A007491 -id:A007491 - cp's OEIS frontend

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

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

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

A144832 Distance from nxtprm(n^2) to (n+1)^2 in A144831 is prime.

Original entry on oeis.org

2, 5, 7, 11, 17, 17, 23, 29, 31, 41, 47, 67, 71, 71, 89, 89, 97, 113, 109, 107, 127, 131, 137, 157, 167, 173, 173, 191, 197, 193, 197, 227, 233, 227, 251, 257, 271, 293, 271, 307, 313, 317, 331, 349, 353, 383, 383, 409, 419, 431, 449, 463, 467, 487, 503, 509

Offset: 1

Enoch Haga, Sep 22 2008

			a(2)=5 because 3^2=9 and 4^2=16. Nxtprm(3^2)=11 and 16-11=5, a prime.

A007491 A053000 A144831

Terms in this sequence are from A144831 iff the distance from nxtprm n^2 to (n+1)^2 is prime.

A158061 a(1)=2, a(n+1) is the smallest prime > n^smallest digit of a(n).

Original entry on oeis.org

2, 2, 5, 251, 5, 3137, 7, 823547, 67, 531457, 11, 13, 13, 17, 17, 17, 17, 19, 19, 23, 401, 2, 487, 279847, 577, 9765629, 677, 387420499, 2, 853, 27011, 2, 1031, 2, 1163, 37, 46663, 50671, 2, 1523, 41, 43, 74093, 2, 1949, 47, 4477457, 4879687, 5308417, 2

Offset: 1

Juri-Stepan Gerasimov, Mar 12 2009

			2, 2(>1=1^2), 5(>4=2^2), 251(>241=3^5), 5(>4=4^1), 3137(>3125=5^5), 7(>6=6^1).

Cf. A007491, A156615.

nxt[{n_,a_}]:={n+1,NextPrime[(n+1)^Min[IntegerDigits[a]]]}; Join[ {2},NestList[ nxt,{1,2},50][[All,2]]] (* Harvey P. Dale, Nov 18 2021 *)

More terms from R. J. Mathar, Mar 17 2009

Edited by Charles R Greathouse IV, Mar 25 2010

A181616 a(1)=5; thereafter a(2n) = nextprime(a(2n-1)^2), a(2n+1) = nextprime(floor(2*a(2n)/(a(2n-1) + 1))) where nextprime(.) is A007918(.).

Original entry on oeis.org

5, 29, 11, 127, 23, 541, 47, 2213, 97, 9413, 193, 37253, 389, 151337, 787, 619373, 1579, 2493259, 3163, 10004573, 6329, 40056253, 12659, 160250297, 25321, 641153069, 50647, 2565118639, 101293, 10260271859, 202591, 41043113401, 405199

Offset: 1

Bill McEachen, Jan 30 2011

nonn

This gives a sawtooth log plot a bit reminiscent of Goldbach's comet, with wave frequency and amplitude increasing indefinitely. I started at 5 for no particular reason.

The two "lines" in the graph approach ratio 2.0 and 4.0 respectively for consecutive terms. The two are then (5, 11, 23, 47, ...) and (29, 127, 541, 2213, ...). - Bill McEachen, Sep 27 2013

			Beginning at 5 (n=1), a(2) via nextprime(5^2) = 29.
Divisor = ceiling(5/2) = 3 so a(3) = nextprime(floor(29/3)) = 11.
Then repeat: a(4) via nextprime(11^2) = 127.
Divisor = ceiling(11/2) = 6 so a(5) = nextprime(floor(127/6)) = 23.

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

A007491 := proc(n) nextprime(n^2) ; end proc:
A181616 := proc(n) option remember; if n = 1 then 5; elif type(n,'even') then A007491(procname(n-1)) ; else 2*procname(n-1)/(procname(n-2)+1) ; nextprime(floor(%)) ; end if; end proc: # R. J. Mathar, Feb 09 2011

a[1] = 5; a[n_] := a[n] = If[OddQ@ n, NextPrime[ a[n - 1]/Ceiling[ a[n - 2]/2]], NextPrime[ a[n - 1]^2]]; Array[a, 33]

\\ example call newseq9(2,50) to use square power, 1st 50 terms
\\  I never tried any power but 2
newseq9(a,iend)=
{
a=floor(a);
if(a<2,a=2);
i5=5;
print(i5);
for(n=1,iend,
  i6=nextprime(i5^a);
  b=ceil(i5/2);   \\ vary as f{i5}
  i7=nextprime(floor(i6/b));
  print(i6);
  print(i7);
  i5=i7
);  \\end FOR
print("Designed pgm exit (a,b) ...",a," , ",b);
}

A187872 Second smallest prime after 2^n.

Original entry on oeis.org

3, 5, 7, 13, 19, 41, 71, 137, 263, 523, 1033, 2063, 4111, 8219, 16417, 32779, 65539, 131111, 262151, 524341, 1048589, 2097211, 4194329, 8388619, 16777289, 33554473, 67108913, 134217773, 268435463, 536870951, 1073741831, 2147483693

Offset: 0

Vladimir Joseph Stephan Orlovsky, Mar 14 2011

nonn

			2^2=4, second smallest prime=7;
2^3=8, second smallest prime=13; ..

Robert Israel, Table of n, a(n) for n = 0..3318

Cf. A007491, A014210.

seq(nextprime(nextprime(2^n)),n=0..100); # Robert Israel, Nov 04 2020

Mathematica
```
NextPrime[2^Range[0,100], 2]
```

a(n) = nextprime(nextprime(2^n+1)+1); \\ Michel Marcus, Nov 05 2020

A187873 Second smallest prime after n^2.

