A056892 -id:A056892

A176032

Absolute values of A106044-A056892.

+20
3

1, 1, 3, 1, 3, 1, 7, 3, 5, 3, 1, 11, 3, 1, 9, 7, 5, 9, 11, 3, 1, 13, 15, 3, 13, 19, 15, 7, 3, 5, 11, 3, 9, 13, 15, 11, 1, 13, 21, 19, 7, 3, 17, 21, 27, 23, 1, 25, 27, 23, 15, 3, 1, 21, 31, 19, 7, 3, 9, 17, 21, 27, 1, 9, 13, 21, 23, 11, 9, 13, 21, 33, 27, 15, 3, 5, 17, 33, 39, 23, 3, 1, 21, 25

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

OFFSET

1,3

COMMENTS

A106044 2,1,4,2,5,3,8,6,2,7,5,12,8,6,2,11,.. A056892 1,2,1,3,2,4,1,3,7,4,6,1,5,7,11,4,10,.. 2-1=1,2-1=1,4-1=3,3-2=1,5-2=3,...

LINKS

Table of n, a(n) for n=1..84.

MATHEMATICA

f[n_]:=Floor[Sqrt[n]]; lst={}; Do[p=Prime[n]; AppendTo[lst, Abs[((f[p]+1)^2-p)-(p-f[p]^2)]], {n, 3*5!}]; lst

CROSSREFS

Cf. A056892, A106044

KEYWORD

nonn

AUTHOR

Vladimir Joseph Stephan Orlovsky, Apr 06 2010

STATUS

approved

A320688

Sum of the square excess A056892 of the primes between two squares.

+20
1

3, 4, 6, 11, 10, 24, 26, 34, 26, 33, 50, 67, 72, 46, 70, 109, 96, 132, 122, 153, 132, 145, 174, 229, 208, 175, 194, 287, 232, 244, 338, 267, 276, 345, 374, 239, 392, 396, 424, 390, 484, 373, 514, 563, 618, 424, 654, 821, 442, 557, 890, 814, 668, 741, 580, 642, 990, 811, 982, 968, 772

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

OFFSET

1,1

COMMENTS

Consider the primes p1,...,pK between two squares n^2 and (n+1)^2, and take the sum of the differences: (p1 - n^2) + ... + (pK - n^2). Obviously this equals (sum of these primes) - (number of these primes) * n^2.

LINKS

Table of n, a(n) for n=1..61.

FORMULA

a(n) = A108314(n) - A014085(n)*A000290(n), where A000290(n) = n^2.

PROG

(PARI) a(n, s=0)={forprime(p=n^2, (n+1)^2, s+=p-n^2); s}

CROSSREFS

Cf. A014085, A000290, A108314, A106044.

Row sums of A056892, read as a table.

Equals A108314 - A014085 * A000290.

KEYWORD

nonn

AUTHOR

M. F. Hasler, Oct 19 2018

STATUS

approved

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

A056896

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

+10
4

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

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

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..70.

FORMULA

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

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

MATHEMATICA

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

CROSSREFS

Cf. A000040, A002496, A056892-A056898.

KEYWORD

nonn

AUTHOR

Henry Bottomley, Jul 05 2000

EXTENSIONS

Example corrected by Harvey P. Dale, Apr 01 2011

STATUS

approved

A056897

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

+10
4

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

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

OFFSET

0,8

LINKS

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

FORMULA

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

EXAMPLE

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

MATHEMATICA

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 *)

CROSSREFS

Cf. A000040, A002496, A056892-A056898.

KEYWORD

nonn

AUTHOR

Henry Bottomley, Jul 05 2000

STATUS

approved

A104492

Cube excess of the n-th prime.

+10
4

1, 2, 4, 6, 3, 5, 9, 11, 15, 2, 4, 10, 14, 16, 20, 26, 32, 34, 3, 7, 9, 15, 19, 25, 33, 37, 39, 43, 45, 49, 2, 6, 12, 14, 24, 26, 32, 38, 42, 48, 54, 56, 66, 68, 72, 74, 86, 7, 11, 13, 17, 23, 25, 35, 41, 47, 53, 55, 61, 65, 67, 77, 91, 95, 97, 101, 115, 121, 4, 6, 10, 16, 24, 30

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

OFFSET

1,2

LINKS

Harvey P. Dale, Table of n, a(n) for n = 1..1000

FORMULA

a(n) = A055400(A000040(n)).

a(n) = prime(n) - floor(prime(n)^(1/3))^3. - Jon E. Schoenfield, Jan 17 2015

EXAMPLE

a(48) = 7 because the 48th prime is 223 and 223 - 6^3 = 7, while 223 - 7^3 = -120.

MATHEMATICA

lst={}; Do[p=Prime[n]; s=p^(1/3); f=Floor[s]; a=f^3; d=p-a; AppendTo[lst, d], {n, 6!}]; lst (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)

#-Floor[Surd[#, 3]]^3&/@Prime[Range[80]] (* Harvey P. Dale, Feb 15 2018 *)

CROSSREFS

Cf. A000040, A053186, A055400, A056892, A102821.

KEYWORD

easy,nonn

AUTHOR

Jonathan Vos Post, Mar 10 2005

STATUS

approved

A056894

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

+10
3

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

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

OFFSET

1,3

LINKS

Table of n, a(n) for n=1..55.

FORMULA

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

EXAMPLE

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

CROSSREFS

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

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

CROSSREFS

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

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

KEYWORD

nonn

AUTHOR

Henry Bottomley, Jul 05 2000

STATUS

approved

A102821

Numbers n for which the square excess of n-th prime is prime.

+10
3

2, 4, 5, 8, 9, 13, 14, 15, 19, 20, 23, 27, 28, 30, 35, 36, 37, 38, 39, 46, 49, 56, 57, 67, 68, 69, 71, 81, 83, 86, 93, 94, 96, 98, 107, 108, 109, 111, 112, 113, 114, 124, 128, 138, 139, 142, 144, 155, 156, 157, 158, 159, 160, 161, 162, 173, 178, 182, 192, 195, 196, 199

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

OFFSET

0,1

LINKS

Table of n, a(n) for n=0..61.

EXAMPLE

7 - 2^2 = 3 is the square excess (see A056892) of 7 and it is prime. 7 is the 4th prime, so 4 is in the sequence.

MATHEMATICA

Select[Range[200], PrimeQ[Prime[#]-Floor[Sqrt[Prime[#]]]^2]&] (* Harvey P. Dale, Jul 06 2014 *)

CROSSREFS

Cf. A056892.

KEYWORD

easy,nonn

AUTHOR

Olaf Voß, Feb 27 2005

STATUS

approved