A053001 -id:A053001 - cp's OEIS frontend

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

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)

A090119 a(n) = nextprime(A090117(n)), the smallest prime following squares listed in A090117 and also the distance of a(n) from the preceding prime is 2*n.

Original entry on oeis.org

5, 11, 29, 367, 149, 631, 127, 1949, 541, 907, 3251, 1693, 2503, 10427, 5779, 10831, 10007, 22229, 30631, 25301, 121123, 76207, 93047, 157627, 212557, 35729, 119027, 1121509, 190979, 672439, 693943, 1004027, 259099, 1646101, 675713, 1207841

Offset: 1

Labos Elemer, Jan 09 2004

nonn

			a(7) = 127 because 127-113 = 14 = 2*7 and 121 = 11^2 is between {127,113} closest primes to 121 a suitable square number. Also 127 is the smallest prime with this property.

Cf. A090116, A090117, A090118, A007917, A007918, A000720, A000040, A053001, A007491, A000290.

pre[x_] := Prime[PrimePi[x]]; nex[x_] := Prime[PrimePi[x]+1]; de[x_] := Prime[PrimePi[x]+1]-Prime[PrimePi[x]]; de[1] = 0; t=Table[de[w^2], {w, 1, 50000}]; mt=Table[Min[Flatten[Position[t, 2*j]]], {j, 1, 100}]; Table[nex[Part[mt, j]^2], {j, 1, Length[mt]}]

a(n) = nextprime(A090117(n)) = nextprime(A090116(n)^2).

a(n) = A007918(A090117(n)) = prime(1+pi(A090117(n))).

Name corrected by Jason Yuen, Jun 23 2025

A065384 Largest prime <= n * (n + 1) / 2.

Original entry on oeis.org

3, 5, 7, 13, 19, 23, 31, 43, 53, 61, 73, 89, 103, 113, 131, 151, 167, 181, 199, 229, 251, 271, 293, 317, 349, 373, 401, 433, 463, 491, 523, 557, 593, 619, 661, 701, 739, 773, 811, 859, 887, 941, 983, 1033, 1069, 1123, 1171, 1223, 1259, 1321, 1373, 1429, 1483

Offset: 2

Reinhard Zumkeller, Nov 05 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 2..1000

Cf. A065382, A053001, A064383, A000217.

PrimePrev[n_]:=Module[{k=n},While[ !PrimeQ[k],k-- ];k];f[n_]:=n*(n+1)/2;lst={};Do[AppendTo[lst,PrimePrev[f[n]]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Feb 26 2010 *)

{ for (n=2, 1000, write("b065384.txt", n, " ", precprime(n*(n + 1)/2)) ) } [Harry J. Smith, Oct 17 2009]

A090120 Numbers k such that nextprime(k^2) - prevprime(k^2) = 4.

Original entry on oeis.org

3, 4, 9, 10, 14, 15, 20, 21, 26, 33, 40, 110, 117, 124, 146, 206, 237, 250, 273, 303, 309, 326, 340, 350, 387, 429, 436, 440, 441, 447, 470, 513, 561, 573, 609, 634, 686, 704, 807, 897, 920, 1004, 1035, 1054, 1060, 1071, 1113, 1124, 1143, 1156, 1233, 1239

Offset: 1

Labos Elemer, Jan 09 2004

nonn

Note that the gap = 4 is partitioned either as 2+2 or as 3+1; 1+3 never occurs since n^2-1 is composite if n>2.

			k = 3 is a term since, k^2 = 9 is surrounded by the closest primes: {7,[9],11}.
k = 10 is a term since k^2 = 100 is surrounded by {97,[100],101}.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A090116, A090117, A090118, A090119, A007917, A007918, A000720, A000040, A053001, A007491, A000290.

Select[Range[3,1500], NextPrime[#^2] == NextPrime[#^2, -1] + 4 &] (* Giovanni Resta, May 26 2018 *)

isok(n) = nextprime(n^2) - precprime(n^2) == 4; \\ Michel Marcus, May 26 2018

Solutions to {x; A007918(x^2)-A007917(x^2) = 4}.

A132657 a(n) is the product of the least prime > n^2 and the greatest prime < (n+1)^2.

Original entry on oeis.org

6, 35, 143, 391, 899, 1739, 3233, 5293, 8051, 11413, 17653, 24883, 33389, 43931, 56977, 72731, 92881, 118829, 145699, 176039, 212197, 254701, 308911, 357163, 424663, 492179, 566609, 660293, 756611, 864371, 987307, 1120697, 1257923

Offset: 1

Jonathan Vos Post, Nov 15 2007

			a(1) = 6 = 2*3 = (smallest prime in [1^2,2^2]) * (largest prime in [1^2,2^2]).
a(2) = 35 = 5*7 = (smallest prime in [2^2,3^2]) * (largest prime in [2^2,3^2]).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A001358, A007491, A014085, A053000, A053001, A053607, A077766, A077767, A132435.

seq(nextprime(n^2)*prevprime((n+1)^2,n=1..100); # Robert Israel, Jan 26 2020

Table[Prime[PrimePi[n^2] + 1]*Prime[PrimePi[(n + 1)^2]], {n, 1, 40}] (* Stefan Steinerberger, Nov 20 2007 *)
NextPrime[#[[1]]]NextPrime[#[[2]],-1]&/@Partition[Range[40]^2,2,1] (* Harvey P. Dale, Aug 27 2022 *)

for(n=1,33,print1(nextprime(n^2)*precprime((n+1)^2),", ")) \\ Hugo Pfoertner, Jan 26 2020

a(n) = A007491(n) * A053001(n+1).