Original entry on oeis.org

3, 3, 7, 13, 19, 31, 41, 59, 71, 89, 103, 131, 151, 179, 199, 229, 263, 307, 337, 373, 409, 449, 491, 547, 587, 641, 683, 739, 797, 857, 911, 971, 1033, 1093, 1171, 1231, 1301, 1381, 1451, 1531, 1607, 1697, 1783, 1867, 1951, 2029

Offset: 0

Vladimir Joseph Stephan Orlovsky, Mar 14 2011

nonn

From Robert Israel, Dec 18 2018: (Start)
Oppermann's conjecture implies a(n) < (n+1)^2 for n > 0.
For n > 1, a(n) >= n^2 + 3, with equality for n in A080149. (End)

			2^2=4, second smallest prime=7;
3^2=9, second smallest prime=13; ..

Robert Israel, Table of n, a(n) for n = 0..10000
Wikipedia, Oppermann's conjecture

Cf. A007491, A014210, A080149.

seq(nextprime(nextprime(n^2)),n=0..50); # Robert Israel, Dec 18 2018

Mathematica
```
NextPrime[Range[0,100]^2, 2]
```

A236627 Number of positive integers <= sqrt(n) not dividing n.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 2, 0, 2, 1, 1, 1, 3, 1, 3, 1, 2, 2, 3, 0, 3, 3, 3, 2, 4, 1, 4, 2, 3, 3, 3, 1, 5, 4, 4, 2, 5, 2, 5, 3, 3, 4, 5, 1, 5, 4, 5, 4, 6, 3, 5, 3, 5, 5, 6, 1, 6, 5, 4, 4, 6, 4, 7, 5, 6, 4, 7, 2, 7, 6, 5, 5, 6, 4, 7, 3, 6, 7, 8, 3, 7, 7, 7, 5, 8, 3, 7, 6, 7, 7, 7, 3, 8, 6, 6, 5, 9

Offset: 1

Omar E. Pol, Feb 05 2014

nonn

It appears that the indices of the zeros in the sequence are in A018253.

Omar E. Pol, , Three illustrations of A000005 giving the structure of this sequence, (2013)

Cf. A000005, A000196, A010052, A018253, A038548, A007491, A160812, A161205, A212119, A228813, A228814, A229940, A229942, A229944, A229950, A229951.

a(n) = sum(i=1, sqrtint(n), (n % i) != 0); \\ Michel Marcus, Mar 16 2014

a(n) = A000196(n) - A038548(n) = A160812(n)/2.

A331698 a(n) = (1/2) * ((greatest prime < (n+1)^2) - (least prime > n^2)) for n >= 2.

Original entry on oeis.org

1, 1, 3, 1, 5, 4, 6, 7, 6, 6, 9, 10, 13, 12, 13, 12, 14, 15, 19, 18, 18, 15, 21, 21, 25, 20, 26, 17, 23, 27, 28, 31, 30, 31, 35, 33, 32, 37, 34, 33, 35, 36, 34, 43, 39, 42, 45, 33, 45, 45, 48, 45, 53, 42, 46, 55, 49, 51, 56, 53, 60, 52, 60, 60, 63, 64, 61, 53

Offset: 2

Hugo Pfoertner, Jan 27 2020

nonn

			a(2) = 1 because 7 is the greatest prime < 3^2 and 5 is the least prime > 2^2. (7-5)/2 = 1.

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

Cf. A007491, A053001, A132657.

for(n=2, 69, print1((precprime((n+1)^2)-nextprime(n^2))/2, ", "))

A379444 a(n) is the difference between the least prime > (n+1)^2 and the largest prime < n^2, divided by 2.

Original entry on oeis.org

4, 5, 8, 7, 11, 10, 11, 11, 15, 18, 17, 15, 17, 17, 21, 24, 25, 21, 23, 24, 31, 27, 30, 29, 30, 30, 40, 34, 40, 39, 35, 38, 38, 37, 41, 40, 42, 45, 48, 54, 51, 51, 47, 56, 50, 51, 57, 52, 66, 57, 60, 57, 64, 57, 65, 71, 65, 69, 67, 64, 78, 66, 68, 69, 72, 77, 81

Offset: 2

Hugo Pfoertner, Dec 23 2024

2*a(n) would be the gap needed between consecutive primes to provide a counterexample to Legendre's conjecture that there is always a prime between n^2 and (n+1)^2. The gaps actually observed are significantly smaller; see A378904 for comparison.

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

Cf. A001223, A007491, A053001, A058043, A350100, A378904.

a[n_]:=(NextPrime[(n+1)^2] - NextPrime[n^2,-1])/2; Array[a,67,2] (* Stefano Spezia, Jan 24 2025 *)

a379444(n) = (nextprime((n+1)^2) - precprime(n^2))/2

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

a(n) >= n + 2.

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

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.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

PARI

PARI

Formula

Extensions

A144832 Distance from nxtprm(n^2) to (n+1)^2 in A144831 is prime.

Views

Author

Keywords

Examples

Crossrefs

Formula

A158061 a(1)=2, a(n+1) is the smallest prime > n^smallest digit of a(n).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Extensions

A181616 a(1)=5; thereafter a(2n) = nextprime(a(2n-1)^2), a(2n+1) = nextprime(floor(2*a(2n)/(a(2n-1) + 1))) where nextprime(.) is A007918(.).

Views

Author

Keywords

Comments

Examples

Links

Programs

Maple

Mathematica

PARI

A187872 Second smallest prime after 2^n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A187873 Second smallest prime after n^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A236627 Number of positive integers <= sqrt(n) not dividing n.

Views

Author

Keywords

Comments