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A085042 Smallest multiple of the n-th prime such that every partial sum is a prime, or 0 if no such number exists. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified April 7 14:55 EDT 2020. Contains 333305 sequences. (Running on oeis4.)