A085415 -id:A085415 - cp's OEIS frontend

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

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

A085418 Primes reached in A085417.

Original entry on oeis.org

3, 5, 17, 11, 29, 19, 41, 127, 53, 89, 67, 107, 83, 277, 113, 233, 113, 79, 127, 157, 139, 101, 181, 113, 293, 127, 313, 257, 199, 239, 223, 163, 239, 173, 257, 467, 271, 541, 461, 383, 349, 223, 563, 787, 383, 389, 1021, 271, 647, 1489, 389, 509, 1789, 773, 983

Offset: 1

Zak Seidov, Jun 30 2003

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

Cf. A085415, A085416, A085417.

A249112 Second smallest k > 0 such that n+(1+2+...+k) is prime.

Original entry on oeis.org

3, 2, 7, 2, 8, 10, 4, 5, 7, 2, 8, 10, 4, 5, 16, 2, 8, 10, 7, 6, 16, 5, 8, 22, 7, 5, 16, 2, 15, 22, 4, 6, 7, 9, 8, 13, 4, 5, 19, 2, 11, 10, 7, 5, 16, 5, 8, 13, 12, 6, 7, 5, 8, 22, 7, 5, 16, 2, 15, 13, 4, 9, 16, 5, 8, 13, 8, 5, 7, 2, 11, 10, 4, 14, 16, 6, 8

Offset: 1

Gil Broussard, Oct 21 2014

Take the counting numbers and continue adding 1, 2, ..., a(n) until reaching a second prime.
Conjecturally (Hardy & Littlewood conjecture F), a(n) exists for all n. - Charles R Greathouse IV, Oct 21 2014
It appears that the minimum value reached by a(n) is 2, and this occurs for n= 2, 4, 10, 16, 28, 40, 58, 70, ... Is this A144834? - Michel Marcus, Oct 26 2014

			a(3)=7 because 3+1+2+3+4+5+6+7=31 and one partial sum is prime.
a(4)=2 because 4+1=5 and 4+1+2=7.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A085415, A249113.

Table[k = 0; Do[k++; While[! PrimeQ[n + Total@ Range@ k], k++], {x, 2}]; k, {n, 77}] (* Michael De Vlieger, Jan 03 2016 *)

a(n)=my(k, s=2); while(s, if(isprime(n+=k++), s--)); k \\ Charles R Greathouse IV, Oct 21 2014

a(n,s=2)=my(k);until(isprime(n+=k++)&&!s--,);k \\ allows one to get A249113(n) as a(n,3). - M. F. Hasler, Oct 21 2014

n+A000217(k) is prime for k=a(n) and exactly one smaller positive value. - M. F. Hasler, Oct 21 2014

A249113 Take n and successively add 1, 2, ..., a(n) until reaching a prime for the third time.

Original entry on oeis.org

4, 5, 16, 5, 11, 13, 8, 6, 19, 6, 12, 13, 7, 9, 28, 5, 11, 13, 12, 17, 19, 6, 11, 25, 8, 6, 28, 5, 20, 37, 7, 14, 19, 10, 11, 34, 8, 6, 40, 6, 20, 25, 8, 9, 31, 6, 11, 25, 19, 21, 19, 6, 12, 25, 16, 9, 28, 5, 20, 22, 7, 14, 40, 9, 11, 34, 19, 6, 52, 17, 12

Offset: 1

Gil Broussard, Oct 21 2014

Conjecturally (Hardy & Littlewood conjecture F), a(n) exists for all n. - Charles R Greathouse IV, Oct 21 2014

			a(1)=4 because 1+1+2+3+4=11 and exactly two partial sums are prime (2,7).
a(2)=5 because 2+1+2+3+4+5=17 and exactly two partial sums are prime (3,5).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A085415, A249112.

Table[k = 0; Do[k++; While[! PrimeQ[n + Total@ Range@ k], k++], {x, 3}]; k, {n, 71}] (* Michael De Vlieger, Jan 03 2016 *)

a(n)=my(k,s=3); while(s,if(isprime(n+=k++),s--));k \\ Charles R Greathouse IV, Oct 21 2014

a(n,s=3)=my(k);until(isprime(n+=k++)&&!s--,);k \\ allows one to get A249112(n) as a(n,2). - M. F. Hasler, Oct 21 2014

n+A000217(k) is prime for k=a(n) and exactly two smaller positive values. - M. F. Hasler, Oct 21 2014

A249114 Take the counting numbers and continue adding 1, 2, ..., a(n) until one reaches a fourth prime.

Original entry on oeis.org

7, 6, 19, 10, 12, 25, 11, 9, 40, 13, 15, 25, 11, 17, 67, 6, 15, 22, 15, 18, 43, 9, 12, 34, 12, 9, 31, 9, 32, 58, 8, 21, 28, 14, 12, 37, 11, 9, 55, 13, 23, 46, 11, 14, 43, 10, 15, 34, 24, 26, 28, 9, 15, 37, 23, 18, 40, 6, 24, 61, 8, 18, 43, 22, 27, 37, 20, 9

Offset: 1

Gil Broussard, Oct 21 2014

Conjecturally (Hardy & Littlewood conjecture F), a(n) exists for all n. - Charles R Greathouse IV, Oct 21 2014
It appears that the minimum value reached by a(n) is 6. This occurs for n=2, 16, 58, 136, 178, 418, 598, 808, ... . - Michel Marcus, Oct 26 2014
The conjecture in the previous line is true - if n is odd, then n+1 is even, n+3 is even, n+6 and n+10 are odd, etc., so a(n)>6. If n is even, then +1 and +3 are odd, +6, +10 are even, so the fourth prime can be first for a(n)=6. - Jon Perry, Oct 29 2014
Conjecture: a(n) is odd approximately 50% of the time. - Jon Perry, Oct 29 2014

			a(1) = 7 because 1+1+2+3+4+5+6+7 = 29 and exactly three partial sums are prime (2,7,11).
a(2) = 6 because 2+1+2+3+4+5+6 = 23 and exactly three partial sums are prime (3,5,17).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A085415, A249112, A249113.

f:= proc(n) local j,count;
      count:= 0;
      for j from 1 do
        if isprime(n + j*(j+1)/2) then
           count:= count+1;
           if count = 4 then return j fi
        fi
      od
end proc:
seq(f(n),n=1..100); # Robert Israel, Oct 29 2014

a[n_] := Module[{j, cnt = 0}, For[j = 1, True, j++, If[PrimeQ[n+j(j+1)/2], cnt++; If[cnt == 4, Return[j]]]]];
Array[a, 100] (* Jean-François Alcover, Oct 03 2020, after Maple *)

a(n)=my(k, s=4); while(s, if(isprime(n+=k++), s--)); k \\ Charles R Greathouse IV, Oct 21 2014

a(n) = Min_{k>0 | { n+A000217(j), j=1...k} contains four primes}. - M. F. Hasler, Oct 29 2014

cp's OEIS Frontend

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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A085418 Primes reached in A085417.

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A249112 Second smallest k > 0 such that n+(1+2+...+k) is prime.

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A249113 Take n and successively add 1, 2, ..., a(n) until reaching a prime for the third time.

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A249114 Take the counting numbers and continue adding 1, 2, ..., a(n) until one reaches a fourth prime.

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