A007351 -id:A007351 - cp's OEIS frontend

A118480 (n-th 4k+1 prime minus n-th 4k+3 prime minus two)/4.

0, 1, 1, 2, 3, 2, 2, 3, 3, 5, 6, 5, 6, 2, 7, 5, 6, 8, 7, 7, 7, 12, 10, 10, 11, 11, 12, 10, 10, 12, 11, 13, 10, 10, 10, 10, 9, 8, 7, 9, 3, 4, 4, 4, 11, 13, 15, 17, 19, 19, 22, 19, 16, 13, 17, 16, 15, 16, 14, 17, 16, 21, 24, 19, 19, 13, 17, 17, 19, 19, 16, 11, 13, 13, 22, 19, 19, 17, 22

Offset: 1

Clark Kimberling and Robert G. Wilson v, May 04 2006

sign

Zero occurs infinitely often as do the negative numbers.
First occurrence of a(n) beginning with 0: 1, 2, 4, 5, 42, 10, 11, 15, 18, 37, 23, 25, 22, 32, 59, 47, 53, 48, 83, 49, 110, 62, 51, 82, 63, 170, ...,
The first negative term is at n=1473. - T. D. Noe, Apr 09 2009

T. D. Noe, Table of n, a(n) for n=1..2000

Cf. A022757, A007350, A007351, A038691.

(Select[1 + 4Range@245, PrimeQ@# &] - Select[ -1 + 4Range@225, PrimeQ@# &] - 2)/4

a(n) = (A002144(n) - A002145(n) - 2)/4.

A122035 Primes p = Prime[m] such that polynomial (1 + Sum[x^Prime[k],{k,1,m}]) factors over the integers.

Original entry on oeis.org

5, 17, 41, 461

Offset: 1

Alexander Adamchuk, Sep 13 2006

Corresponding numbers m such that a(n) = Prime[m] are {3,7,13,89,...}. All 4 listed initial terms of a(n) coincide with A007351[n+1].
The polynomial is divisible by x^2+1 if and only if p is a member of A007351. - David Wasserman, May 20 2008
No other terms below 4175. - Max Alekseyev, May 31 2008

			a(1) = 5 because Factor[1+x^2+x^3+x^5] = (x+1)*(x^2+1)*(x^2-x+1), but polynomials (1+x^2) and (1+x^2+x^3) do not factor over the integers.
a(2) = 17 because Factor[1+x^2+x^3+x^5+x^7+x^11+x^13+x^17] = (x^2+1)*(x^15-x^13+2x^11-x^9+x^7+x^3+1).

Cf. A038691, A007351.

A380877 Primes p where the prime race 12m+1 versus 12m+7 is tied.

Original entry on oeis.org

2, 3, 5, 13, 17, 433, 457, 461

Offset: 1

Ya-Ping Lu, Feb 06 2025

nonn

Primes p such that pi_{12,1}(p) = pi_{12,7}(p), where pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m). For the first 5 billion primes, pi_{12,7}(p) >= pi_{12,1}(p). If exists, a(9) > 122430513841.

Cf. A007351, A068228, A068229, A379989, A380333.

s={};Do[p=Prime[pp];If[Length[Select[Prime[Range[pp]],Mod[#,12]==1&]]==Length[Select[Prime[Range[pp]],Mod[#,12]==7&]],AppendTo[s,p]],{pp,100}];s (* James C. McMahon, Mar 03 2025 *)

from sympy import nextprime; p, d = 2, 0
while p < 500:
    if d == 0: print(p, end = ', ')
    p = nextprime(p); r = p%12
    if r == 7: d += 1
    elif r == 1: d -= 1

cp's OEIS Frontend

A118480 (n-th 4k+1 prime minus n-th 4k+3 prime minus two)/4.

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A122035 Primes p = Prime[m] such that polynomial (1 + Sum[x^Prime[k],{k,1,m}]) factors over the integers.

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A380877 Primes p where the prime race 12m+1 versus 12m+7 is tied.

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