A056929 -id:A056929 - cp's OEIS frontend

A101593 a(n) is the number of m <= 2^n which are in A075190, i.e., such that m^2 is exactly at the center between two consecutive primes, or in other words A056929(m) = 0.

1, 2, 3, 6, 9, 14, 19, 34, 62, 109, 202, 336, 587, 1100, 2003, 3630, 6784, 12607, 23647, 44206, 83510, 157851, 299810, 571264, 1090986, 2088445, 4004347, 7687694, 14788984, 28496850, 54955214, 106159961

Offset: 1

Zak Seidov and M. F. Hasler, Dec 27 2007

A056929(m) = 0 iff m^2 is an interprime <=> m^2 = (A007491(m^2) + A053001(m^2))/2 = average of the next higher and next lower primes.
From M. F. Hasler, Oct 18 2022: (Start)
The ratio a(n+1)/a(n) oscillates between 1.5 and 2 for the first few values, but then seems to converge to a limit between 1.9 and 2; from n = 19 on these ratios appear to be strictly increasing (from 1.87 at n = 19 to 1.92 at n = 27).
At first sight it seems natural that there are twice as many interprimes of the form f(m) when the upper limit on m is increased by a factor of 2, but this might depend on the function f.
If instead of m^2 we consider the same for m^3, then we find the sequence 0, 1, 1, 3, 5, 8, 18, 29, 52, 86, 136, 223, 421, 758, 1376, 2517, 4616, 8714, 16173, 30414, 57583, 109539, ... which follows roughly the same asymptotic behavior. (End)

Cf. A007491, A053001, A056929, A069495, A075190.

PARI
```
a(n)=sum(i=2,2^n,!A056929(i))
```

a(n)=sum(i=2,2^n,nextprime(i^2)+precprime(i^2)==2*i^2)

t=0;vector(15,n,t+=sum(i=1/2<M. F. Hasler, Oct 18 2022] */
for(n=16, 30, print1("/* a("n") = */ ", t += sum(i=2^(n-1)+1,2^n, nextprime(i^2)+precprime(i^2)==2*i^2),", "))

a(1) counts the squares m^2 with m <= 2^n = 2 which are interprimes. The squares 0^2 = 0 and 1^2 = 1 obviously aren't interprimes, so the only such square in that range is m^2 = 2^2 = 4 = (nextprime + precprime)/2 = (3 + 5)/2, so a(1) = 1.

Then for n = 2, up to m <= 2^n = 4 we have the additional squares m^2 = 3^2 = 9 = (7 + 11)/2 (an interprime) and m^2 = 4^2 = 16 <> (13 + 17)/2 = 15, so this m^2 is not an interprime, and a(2) = a(1) + 1 = 2.

a(23)-a(25) from Kevin P. Thompson, Nov 26 2021
a(26)-a(28) from M. F. Hasler, Oct 18 2022
a(29)-a(32) from Bill McEachen, Dec 14 2022

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

Original entry on oeis.org

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

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

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)

A133450 Difference between 4*n^2 and the average of the two prime numbers which bracket this number.

Original entry on oeis.org

0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 1, 2, 1, 4, 3, -2, -2, 2, 1, 1, -4, -5, -5, 1, 10, 1, 3, 7, -2, 0, 4, 0, 3, -5, 4, 0, 2, 12, 0, -9, -2, 6, -6, -3, 3, 0, 2, 1, -3, 10, -9, 1, 10, -3, 1, 0, 4, 2, -2, 5, 1, 1, 8, -12, 5, -1, 8, -2, 0, 0, -3, -1, 1, 2, 8, -4, 12, 3, 4, 5, 1, -2, -10, 0, 10

Offset: 1

Alexander R. Povolotsky, Dec 22 2007

sign

			a(1)=0 because 4 - (3 + 5)/2 = 0
a(2)=1 because 16 - (13 + 17)/2 = 1
a(3)=2 because 36 - (31 + 37)/2 = 2
a(4)=0 because 64 - (61 + 67)/2 = 0
a(5)=1 because 100 - (97 + 101)/2 = 1

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A056929, A101593, A075190.

Table[n^2-(Prime[PrimePi[n^2]]+Prime[PrimePi[n^2]+1])/2,{n,2,200,2}] (* Zak Seidov *)
diff4[n_]:=Module[{x=4n^2},x-(NextPrime[x]+NextPrime[x,-1])/2]; Array[ diff4,90] (* Harvey P. Dale, Aug 31 2017 *)

A133450(n)=4*n^2-(precprime(4*n^2)+nextprime(4*n^2))/2 \\ M. F. Hasler, Dec 26 2007

a(n) = A056929(2n). - M. F. Hasler, Dec 26 2007

Corrected and extended by Zak Seidov, Dec 23 2007

Edited by N. J. A. Sloane, Dec 23 2007

cp's OEIS Frontend

A101593 a(n) is the number of m <= 2^n which are in A075190, i.e., such that m^2 is exactly at the center between two consecutive primes, or in other words A056929(m) = 0.

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A056927 Difference between 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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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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A133450 Difference between 4*n^2 and the average of the two prime numbers which bracket this number.

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