A341765 - OEIS

The OEIS is supported by the many generous donors to the OEIS Foundation.

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give

A341765

Consider gaps between successive odd primes from 3 up to prime(n+2). Let k1 be number of gaps congruent to 2 (mod 6) and let k2 be number of gaps congruent to 4 (mod 6). Then a(n) = k1 - k2.

1, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Theorem A: for all n, a(n) belongs to the set: {1,2}, for proof see A342156.

The indices n for which numbers of 1's and 2's in this sequence are equal are 2, 4, 6, 10, 12, 20, 36, 46, 48 and no other up to n=10^6.

LINKS

Table of n, a(n) for n=1..104.

FORMULA

a(n) = 3 - A039701(n+2). - Andrey Zabolotskiy, Nov 04 2024

MAPLE

a(1)=1 because prime(2+1)-prime(2)=5-3=2 then the gap 2 is congruent to 2 mod 6, then k1=1 and k2=0 so k1 - k2 = 1.

MATHEMATICA

k1 = 0; k2 = 0; cc = {}; Do[

gap = Prime[n + 1] - Prime[n];

If[Mod[gap/2, 3] == 1, k1 = k1 + 1,

If[Mod[gap/2, 3] == 2, k2 = k2 + 1]]; AppendTo[cc, k1 - k2];

If[k1 - k2 == 1, , If[k1 - k2 == 2, , Print[{n, k1 - k2}]]], {n, 2,

105}]; cc

PROG

(PARI) a(n) = {my(vp = vector(n+1, k, prime(k+1)), dp = vector(#vp-1, k, (vp[k+1] - vp[k])/2)); my(s=0); for (k=1, #dp, if ((dp[k]%3)==1, s++); if ((dp[k]%3) == 2, s--)); s; } \\ Michel Marcus, Feb 27 2021

CROSSREFS

Cf. A000230, A001223, A028334, A321856, A341952, A342156, A039701.

Sequence in context: A081129 A022934 A107450 * A104248 A358359 A249973

Adjacent sequences: A341762 A341763 A341764 * A341766 A341767 A341768

KEYWORD

nonn

AUTHOR

Artur Jasinski, Feb 19 2021

EXTENSIONS

Name edited by Andrey Zabolotskiy, Nov 04 2024

STATUS

approved