A274025 Primes p such that Sum_{k=primes<p} (k mod p) and Sum_{k=primes<p} (p mod k) are both prime.

5, 691, 2399, 3433, 5099, 6217, 7451, 9007, 10253, 10883, 16561, 21839, 23189, 25679, 26501, 30661, 39097, 41443, 43591, 48119, 50893, 56009, 60293, 64927, 65537, 78979, 79829, 85853, 98669, 100403, 105491, 115981, 124783, 140557, 142547, 148013, 149953, 164113, 166219, 169249

Offset

1,1

Comments

As 0 < k < p, k mod p = k, so Sum_{k = primes<p} (k mod p) = A007504(A000720(A151799(p))) for p > 3. - David A. Corneth, Jun 07 2016

Links

Robert Israel, Table of n, a(n) for n = 1..1003

Example

2 mod 5 = 2, 3 mod 5 = 3 and 2 + 3 = 5 is prime;
5 mod 2 = 1, 5 mod 3 = 2 and 1 + 2 = 3 is prime.

Maple

with(numtheory): P:=proc(q) local a, b, j, k, n; for j from 1 to q do n:=ithprime(j); a:=0; b:=0; for k from 1 to n-1 do
if isprime(k) then a:=a+k; b:=b+(n mod k); fi; od;
if isprime(a) and isprime(b) then print(n); fi; od; end: P(10^6);
# Alternative:
N:= 10^6: # to get all entries <= N
Primes:= select(isprime, [2, seq(i, i=3..N, 2)]):
PS:= ListTools:-PartialSums(Primes):
count:= 0:
for i from 2 to nops(Primes) do
   n := Primes[i];
   if isprime(PS[i-1]) and isprime(add(n mod Primes[j], j=1..i-1)) then
     count:= count+1;
     A[count]:= n;
   fi
od:
seq(A[i], i=1..count); # Robert Israel, Jun 07 2016

Prog

(PARI) is(n) = {if(isprime(n), my(nk, kn, u=prime(primepi(n-1)));
forprime(k=2, u, kn+=k; nk+=n%k); isprime(kn)&&isprime(nk), 0)} \\ David A. Corneth, Jun 07 2016

Crossrefs

Cf. A000040, A000720, A007504, A151799.

Sequence in context: A000367 A176546 A092133 * A281585 A071772 A201005

Adjacent sequences: A274022 A274023 A274024 * A274026 A274027 A274028

Keyword

nonn,easy

Author

Paolo P. Lava, Jun 07 2016

Status

approved