A056898 -id:A056898 - cp's OEIS frontend

A056892 a(n) = square excess of the n-th prime.

1, 2, 1, 3, 2, 4, 1, 3, 7, 4, 6, 1, 5, 7, 11, 4, 10, 12, 3, 7, 9, 15, 2, 8, 16, 1, 3, 7, 9, 13, 6, 10, 16, 18, 5, 7, 13, 19, 23, 4, 10, 12, 22, 24, 1, 3, 15, 27, 2, 4, 8, 14, 16, 26, 1, 7, 13, 15, 21, 25, 27, 4, 18, 22, 24, 28, 7, 13, 23, 25, 29, 35, 6, 12, 18, 22, 28, 36, 1, 9, 19, 21

Offset: 1

Henry Bottomley, Jul 05 2000

nonn

			a(5) = 2 since the 5th prime is 11 = 3^2 + 2.
From _M. F. Hasler_, Oct 19 2018: (Start)
Written as a table, starting a new row when a square is reached, the sequence reads:
  1, 2,    // = 2 - 1, 3 - 1 = {primes between 1^2 = 1 and 2^2 = 4} - 1
  1, 3,     // = 5 - 4, 7 - 4 = {primes between 2^2 = 4 and 3^2 = 9} - 4
  2, 4,      // = 11 - 9, 13 - 9 = {primes between 3^2 = 9 and 4^2 = 16} - 9
  1, 3, 7,    // = 17 - 16, 19 - 16, 23 - 16 = {primes between 16 and 25} - 16
  4, 6,        // = 29 - 25, 31 - 25 = {primes between 5^2 = 25 and 6^2 = 36} - 25
  1, 5, 7, 11,  // = {37, 41, 43, 47: primes between 6^2 = 36 and 7^2 = 49} - 36
  4, 10, 12,    // = {53, 59, 61: primes between 7^2 = 49 and 8^2 = 64} - 49
  3, 7, 9, 15,  // = {67, 71, 73, 79: primes between 8^2 = 64 and 9^2 = 81} - 64
  2, 8, 16,     // = {83, 89, 97: primes between 9^2 = 81 and 10^2 = 100} - 81
  etc. (End)

M. F. Hasler, Table of n, a(n) for n = 1..10000

Cf. A000040, A000196, A002496, A048760, A053186, A056893 .. A056898.

When written as a table, row lengths are A014085, and row sums are A108314 - A014085 * A000290 = A320688.

lst={};Do[p=Prime[n];s=p^(1/2);f=Floor[s];a=f^2;d=p-a;AppendTo[lst,d],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)
#-Floor[Sqrt[#]]^2&/@Prime[Range[90]] (* Harvey P. Dale, Jul 06 2014 *)

A056892(n)={my(p=prime(n));p-sqrtint(p)^2} \\ M. F. Hasler, Oct 04 2009

a(n) = A053186(A000040(n)).

a(n) = A000040(n) - A000006(n)^2. - M. F. Hasler, Oct 04 2009

A056893 Smallest prime with square excess of n.

Original entry on oeis.org

2, 3, 7, 13, 41, 31, 23, 89, 73, 59, 47, 61, 113, 239, 79, 97, 593, 139, 163, 461, 277, 191, 167, 193, 281, 251, 223, 317, 353, 991, 431, 761, 433, 563, 359, 397, 521, 479, 439, 569, 617, 571, 619, 773, 829, 887, 947, 673, 1493, 1571, 727, 1013, 953, 1279

Offset: 1

Henry Bottomley, Jul 05 2000

nonn

			a(4)=13 since 13=3^2+4, while 2, 3, 5, 7 and 11 have square excesses of 1, 2, 1, 3 and 3 respectively.

Cf. A053186, A000196, A002496, A048760, A056892-A056898.

A056893 := proc(n)
    local p ;
    p :=2 ;
    while A053186(p) <> n do
        p := nextprime(p) ;
    end do:
    return p ;
end proc: # R. J. Mathar, Jul 28 2013

A056893(n)={
    local(p=2) ;
    while( A053186(p)!=n,
        p=nextprime(p+1)
    ) ;
    return(p)
} /* R. J. Mathar, Jul 28 2013 */

a(n) =n+A056894(n).

a(n) = min{p in A000040: A053186(p) = n}. - R. J. Mathar, Jul 28 2013

A056896 Smallest prime which can be written as k^2 + n for k >= 0.

Original entry on oeis.org

2, 2, 3, 5, 5, 7, 7, 17, 13, 11, 11, 13, 13, 23, 19, 17, 17, 19, 19, 29, 37, 23, 23, 73, 29, 107, 31, 29, 29, 31, 31, 41, 37, 43, 71, 37, 37, 47, 43, 41, 41, 43, 43, 53, 61, 47, 47, 73, 53, 59, 67, 53, 53, 79, 59, 137, 61, 59, 59, 61, 61, 71, 67, 73, 101, 67, 67, 149, 73, 71

Offset: 1

Henry Bottomley, Jul 05 2000

			a(8)=17 because 17=3^2+8.

