A085418 -id:A085418 - cp's OEIS frontend

A085415 Take prime[n] and continue adding 1, 2, ..., a(n) until one reaches a prime.

1, 4, 3, 3, 3, 3, 3, 4, 3, 12, 3, 3, 3, 4, 3, 3, 12, 3, 3, 8, 3, 4, 3, 12, 3, 3, 3, 3, 7, 8, 4, 3, 8, 4, 12, 3, 3, 4, 3, 3, 12, 4, 3, 3, 8, 7, 7, 3, 3, 4, 3, 12, 4, 3, 3, 3, 12, 3, 3, 8, 4, 11, 3, 3, 8, 8, 3, 4, 3, 4, 3, 15, 3, 3, 4, 3, 12, 8, 11, 4, 24, 4, 8, 3, 4, 3, 15, 3, 3, 7, 8, 12, 8, 11, 4, 3, 12, 8

Offset: 1

Zak Seidov, Jun 29 2003

Resulting primes in A085416. See also A085417, A085418.

Prime[n] plus a triangular number is prime. - Harvey P. Dale, Jun 12 2013

			a(2)=4 because prime[2]+(1+2+3+4)=3+10=13 is a prime

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

Cf. A085416, A085417, A085418.

Flatten[(Sqrt[1+8#]-1)/2&/@With[{trnos=Accumulate[Range[30]]}, Table[ Select[ trnos,PrimeQ[Prime[n]+#]&,1],{n,100}]]] (* Harvey P. Dale, Jun 12 2013 *)

Prime[n]+m*(1+m)/2 is a prime for some m>0.

A085416 Take prime[n] and continue adding 1,2,..., A085415(n) until one reaches a prime a(n).

Original entry on oeis.org

3, 13, 11, 13, 17, 19, 23, 29, 29, 107, 37, 43, 47, 53, 53, 59, 137, 67, 73, 107, 79, 89, 89, 167, 103, 107, 109, 113, 137, 149, 137, 137, 173, 149, 227, 157, 163, 173, 173, 179, 257, 191, 197, 199, 233, 227, 239, 229, 233, 239, 239, 317, 251, 257, 263, 269, 347

Offset: 1

Zak Seidov, Jun 29 2003

Primes resulting in procedure of A085415. See also A085417, A085418.

			a(2)=13 because prime[2]=3 and 3+(1+2+3+4)=3+10=13 is a prime.

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

Cf. A085415, A085417, A085418.

Table[SelectFirst[Prime[n]+Accumulate[Range[20]],PrimeQ],{n,60}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 11 2019 *)

A085417 Take prime[n] and continue adding n,n+1,..., n+a(n)-1 until one reaches a prime.

Original entry on oeis.org

1, 1, 3, 1, 3, 1, 3, 9, 3, 5, 3, 5, 3, 12, 4, 9, 3, 1, 3, 4, 3, 1, 4, 1, 7, 1, 7, 5, 3, 4, 3, 1, 3, 1, 3, 8, 3, 9, 7, 5, 4, 1, 8, 12, 4, 4, 15, 1, 8, 21, 3, 5, 24, 9, 12, 8, 3, 4, 3, 9, 11, 4, 3, 5, 48, 1, 7, 33, 3, 1, 3, 1, 15, 12, 3, 5, 8, 5, 3, 36, 19, 1, 3, 5, 11, 5, 12, 5, 4, 4, 3, 1, 3, 5, 3, 1, 15, 1

Offset: 1

Zak Seidov, Jun 30 2003

Primes obtained are in A085418. See also A085415, A085416.

No terms == 2 (mod 4). - Robert Israel, Mar 24 2023

			a(3)=3 because prime[3]=5 and 5+(3+4+5)=17= is a prime A085418(3).

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

Cf. A085415, A085416, A085418.

f:= proc(n) local m,k,x;
  m:= ithprime(n) - (n-1)*n/2;
  for k from n do
    x:= k*(k+1)/2 + m;
    if isprime(x) then return k+1-n fi
  od;
end proc:
map(f, [$1..100]); # Robert Israel, Mar 24 2023

cp's OEIS Frontend

A085415 Take prime[n] and continue adding 1, 2, ..., a(n) until one reaches a prime.

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A085416 Take prime[n] and continue adding 1,2,..., A085415(n) until one reaches a prime a(n).

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A085417 Take prime[n] and continue adding n,n+1,..., n+a(n)-1 until one reaches a prime.

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