A038691 -id:A038691 - cp's OEIS frontend

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

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.

A349519 a(n)=x is the least prime with pi(x,4,3) - pi(x,4,1) = 1-n where pi(x,4,k) is the number of primes 4*j + k <= x.

Original entry on oeis.org

2, 26861, 616897, 616909, 616933, 623641, 623653, 623669, 623681, 12315529, 12315581, 12315613, 12315617, 12362653, 12362657, 12362717, 12362741, 12362981, 12362989, 12365033, 12365057, 12365153, 12365173, 12365201, 12366589, 951821281

Offset: 1

Gerhard Kirchner, Nov 20 2021

nonn

The difference d(x) = pi(x,4,3) - pi(x,4,1) changes sign infinitely often, see link "Prime Quadratic Effect". But this does not say anything about the amplitudes of these oscillations. For diagrams, see A349518, "Oscillations of d(x)". If d(x) has no lower limit, the current sequence is infinite. Regarding the upper limit, see A349518.

Note the gaps between 2, 26861 and 616897, 623681 and 12315529, 12366589 and 951821281.

			primes 4*j+1: 5, 13, 17, ...
       4*j+3: 3, 7, 11, ...
d(x) = pi(x,4,3) - pi(x,4,1)
.
  n     x  pi(x,4,3) pi(x,4,1)   d(x)=1-n?
  - -----  --------- ---------  -----------
  1     2       0         0      0=0  true   a(1)  = 2
  2     3       1         0      1=-1 false  a(2) != 3
  2     5       1         1      2=-1 false  a(2) != 5
  ...........................
  2 26861    1472      1473     -1=-1 true   a(3)  = 26861

Eric Weisstein's World of Mathematics, Prime Quadratic Effect.

Cf. A038691, A007350.

block(w:[2],  su:0, sum:0, n:1, p:2, nmax: 25,
  /* returns nmax terms */
while n

cp's OEIS Frontend

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

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