Cf. A000040, A002496, A056892-A056898.

Table[k = 0; While[p = n + k^2; ! PrimeQ[p], k++]; p, {n, 100}] (* T. D. Noe, Apr 01 2011 *)

a(n) = A056897(n)+n = A056898(n)^2+n.

For p a prime: a(p)=p (and a(p-1)=p if p<>3).

Example corrected by Harvey P. Dale, Apr 01 2011

A056897 Smallest square where a(n)+n is prime.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 9, 4, 1, 0, 1, 0, 9, 4, 1, 0, 1, 0, 9, 16, 1, 0, 49, 4, 81, 4, 1, 0, 1, 0, 9, 4, 9, 36, 1, 0, 9, 4, 1, 0, 1, 0, 9, 16, 1, 0, 25, 4, 9, 16, 1, 0, 25, 4, 81, 4, 1, 0, 1, 0, 9, 4, 9, 36, 1, 0, 81, 4, 1, 0, 1, 0, 9, 4, 25, 36, 1, 0, 9, 16, 1, 0, 25, 4, 81, 16, 1, 0, 49, 16, 9, 4, 9

Offset: 0

Henry Bottomley, Jul 05 2000

nonn

			a(8)=9 since 9 is a square and 9+8=7 which is a prime

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A000040, A002496, A056892-A056898.

With[{sqs=Range[0,20]^2},Table[SelectFirst[sqs,PrimeQ[n+#]&],{n,100}]] (* The program uses the SelectFirst function from Mathematica version 10 *) (* Harvey P. Dale, May 07 2016 *)

a(n) =A056896(n)-n =A056898(n)^2

A056894 If the smallest prime with a square excess of n is p then a(n)=p-n.

Original entry on oeis.org

1, 1, 4, 9, 36, 25, 16, 81, 64, 49, 36, 49, 100, 225, 64, 81, 576, 121, 144, 441, 256, 169, 144, 169, 256, 225, 196, 289, 324, 961, 400, 729, 400, 529, 324, 361, 484, 441, 400, 529, 576, 529, 576, 729, 784, 841, 900, 625, 1444, 1521, 676, 961, 900, 1225, 784

Offset: 1

Henry Bottomley, Jul 05 2000

nonn

			a(4)=9 because the smallest prime with a square excess of 4 is 13 and 13-4=9

Cf. A000040, A000196, A002496, A048760, A053186, A056892-A056898.

a(n) = A056893(n) - n = A048760(A056893(n)) = A056895(n)^2

A056895 If the smallest prime with a square excess of n is p then a(n)^2 = p - n.

Original entry on oeis.org

1, 1, 2, 3, 6, 5, 4, 9, 8, 7, 6, 7, 10, 15, 8, 9, 24, 11, 12, 21, 16, 13, 12, 13, 16, 15, 14, 17, 18, 31, 20, 27, 20, 23, 18, 19, 22, 21, 20, 23, 24, 23, 24, 27, 28, 29, 30, 25, 38, 39, 26, 31, 30, 35, 28, 45, 34, 31, 42, 31, 34, 33, 32, 33, 36, 35, 34, 75, 40, 37, 36, 41, 48, 45

Offset: 1

Henry Bottomley, Jul 05 2000

nonn

			a(4)=3 because the smallest prime with a square excess of 4 is 13 and 13 - 4 = 3^2.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A000040, A000196, A002496, A048760, A053186.

Cf. A056892, A056893, A056894, A056896, A056897, A056898.

a = {}; Do[p = 2; While[n != p - (r = Floor@Sqrt[p])^2, p = NextPrime[p]]; AppendTo[a, r], {n, 74}]; a (* Ivan Neretin, May 02 2019 *)

a(n) = {my(p=2); while(n != p-sqrtint(p)^2, p = nextprime(p+1)); sqrtint(p - n);} \\ Michel Marcus, May 05 2019

a(n) = sqrt(A056893(n)-n) = A000196(A056893(n)) = sqrt(A056894(n)).

cp's OEIS Frontend

A056892 a(n) = square excess of the n-th prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A056893 Smallest prime with square excess of n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

PARI

Formula

A056896 Smallest prime which can be written as k^2 + n for k >= 0.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

A056897 Smallest square where a(n)+n is prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A056894 If the smallest prime with a square excess of n is p then a(n)=p-n.

Views

Author

Keywords

Examples

Crossrefs

Formula

A056895 If the smallest prime with a square excess of n is p then a(n)^2 = p - n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula