A065376 -id:A065376 - cp's OEIS frontend

A065377 Primes not of the form p + k^2, with p prime and k > 0.

Original entry on oeis.org

2, 5, 13, 31, 37, 61, 127, 379, 439, 571, 829, 991, 1549, 3319, 7549

Offset: 1

Reinhard Zumkeller, Nov 03 2001

nonn

Probably finite and 7549 is the last entry. - Robert G. Wilson v, Nov 05 2001

No more terms below 10^9. - Mauro Fiorentini, Mar 02 2020

Cf. A000040. Complement of A065376.

N:= 10^6: # to get all entries <= N
Primes:= select(isprime,{2,seq(2*i+1,i=1..floor((N-1)/2))}):
A:= NULL:
for i from 1 to nops(Primes) do
for k from floor(sqrt(Primes[i])) to 1 by -1 do
   if isprime(Primes[i] - k^2) then break fi
od:
if k = 0 then A:= A, Primes[i] fi;
od:
A; # Robert Israel, Sep 03 2014

Do[ k = 1; p = Prime[n]; While[k^2 < p && !PrimeQ[p - k^2], k++ ]; If[k^2 > p, Print[p]], {n, 1, 10^6} ]
Module[{nn=1000,pr},pr=Flatten[Table[Prime[n]+Range[nn]^2,{n,nn}]];Complement[Prime[Range[nn]],pr]] (* Harvey P. Dale, May 30 2014 *)

is(p)=forstep(m=2,sqrtint(p),2,if(isprime(p-m^2),return(0)));isprime(p) && (p==2 || !issquare(p-2)) \\ Charles R Greathouse IV, Jun 04 2012

Offset corrected by Charles R Greathouse IV, May 29 2012

A065396 Primes of the form p + k*(k+1) / 2, p prime and k > 0.

Original entry on oeis.org

3, 5, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283

Offset: 1

Reinhard Zumkeller, Nov 05 2001

nonn

a(2) = 11 = 5 + 3*(3+1) / 2.

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

A000040, A000217, A065376, A065397.

Module[{upto=300,trs},trs=Accumulate[Range[(Sqrt[1+8upto]-1)/2]];Select[ Flatten[ Table[p+trs,{p,Prime[Range[PrimePi[upto]]]}]],PrimeQ[#] && #<=upto&]]//Union (* Harvey P. Dale, Dec 16 2017 *)

A309179 Primes to which a record size square needs to be added to reach another prime.

Original entry on oeis.org

2, 3, 5, 29, 41, 389, 479, 881, 1931, 3461, 3701, 7589, 9749, 26171, 153089, 405701, 1036829, 1354349, 1516829, 2677289, 4790309, 4990961, 34648631, 46214321, 50583209, 98999969, 305094851, 331498961, 362822099, 4373372351, 11037674441, 12239355719, 16085541359

Offset: 1

Bert Dobbelaere, Jul 15 2019

nonn

a(1) = 2 and r(1) = 1.
For n > 1, a(n) is the smallest prime for which r(n) > r(n-1) exists so that a(n) + r(n)^2 is prime and a(n) + k^2 are composite for 0 < k < r(n).
When omitting the squares in the description, the sequence becomes A002386.

			a(1) = 2; r(1) = 1.
a(2) = 3; 3 + 1^2 is composite, but 3 + 2^2 is prime, so r(2) = 2.
a(3) = 5; 5 + k^2 is composite for 0 < k < 6, but 5 + 6^2 is prime, so r(3) = 6.
The following are primes: 7 + 2^2, 11 + 6^2, 13 + 2^2, 17 + 6^2, 19 + 2^2, 23 + 6^2.
a(4) = 29; 29 + k^2 is composite for 0 < k < 12, but 29 + 12^2 is prime: r(4) = 12.

Cf. A002386, A020495, A065376, A127356, A129314.

f(n) = {k=1; while(!isprime(n+k^2), k++); k;}
lista(NN) = {m=0; forprime(p=1, NN, if(f(p)>m, m=f(p);print1(p,", ")))} \\ Jinyuan Wang, Jul 15 2019

from sympy import isprime, nextprime
n, p, r = 0, 0, 0
while(True):
    p = nextprime(p) ; k = 1
    while not isprime(p + k**2):
        k += 1
    if k > r:
        n += 1 ; r = k
        print("a({}) = {}".format(n,p))

a(30)-a(33) from Giovanni Resta, Jul 16 2019

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A065377 Primes not of the form p + k^2, with p prime and k > 0.

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A065396 Primes of the form p + k*(k+1) / 2, p prime and k > 0.

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A309179 Primes to which a record size square needs to be added to reach another prime.

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PARI

Python

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