A085042 Smallest multiple of the n-th prime such that every partial sum is a prime, or 0 if no such number exists.

2, 3, 0, 14, 22, 26, 34, 38, 138, 232, 62, 148, 164, 516, 94, 318, 1298, 610, 134, 568, 146, 2212, 1494, 356, 1552, 1212, 618, 642, 2616, 1808, 762, 3144, 1096, 1668, 1490, 604, 942, 326, 1002, 2076, 2506, 1448, 2292, 1158, 7092, 4776, 2532, 892, 908, 2290

Offset

1,1

Comments

a(3) = 0, as 2+3 = 5 and 5 + k*5 cannot be a prime unless k = 0. Every term after the third is obviously even.

From Robert Israel, Dec 25 2016: (Start)

a(n) = 0 if and only if Sum_{j<n} a(j) is divisible by prime(n). The "only if" is by Dirichlet's theorem on primes in arithmetic progressions. There are no values of 0 for 3 < n < 5*10^5, but it is conceivable that such values exist. Indeed, heuristically Sum_{j<n} a(j) has probability 1/prime(n) of being divisible by prime(n), and Sum_n 1/prime(n) diverges, so we should expect infinitely many such values (although they will be very rare). (End)

Links

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

Maple

A:= 2, 3:
s:= 5:
for n from 3 to 100 do
  p:= 2*ithprime(n);
  if igcd(p, s) > 1 then
    A:= A, 0;
  else
    for j from 1 do if isprime(s+j*p) then
      A:= A, j*p;
      s:= s+j*p;
      break
    fi od;
  fi
od:
A; # Robert Israel, Dec 25 2016

Mathematica

f[s_List] := Block[{k = 2, p = Prime[1 + Length@ s], t = Plus @@ s}, While[ !PrimeQ[k*p + t], k += 2]; Append[s, p*k]]; Nest[f, {2, 3, 0}, 50] (* Robert G. Wilson v, Dec 25 2016 *)

Crossrefs

Cf. A085043.

Sequence in context: A219341 A059740 A005160 * A096542 A009206 A318146

Adjacent sequences: A085039 A085040 A085041 * A085043 A085044 A085045

Keyword

nonn

Author

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 19 2003

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jun 25 2003

Status

approved