A100672 a(1) = 1; thereafter, a(n) = 1 if n-th prime is 3 mod 4, 0 if n-th prime is 1 mod 4.

1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1

Offset: 1

Author

Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 06 2004

Keywords

• base
• nonn
• easy

Comments

Second least-significant bit in the binary expansion of the n-th prime.

a(n)=1 iff prime(n) is a member of A045326 (equivalently for n>1, iff prime(n)-3 is divisible by 4).

Examples

			a(2)=1 because prime(2)=11_2 (in binary; decimal = 3_10) and its 2^1 bit is 1.
a(3)=0 because prime(3)=101_2 (in binary; decimal = 5_10) and its 2^1 bit is 0.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Fermat's 4n Plus 1 Theorem.
• Eric Weisstein's World of Mathematics, Gaussian Prime.

Crossrefs

Cf. A000040, A045326, A002144, A002145, A039702.

RUNS transform is essentially A091237.

Programs

Maple

A100672 := proc(n)
        if n = 1 then
                1 ;
        else
                ((ithprime(n) mod 4)-1)/2;
        end if;
end proc: # R. J. Mathar, Oct 06 2011

Mathematica

Table[Reverse[RealDigits[Prime[k], 2][[1]]][[2]], {k, 1, 128}]

PARI

for(k=1,105,print1( bittest(prime(k), 1), ", ")) \\ Washington Bomfim, Jan 18 2011

Python

from sympy import prime
def A100672(n): return int(prime(n)>>1&1) # Chai Wah Wu, Jun 23 2023

Formula

a(n) = 1-A098033(n), n>1. - Steven G. Johnson (stevenj(AT)math.mit.edu), Sep 18 2008

a(n) = floor(prime(n)/2) mod 2. - Alois P. Heinz, Jul 16 2024

Extensions

Edxited by N. J. A. Sloane, Jan 11 2025