More terms from Stefan Steinerberger, Nov 20 2007

A243893 a(n) = prime(k-1) with k = n^2 + prime(n)^2.

Original entry on oeis.org

7, 37, 137, 311, 829, 1249, 2269, 2939, 4483, 7411, 8681, 12653, 15877, 17827, 21673, 28087, 35393, 38317, 46957, 53327, 56897, 67493, 75269, 87523, 105143, 115057, 120427, 130811, 136547, 147863, 189067, 202481, 222991, 230393, 267401, 275677

Offset: 1

Freimut Marschner, Jun 14 2014

nonn

prime(k-1) is also the largest prime number < (n^2 + prime(n)^2). Remark : Largest prime number < n^2 is A053001. Largest prime number < n^3 is A077037.

			n=1, 1^2=1, prime(1)^2 = 4, 1 + 4 = 5, 5 - 1= 4, prime(4) = 7 ;
n=2, 2^2=4, prime(2)^2 = 9, 4 + 9= 13, 13 - 1= 12, prime(12) = 37.

Freimut Marschner, Table of n, a(n) for n = 1..63

Cf. A000290 (squares n^2), A000040 (prime(n)), A001248 (prime(n)^2), A106587 (n^2 + prime(n)^2).

a[n_]:=Prime[(n^2 + Prime[n]^2) - 1]; Array[a,36] (* Stefano Spezia, Mar 12 2025 *)

a(n) = prime((n^2 + prime(n)^2) - 1) = prime(A106587(n) - 1).

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.

Original entry on oeis.org

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

A173831 Largest prime < n^4.

Original entry on oeis.org

13, 79, 251, 619, 1291, 2399, 4093, 6553, 9973, 14639, 20731, 28559, 38393, 50599, 65521, 83497, 104971, 130307, 159979, 194479, 234239, 279823, 331769, 390581, 456959, 531383, 614639, 707279, 809993, 923513, 1048573, 1185907, 1336333

Offset: 2

Vladimir Joseph Stephan Orlovsky, Feb 25 2010

nonn

Cf. A014220, A053001, A077037

PrimePrev[n_]:=Module[{k},k=n-1;While[ !PrimeQ[k],k-- ];k];f[n_]:=n^4;lst={};Do[AppendTo[lst,PrimePrev[f[n]]],{n,5!}];lst
NextPrime[Range[2,40]^4,-1] (* Harvey P. Dale, May 05 2018 *)

A180724 a(n) = n^2 + largest prime < n^2.

Original entry on oeis.org

7, 16, 29, 48, 67, 96, 125, 160, 197, 234, 283, 336, 389, 448, 507, 572, 641, 720, 797, 880, 963, 1052, 1147, 1244, 1349, 1456, 1557, 1680, 1787, 1914, 2045, 2176, 2309, 2448, 2587, 2736, 2883, 3032, 3197, 3350, 3523, 3696, 3869, 4042, 4229, 4416, 4601

Offset: 2

Ian Stewart, Sep 18 2010

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

#+NextPrime[#,-1]&/@(Range[2,50]^2) (* Harvey P. Dale, Apr 08 2023 *)

a(n) = n^2 + precprime(n^2-1); \\ Michel Marcus, Aug 23 2013

4 + 3 = 7; 9 + 7 = 16; 16 + 13 = 29; 25 + 23 = 48;
a(n) = n^2+A053001(n). - R. J. Mathar, Sep 19 2010
2n^2 - O(n^1.05) < a(n) < 2n^2. (Probably a much tighter lower bound is true.) - Charles R Greathouse IV, Jan 31 2023

A187918 Largest semiprime < n^2.

Original entry on oeis.org

6, 15, 22, 35, 46, 62, 77, 95, 119, 143, 166, 194, 221, 254, 287, 323, 358, 398, 437, 482, 527, 573, 623, 674, 723, 781, 838, 899, 959, 1018, 1082, 1154, 1219, 1294, 1366, 1441, 1517, 1594, 1679, 1763, 1843, 1934, 2021, 2105, 2206, 2302, 2395, 2498

Offset: 3

Jonathan Vos Post, Mar 15 2011

This is to semiprimes A001358 as A053001 is to primes A000040.

			Offset is 3 because there is no semiprime less than 2^2 = 4 (as 4 is the smallest semiprime).
a(3) = 6 because 6 is the largest semiprime less than 3^2 = 9 (itself a semiprime), with only the prime 7 and the triprime 8 properly in the [6,9] interval.
a(4) = 15 < 16 = 4^2.

Charles R Greathouse IV, Table of n, a(n) for n = 3..10000

Cf. A001358, A053001.

semiPrimeQ[n_] := Total[FactorInteger[n]][[2]] == 2; Table[k = n^2 - 1; While[! semiPrimeQ[k], k--]; k, {n, 3, 100}] (* T. D. Noe, Mar 15 2011 *)

issemi(n)=bigomega(2)==2
a(n)=n*=n; while(!issemi(n--),); n \\ Charles R Greathouse IV, Mar 16 2011

a(n) = MAX{k in A001358 and k < n^2}.

cp's OEIS Frontend

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

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A090119 a(n) = nextprime(A090117(n)), the smallest prime following squares listed in A090117 and also the distance of a(n) from the preceding prime is 2*n.

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A065384 Largest prime <= n * (n + 1) / 2.

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A090120 Numbers k such that nextprime(k^2) - prevprime(k^2) = 4.

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A132657 a(n) is the product of the least prime > n^2 and the greatest prime < (n+1)^2.

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A243893 a(n) = prime(k-1) with k = n^2 + prime(n)^2.

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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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A173831 Largest prime < n^4.