 4) == 1 \\ Charles R Greathouse IV at A004613
lista(nmax) = {my(c = 0, pow = 10, n = 1, nm = nmax + 1); for(k = 1, 10^nmax + 1, if(k > pow, print1(c, ", "); pow *= 10; n++; if(n == nm, break)); if(is1(k), c++));} \\ Amiram Eldar, Jun 03 2024

a(10)-a(11) from Amiram Eldar, Jun 03 2024

A369109 a(n) is the number of pairs of twin primes p and p+2 both less than or equal to 10^n such that p is congruent to 1 modulo 4.

Original entry on oeis.org

1, 4, 19, 105, 604, 4046, 29482, 220419, 1712731, 13706592, 112196635, 935286453

Offset: 1

Stefano Spezia, Jan 13 2024

Gareth A. Jones and Alexander K. Zvonkin, A number-theoretic problem concerning pseudo-real Riemann surfaces, arXiv:2401.00270 [math.NT], 2023. See Table 4 at page 13.

Cf. A001097, A001359, A004613, A006512, A007508, A369105, A369107, A369108, A369111.

a[n_] := Length[Select[Range[10^n-2], PrimeQ[#] && PrimeQ[#+2] && Mod[#,4] == 1 &]]; Array[a,10]

lista(nmax) = {my(prev = 2, c = 0, pow = 10, n = 1, nm = nmax + 1); forprime(p = 3, , if(p > pow, print1(c, ", "); pow *= 10; n++; if(n == nm, break)); if(prev % 4 == 1 && p == prev + 2, c++); prev = p);} \\ Amiram Eldar, Jun 03 2024

a(11)-a(12) from Amiram Eldar, Jun 03 2024

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A369105 Primes p such that p+2 has only prime factors congruent to -1 modulo 4.

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A369107 a(n) is the number of numbers less than or equal to 10^n that are divisible only by primes congruent to 3 mod 4.

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A369108 a(n) is the number of numbers less than or equal to 10^n that are divisible only by primes congruent to 1 mod 4.

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A369109 a(n) is the number of pairs of twin primes p and p+2 both less than or equal to 10^n such that p is congruent to 1 modulo 4.

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Mathematica

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