A199547 Primes p for which pi_{4,3}(p) < pi_{4,1}(p), where pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

26861, 616841, 616849, 616877, 616897, 616909, 616933, 616943, 616951, 616961, 616991, 616997, 616999, 617011, 617269, 617273, 617293, 617311, 617327, 617333, 617339, 617341, 617359, 617369, 617401, 617429, 617453, 617521, 617537, 617689, 617693, 617699, 617717

Offset: 1

Author

Arkadiusz Wesolowski, Dec 09 2011

Keywords

• nonn

Comments

Another version of A007350.

J. E. Littlewood (1914) proved that this sequence is infinite.

a(1) = 26861 was found in 1957 by John Leech.

Prime indices of negative terms in A066520. - Jianing Song, Feb 20 2019

References

• Wacław Sierpiński, O stu prostych, ale trudnych zagadnieniach arytmetyki. Warsaw: PZWS, 1959, p. 22.

Links

• Arkadiusz Wesolowski, Table of n, a(n) for n = 1..1000
• G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 26861

Crossrefs

Cf. A007350, A038691, A038698, A051024, A051025, A066520, A096628.

Programs

Mathematica

lst = {}; For[n = 2; t = 0, n < 50451, n++, t += Mod[p = Prime[n], 4] - 2; If[t < 0, AppendTo[lst, p]]]; lst

Python

from sympy import nextprime; a, p = 0, 2
while p < 617717:
    p=nextprime(p); a += p%4-2
    if a < 0: print(p, end = ', ') # Ya-Ping Lu, Jan 18 2025

Formula

a(n) = prime(A096628(n)). - Jianing Song, Feb 20 2019