A056892 -id:A056892 - cp's OEIS frontend

A176032 Absolute values of A106044-A056892.

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 06 2010

nonn

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,...

Cf. A056892, A106044

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

A320688 Sum of the square excess A056892 of the primes between two squares.

Original entry on oeis.org

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

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

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.

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

Cf. A014085, A000290, A108314, A106044.

Row sums of A056892, read as a table.

R:= NULL: p:= 2: n:= 1: t:= 0:
while n <= 100 do
    t:= t + p-n^2;
    p:= nextprime(p);
    if p > (n+1)^2 then
     R:= R, t; t:= 0; n:= n+1;
    fi:
od:
R; # Robert Israel, Dec 17 2024

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

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

A056898 a(n) = smallest number m such that m^2+n is prime.

Original entry on oeis.org

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

Offset: 1

Henry Bottomley, Jul 05 2000

nonn

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

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

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

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

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

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

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

A104492 Cube excess of the n-th prime.

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Mar 10 2005

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

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

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

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 *)

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

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

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)).

A102821 Numbers n for which the square excess of n-th prime is prime.

Original entry on oeis.org

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

Offset: 0

Olaf Voß, Feb 27 2005

			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.

Cf. A056892.

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

cp's OEIS Frontend

A176032 Absolute values of A106044-A056892.

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A320688 Sum of the square excess A056892 of the primes between two squares.

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PARI

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A056898 a(n) = smallest number m such that m^2+n is prime.

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A056893 Smallest prime with square excess of n.

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

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

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A104492 Cube excess of the n-th prime.

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

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

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