The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A362663

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A362663

a(n) is the partial sum of b(n), which is defined to be the difference between the numbers of primes in (n^2, n^2 + n] and in (n^2 - n, n^2].
(history; published version)

#25 by N. J. A. Sloane at Fri Jun 16 13:44:55 EDT 2023

STATUS

editing

approved

#24 by N. J. A. Sloane at Fri Jun 16 13:44:53 EDT 2023

NAME

a(n) is the partial sum of b(n), which is defined as to be the difference between the numbers of primes in (n^2, n^2 + n] and in (n^2 - n, n^2].

STATUS

proposed

editing

#23 by Michel Marcus at Wed May 24 05:11:33 EDT 2023

STATUS

editing

proposed

#22 by Michel Marcus at Wed May 24 05:09:49 EDT 2023

PROG

(PARI) a(n) = sum(i=1, n, primepi(i^2+i) + primepi(i^2-i) - 2*primepi(i^2)); \\ Michel Marcus, May 24 2023

STATUS

proposed

editing

#21 by Ya-Ping Lu at Tue May 23 07:55:20 EDT 2023

STATUS

editing

proposed

#20 by Ya-Ping Lu at Tue May 23 07:55:14 EDT 2023

COMMENTS

A plot of a(n) for n up to 100000 is given in Links. First negative term is a(177) = -7 and first zero term appears at n = 198. It seems that there are more positive terms than negative ones. For example, for n up to 500000, there are 482085 positive terms, 17903 negative terms, and 12 zero terms.

STATUS

proposed

editing

#19 by Ya-Ping Lu at Thu May 04 08:35:20 EDT 2023

STATUS

editing

proposed

#18 by Ya-Ping Lu at Thu May 04 08:27:58 EDT 2023

COMMENTS

Among the first 350000 terms, there are 12405 negative terms and 10 zero terms. For n <= 350000, the last negative term is a(49834) = -34, the last zero term appears at n = 48834, and a(350000) = 29704.

Conjecture: a(n) > 0 for n > 49834.

STATUS

proposed

editing

Discussion

Thu May 04

08:35

Ya-Ping Lu: Found more negative terms beyond n = 480000.

#17 by Jon E. Schoenfield at Mon May 01 19:10:14 EDT 2023

STATUS

editing

proposed

Discussion

Mon May 01

20:41

Ya-Ping Lu: Thanks, Jon! Just got some more data today and now the conjecture holds for n up to 400000. a(400000)=53218.

#16 by Jon E. Schoenfield at Mon May 01 19:10:12 EDT 2023

EXAMPLE

a(1) = primepi(1^2+1) + primepi(1^2-1) - 2*primepi(1^2) = 1+0-2*0 = 1.

a(2) = a(1) + primepi(2^2+2) + primepi(2^2-2) - 2*primepi(2^2) = 1+3+1-2*2 = 1.

a(3) = a(2) + primepi(3^2+3) + primepi(3^2-3) - 2*primepi(3^2) = 1+5+3-2*4 = 1.

a(4) = a(3) + primepi(4^2+4) + primepi(4^2-4) - 2*primepi(4^2) = 1+8+5-2*6 = 2.

STATUS

proposed

editing