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A224794 Smallest prime p such that 2n+1 = 2p - q^3 for some odd prime q, or 0 if no such prime exists. 2
173, 2459, 17, 67, 19, 113497, 179, 71, 23, 73, 677, 25339, 0, 74453, 29, 79, 31, 1117, 191, 83, 193, 25349, 37, 1123, 197, 89, 41, 3594557, 43, 1129, 3461, 12227, 47, 97, 701, 647551, 3467, 101, 53, 103, 1124087, 647557, 709, 107, 59, 109, 61, 113539, 6133 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

See the odd numbers of the form p^3 - 2q where p and q are primes (A224730, A185046).

a(n) = 0 for n = 13, 62, 171, 364, 665, 1098, ... and 2n+1 = 27, 125, 343, 729, ... are the odd cubes 3^3, 5^3, 7^3, ...

The corresponding primes q are in A224795.

Conjecture: The odd numbers different from a cube are of the form m = 2p - q^3 where p and q are prime numbers.

Remark: if m = c^3 for any odd integer c, then c^3 = 2p - q^3 is impossible because c^3 + q^3 = 2p => (c+q)(c^2 - cq + q^2) with c+q even of the form c+q = 2a => p = a(c^2 - cq + q^2) absurd because p is prime.

LINKS

Michel Lagneau, Table of n, a(n) for n = 1..10000

EXAMPLE

a(4) = 67 because, for (p, q) = (67, 5), 2*4 + 1 = 9 = 2*67 - 5^3 = 134 - 125 = 9.

MAPLE

for n from 3 by 2 to 200 do:jj:=0:for j from 1 to 50000 while (jj=0) do:q:=ithprime(j):p:=(q^3+n)/2:if type(p, prime)=true  then jj:=1: printf(`%d, `, p):else fi:od:if jj=0 then printf(`%d, `, 0):else fi:od:

CROSSREFS

Cf. A185046, A224730, A224795.

Sequence in context: A142134 A142849 A271978 * A209809 A185707 A185701

Adjacent sequences:  A224791 A224792 A224793 * A224795 A224796 A224797

KEYWORD

nonn

AUTHOR

Michel Lagneau, Apr 18 2013

STATUS

approved

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Last modified August 13 05:41 EDT 2020. Contains 336442 sequences. (Running on oeis4.)