A056927 -id:A056927 - cp's OEIS frontend

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

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

A056931 Difference between n-th oblong (promic) number, n(n+1), and the average of the smallest prime greater than n^2 and the largest prime less than (n+1)^2.

Original entry on oeis.org

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

Offset: 2

Henry Bottomley, Jul 12 2000

a(1)=-0.5 which is not an integer

			a(4)=0 because smallest prime greater than 4^2 is 17, largest prime less than 5^2 is 23, average of 17 and 23 is 20 and 4*5-20=0

Cf. A002378, A007491, A053000, A053001, A056927, A056928, A056929, A056930.

with(numtheory): A056931 := n-> n*(n+1)-(prevprime((n+1)^2)+nextprime(n^2))/2);

a(n) =A002378(n)-(A007491(n)+A053001(n+1))/2 =A002378(n)-A056930(n).

More terms from James Sellers, Jul 13 2000

A060272 Distance from n^2 to closest prime.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 6, 5, 2, 1, 2, 1, 4, 7, 2, 1, 2, 3, 6, 1, 6, 1, 2, 3, 2, 7, 6, 3, 2, 3, 2, 1, 2, 3, 2, 1, 12, 5, 2, 3, 2, 3, 2, 5, 2, 3, 8, 3, 6, 1, 2, 1, 2, 3, 10, 7, 2, 3, 2, 3, 4, 1, 4, 3, 2, 3, 2, 5, 4, 1, 2, 3, 2, 5, 6, 3, 2, 5, 6, 1, 4, 3, 4, 3, 2, 1, 6, 3, 2, 1, 4, 5, 4, 3, 2, 7, 8, 5, 2

Offset: 1

Labos Elemer, Mar 23 2001

nonn

			n=1: n^2=1 has next prime 2, so a(1)=1;
n=11: n^2=121 is between primes {113,127} and closer to 127, thus a(11)=6.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A007491, A053001, A058043, A056927, A053000, A051700, A060268-A060269, A060272, A059959, A059790.

seq((s-> min(nextprime(s)-s, `if`(s>2, s-prevprime(s), [][])))(n^2), n=1..256);  # edited by Alois P. Heinz, Jul 16 2017

Table[Function[k, Min[k - #, NextPrime@ # - k] &@ If[n == 1, 0, Prime@ PrimePi@ k]][n^2], {n, 103}] (* Michael De Vlieger, Jul 15 2017 *)
Min[#-NextPrime[#,-1],NextPrime[#]-#]&/@(Range[110]^2) (* Harvey P. Dale, Jun 26 2021 *)

a(n) = if (n==1, nextprime(n^2) - n^2, min(n^2 - precprime(n^2), nextprime(n^2) - n^2)); \\ Michel Marcus, Jul 16 2017

a(n) = abs(A000290(n) - A113425(n)) = abs(A000290(n) - A113426(n)). - Reinhard Zumkeller, Oct 31 2005

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)

A309726 Numbers k such that k^2 - 12 is prime.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 25, 29, 35, 41, 49, 53, 59, 61, 79, 85, 91, 95, 97, 103, 107, 113, 119, 121, 137, 139, 145, 149, 163, 169, 173, 179, 181, 185, 191, 205, 209, 227, 233, 235, 245, 251

Offset: 1

Daniel Starodubtsev, Aug 14 2019

nonn

All terms are odd and not divisible by 3.

			11 is in the sequence because 11^2 - 12 = 109, which is prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A028870, A028873, A028876, A028879, A028882, A028885, A186815, A090696, A296507.

Cf. A056927.

Select[Range[5,301,2],PrimeQ[#^2-12]&] (* Harvey P. Dale, Dec 23 2019 *)

select(n->isprime(n^2-12), [1..1000]) \\ Andrew Howroyd, Aug 14 2019

If A056927(k) = 12, then k is a term. - A.H.M. Smeets, Aug 15 2019

A187409 n^2 + nextprime(n^2).

Original entry on oeis.org

3, 9, 20, 33, 54, 73, 102, 131, 164, 201, 248, 293, 342, 393, 452, 513, 582, 655, 728, 801, 884, 971, 1070, 1153, 1256, 1353, 1462, 1571, 1694, 1807, 1928, 2055, 2180, 2319, 2454, 2593, 2742, 2891, 3044, 3201, 3374, 3541, 3710, 3885, 4052, 4245, 4422, 4613

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 09 2011

nonn

			1^2+2=3, 2^2+5=9, 3^2+11=20,..

Cf. A053000, A056927, A007491.

Table[n2=n^2; NextPrime[n2]+n2, {n,100}]
#+NextPrime[#]&/@(Range[100]^2) (* Harvey P. Dale, Sep 20 2022 *)

A253474 Numbers n such that the difference between n^2 and largest prime less than n^2 is not prime.

Original entry on oeis.org

2, 11, 17, 23, 25, 31, 39, 41, 45, 51, 53, 56, 57, 59, 65, 67, 73, 76, 79, 81, 83, 85, 87, 91, 95, 97, 99, 100, 101, 105, 109, 111, 113, 115, 123, 125, 129, 133, 137, 141, 143, 147, 149, 151, 153, 154, 157, 159, 163, 165, 167, 170, 171, 175, 179, 181, 185, 187, 189, 193, 195, 197, 199, 201, 203, 207, 209, 213, 215, 219, 221, 225

Offset: 1

Carlos Eduardo Olivieri, Mar 17 2015

Indices of terms in A056927 that are not prime.

			a(1) = 2, since 2^2 - 3 = 1.
a(2) = 11, since 11^2 - 113 = 8.
a(3) = 17, since 17^2 - 283 = 6.
a(4) = 23, since 23^2 - 523 = 6.

Richard L. Francis, Between consecutive squares, Missouri Journal of Mathematical Sciences, Volume 16 Issue #1 - Winter 2004.

Cf. A000040, A000290, A056927, A053001.

f[n_] := n^2 - NextPrime[n^2, -1]; Select[Range[2, 230], !PrimeQ[f[#]] &]

lista(nn) = for (n=2, nn, if (!isprime(n^2-precprime(n^2)), print1(n, ", "))); \\ Michel Marcus, Mar 22 2015

cp's OEIS Frontend

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

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A056931 Difference between n-th oblong (promic) number, n(n+1), and the average of the smallest prime greater than n^2 and the largest prime less than (n+1)^2.

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A060272 Distance from n^2 to closest prime.

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

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

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A309726 Numbers k such that k^2 - 12 is prime.

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A187409 n^2 + nextprime(n^2).

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