A172104 -id:A172104 - cp's OEIS frontend

A172105 Numbers k such that k-th partial sum of A167020 - k-th partial sum of A167021 = 0 (or A172103(k) - A172104(k) = 0).

Original entry on oeis.org

1, 2, 3, 6, 7, 13, 27, 37, 38

Offset: 1

Juri-Stepan Gerasimov, Jan 25 2010

No further terms between 38 and 1500000. [From R. J. Mathar, May 02 2010]

Cf. A167020, A167021, A172103, A172104.

Corrected (13 inserted, terms in the range 297 to 316 removed) by R. J. Mathar, May 02 2010

A323011 a(n) = A172103(n) - A172104(n).

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 0, 1, 2, 2, 1, 1, 0, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 2, 2, 1, 0, 1, 2, 2, 2, 2, 2, 2, 1, 1, 0, 0, 1, 1, 1, 2, 3, 4, 4, 3, 3, 3, 4, 4, 3, 3, 4, 4, 3, 2, 2, 2, 3, 4, 3, 2, 1, 2, 3, 2, 3, 2, 2, 2, 2, 2, 1, 2, 3, 2, 2, 3, 3, 4, 3, 4, 3, 4, 5, 5, 5, 5, 5, 4

Offset: 1

Torlach Rush, Jan 01 2019

sign

			a(1) = A172103(1) - A172104(1) = 0.
a(2) = A172103(2) - A172104(2) = 0.
a(3) = A172103(3) - A172104(3) = 0.
a(4) = A172103(4) - A172104(4) = 1.

R. H. Hudson and A. Brauer, On the exact number of primes in the arithmetic progressions 4n +/- 1 and 6n +/- 1, J. reine angew. Math., 291 (1977), 23-29.

Cf. A167020, A167021, A172103, A172104.

f:= proc(t) `if`(isprime(6*t-1),1,0) - `if`(isprime(6*t+1),1,0) end proc:
ListTools:-PartialSums(map(f, [$1..100])); # Robert Israel, Feb 19 2019

Accumulate@ Boole@ PrimeQ[6 Range@ # - 1] - Accumulate@ Boole@ PrimeQ[6 Range@ # + 1] &@ 60 (* Michael De Vlieger, Jan 27 2019 *)

isp(n) = isprime(6*n+1); \\ A167021
ism(n) = isprime(6*n-1); \\ A167020
psisp(n) = sum(k=1, n, isp(k)); \\ A172104
psism(n) = sum(k=1, n, ism(k)); \\ A172103
a(n) = psism(n) - psisp(n); \\ Michel Marcus, Jan 18 2019

More terms from Michel Marcus, Feb 01 2019

A172103 Partial sums of A167020 where A167020(n)=1 iff 6*n-1 is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 5, 6, 7, 8, 9, 9, 10, 10, 11, 12, 12, 13, 14, 15, 15, 15, 16, 17, 17, 18, 18, 18, 19, 20, 21, 21, 22, 23, 23, 23, 23, 23, 24, 25, 26, 26, 27, 28, 29, 30, 30, 31, 31, 32, 32, 32, 33, 34, 34, 34, 34, 34, 35, 36, 37, 37, 37, 37, 38, 39, 39, 40, 40, 40, 41, 41, 42, 42

Offset: 1

Juri-Stepan Gerasimov, Jan 25 2010

nonn

Where A167020 is the characteristic sequence: A167020(n)=1 iff 6*n-1 is prime.

Cf. A007528, A167020, A172104.

Accumulate[Table[If[PrimeQ[6 n-1],1,0],{n,100}]] (* Harvey P. Dale, Dec 27 2022 *)

ism(n) = isprime(6*n-1); \\ A167020
a(n) = sum(k=1, n, ism(k)); \\ Michel Marcus, Feb 06 2019

Entries checked by R. J. Mathar, Apr 14 2010

A340763 Number of primes p <= n that are congruent to 1 modulo 3.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 1

Jianing Song, Apr 28 2021

			There are 11 primes <= 100 that are congruent to 1 modulo 3, namely 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, so a(100) = 11.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A002476, A340764, A172104, A066339, A066490.

a(n) = sum(i=1, n, isprime(i) && (i%3==1))

a(n) + A340764(n) = A000720(n) - 1 for n >= 3.

A340767 Number of primes p <= 6*n + 5 that are congruent to 2 modulo 3.

Original entry on oeis.org

2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 10, 11, 11, 12, 13, 13, 14, 15, 16, 16, 16, 17, 18, 18, 19, 19, 19, 20, 21, 22, 22, 23, 24, 24, 24, 24, 24, 25, 26, 27, 27, 28, 29, 30, 31, 31, 32, 32, 33, 33, 33, 34, 35, 35, 35, 35, 35, 36, 37, 38, 38, 38, 38, 39, 40, 40, 41, 41, 41, 42, 42

Offset: 0

Jianing Song, Apr 28 2021

			There are 14 primes <= 6*16+5 = 101 that are congruent to 2 modulo 3, namely 2, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, so a(16) = 14.

Jianing Song, Table of n, a(n) for n = 0..10000

Cf. A003627, A167020, A340764, A172104, A296021, A296020.

a(n) = sum(i=1, 6*n+5, isprime(i) && (i%3==2))

a(n) = A340764(6*n+5).

a(n) = 1 + Sum_{k=0..n+1} A167020(k).

cp's OEIS Frontend

A172105 Numbers k such that k-th partial sum of A167020 - k-th partial sum of A167021 = 0 (or A172103(k) - A172104(k) = 0).

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A323011 a(n) = A172103(n) - A172104(n).

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A172103 Partial sums of A167020 where A167020(n)=1 iff 6*n-1 is prime.

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Mathematica

PARI

Extensions

A340763 Number of primes p <= n that are congruent to 1 modulo 3.

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PARI

Formula

A340767 Number of primes p <= 6*n + 5 that are congruent to 2 modulo 3.

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PARI

Formula