A056931 -id:A056931 - cp's OEIS frontend

A056927 Difference between n^2 and largest prime less than n^2.

1, 2, 3, 2, 5, 2, 3, 2, 3, 8, 5, 2, 3, 2, 5, 6, 7, 2, 3, 2, 5, 6, 5, 6, 3, 2, 11, 2, 13, 8, 3, 2, 3, 2, 5, 2, 5, 10, 3, 12, 5, 2, 3, 8, 3, 2, 7, 2, 23, 8, 5, 6, 7, 2, 15, 20, 3, 12, 7, 2, 11, 2, 3, 6, 7, 6, 3, 2, 11, 2, 5, 6, 5, 2, 27, 2, 5, 12, 3, 8, 5, 6, 13, 6, 3, 8, 3, 2, 7, 8, 3, 2, 5, 12, 7, 6, 3

Offset: 2

Henry Bottomley, Jul 12 2000

Legendre's conjecture (still open) that there is always a prime between n^2 and (n+1)^2 is equivalent to the conjecture that a(n) < 2n-1 for all n>1.

Will the most common subsequence seen be (2,3,2)? - Bill McEachen, Jan 30 2011

			a(4)=3 because largest prime less than 4^2 is 13 and 16-13=3.

T. D. Noe, Table of n, a(n) for n=2..10000

Cf. A053001, A056929, A056931.

A056927 := n-> n^2-prevprime(n^2); seq(A056927(n), n=2..100);

Table[n2=n^2;n2-NextPrime[n2,-1],{n,2,100}] (* Vladimir Joseph Stephan Orlovsky, Mar 09 2011 *)

{my(maxx=10000);n=2;ptr=2;while(n<=maxx,q=n^2;pp=precprime(q); diff=q-pp;print(ptr,"  ",diff);n++;ptr++ );} \\ Bill McEachen, May 07 2014

a(n) = A000290(n)-A053001(n).

More terms from James Sellers, Jul 13 2000

A056929 Difference between n^2 and average of smallest prime greater than n^2 and largest prime less than n^2.

Original entry on oeis.org

0, 0, 1, -1, 2, -1, 0, 0, 1, 1, 0, -1, 1, 0, 2, 1, 0, -2, 1, 0, 1, -3, 2, 0, 1, -1, 4, -5, 3, 1, -2, 0, -2, -1, 2, -1, 1, 4, 1, 0, -4, -5, -5, 3, -5, -1, 1, -4, 10, 0, 1, -2, 3, -5, 7, 9, -2, 1, 0, -2, 4, -9, 0, 1, 3, 1, -5, -10, 4, -4, 0, 1, 2, -6, 12, -4, 0, 3, -9, 3, -2, -2, 6, 1, -6, 2, -3

Offset: 2

Henry Bottomley, Jul 12 2000

Conjecture: the most frequent value will be 1 (including sequence variants with any even power n^2k). - Bill McEachen, Dec 12 2022

			a(4)=1 because smallest prime greater than 4^2 is 17, largest prime less than 4^2 is 13, average of 17 and 13 is 15 and 16-15=1.

Hugo Pfoertner, Table of n, a(n) for n = 2..10000

Cf. A007491, A053000, A053001, A056927, A056928, A056930, A056931.

with(numtheory): A056929 := n-> n^2-(prevprime(n^2)+nextprime(n^2))/2);

Array[# - Mean@ {NextPrime[#], NextPrime[#, -1]} &[#^2] &, 87, 2] (* Michael De Vlieger, May 20 2018 *)

a(n) = n^2 - (nextprime(n^2) + precprime(n^2))/2; \\ Michel Marcus, May 20 2018

a(n) = A000290(n) - A056928(n).

a(n) = (A056927(n) - A053000(n))/2.

More terms from James Sellers, Jul 13 2000

A056928 Average of the smallest prime greater than n^2 and the largest prime less than n^2.

Original entry on oeis.org

4, 9, 15, 26, 34, 50, 64, 81, 99, 120, 144, 170, 195, 225, 254, 288, 324, 363, 399, 441, 483, 532, 574, 625, 675, 730, 780, 846, 897, 960, 1026, 1089, 1158, 1226, 1294, 1370, 1443, 1517, 1599, 1681, 1768, 1854, 1941, 2022, 2121, 2210, 2303, 2405, 2490

Offset: 2

Henry Bottomley, Jul 12 2000

			a(4)=15 because the smallest prime greater than 4^2 is 17, the largest prime less than 4^2 is 13, and the average of 17 and 13 is 15.

Hugo Pfoertner, Table of n, a(n) for n = 2..10000

Cf. A007491, A053000, A053001, A056927, A056929, A056930, A056931.

Table[n2=n^2;(NextPrime[n2,-1]+NextPrime[n2])/2,{n,2,100}] (* Vladimir Joseph Stephan Orlovsky, Mar 09 2011 *)

a(n) = (nextprime(n^2) + precprime(n^2))/2; \\ Michel Marcus, May 20 2018

a(n) = (A007491(n) - A053001(n))/2.
a(n) = A000290(n) + (A053000(n) - A056927(n))/2.
a(n) = A000290(n) - A056929(n).

A056930 Average of smallest prime greater than n^2 and largest prime less than (n+1)^2.

Original entry on oeis.org

6, 12, 20, 30, 42, 57, 73, 90, 107, 133, 158, 183, 210, 239, 270, 305, 345, 382, 420, 461, 505, 556, 598, 652, 702, 753, 813, 870, 930, 994, 1059, 1122, 1193, 1260, 1332, 1406, 1479, 1560, 1635, 1726, 1812, 1897, 1983, 2070, 2168, 2255, 2354, 2444

Offset: 2

Henry Bottomley, Jul 12 2000

a(1)=2.5 which is not an integer

			a(4)=1 because smallest prime greater than 4^2 is 17, largest prime less than 5^2 is 23 and average of 17 and 23 is 20

Harvey P. Dale, Table of n, a(n) for n = 2..1000

Cf. A002378, A007491, A053000, A053001, A056927, A056928, A056929, A056931.

Table[Mean[{NextPrime[n^2],NextPrime[(n+1)^2,-1]}],{n,2,50}] (* Harvey P. Dale, May 10 2019 *)

a(n) =(A007491(n)+A053001(n+1))/2 =A002378(n)-A056931(n)

cp's OEIS Frontend

A056927 Difference between n^2 and largest prime less than n^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A056929 Difference between n^2 and average of smallest prime greater than n^2 and largest prime less than n^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A056928 Average of the smallest prime greater than n^2 and the largest prime less than n^2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A056930 Average of smallest prime greater than n^2 and largest prime less than (n+1)^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula