A056894 -id:A056894

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A056898

a(n) = smallest number m such that m^2+n is prime.

+10
7

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,8

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

FORMULA

a(n) = sqrt(A056896(n)-n) = sqrt(A056897(n)).

For p a prime: a(p) = 0 (and a(p-1) = 1 if p<>3).

EXAMPLE

a(8) = 3 since 3^2+8 = 17 which is prime.

PROG

(PARI) A056898(n) = { my(m=0); while(!isprime((m*m)+n), m++); (m); }; \\ Antti Karttunen, Mar 04 2018

CROSSREFS

Cf. A000040, A002496, A056892, A056893, A056894, A056895, A056896, A056897.

KEYWORD

nonn

AUTHOR

Henry Bottomley, Jul 05 2000

STATUS

approved

A056893

Smallest prime with square excess of n.

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..54.

FORMULA

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

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

EXAMPLE

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.

MAPLE

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

PROG

(PARI) A056893(n)={

local(p=2) ;

while( A053186(p)!=n,

p=nextprime(p+1)

) ;

return(p)

} /* R. J. Mathar, Jul 28 2013 */

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Henry Bottomley, Jul 05 2000

STATUS

approved

A056895

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

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

FORMULA

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

EXAMPLE

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

MATHEMATICA

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 *)

PROG