A224730 -id:A224730 - cp's OEIS frontend

A185046 Smallest prime p such that 2n+1 = p^3 - 2q for some odd prime q, or 0 if no such prime exists.

5, 3, 5, 7, 13, 3, 13, 3, 5, 3, 3, 11, 0, 7, 5, 19, 37, 11, 5, 7, 5, 7, 37, 11, 5, 31, 53, 31, 13, 23, 5, 7, 5, 7, 13, 23, 13, 19, 5, 7, 421, 47, 5, 7, 5, 11, 13, 11, 5, 43, 5, 11, 61, 23, 5, 19, 5, 7, 5, 5, 53, 7, 17, 7, 13, 11, 13, 7, 113, 7, 373, 11, 17, 7

Offset: 1

Michel Lagneau, Apr 17 2013

nonn

a(n) = 0 for n = 13, 171, 364, 1098, 2456, 3429, 6083, 7812, 9841, 12194, 14895, 17968,... and 2n+1 = 27, 343, 729,... is a class of cubes.
The corresponding primes q are in A224730.
Conjecture: The odd numbers different from a cube are of the form m = p^3 - 2q where p and q are prime numbers.
Remark: Its converse is false: there exists cubes m = c^3 that are in the sequence with the form c^3 = p^3 - 2q, where p-c = 2, and q of the form x^2 +x*y+y^2 (see A007645). For example: 5^3 = 7^3 - 2*109.

			a(4) = 7 because, for (p, q) = (7, 167) => 2*4+1 = 9 = 7^3 - 2*167 = 343 - 334 = 9.

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

Cf. A007645, A224730.

for n from 3 by 2 to 200 do:
      jj:=0:
          for j from 1 to 10000 while (jj=0) do:
             p:=ithprime(j):q:=(p^3-n)/2:
             if q> 0 and type(q,prime)=true
             then
             jj:=1:printf(`%d, `,p):
             else
             fi:
         od:
            if jj=0 then
            printf(`%d, `,0):
            else
            fi:
     od:

A224794 Smallest prime p such that 2n+1 = 2p - q^3 for some odd prime q, or 0 if no such prime exists.

Original entry on oeis.org

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

Offset: 1

Michel Lagneau, Apr 18 2013

nonn

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.

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

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

Cf. A185046, A224730, A224795.

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:

A224795 Smallest prime q such that 2n+1 = 2p - q^3 for some odd prime p, or 0 if no such prime exists.

Original entry on oeis.org

7, 17, 3, 5, 3, 61, 7, 5, 3, 5, 11, 37, 0, 53, 3, 5, 3, 13, 7, 5, 7, 37, 3, 13, 7, 5, 3, 193, 3, 13, 19, 29, 3, 5, 11, 109, 19, 5, 3, 5, 131, 109, 11, 5, 3, 5, 3, 61, 23, 5, 7, 13, 3, 13, 7, 29, 3, 29, 3, 73, 7, 0, 331, 5, 3, 61, 7, 5, 3, 53, 59, 13, 11, 5, 3

Offset: 1

Michel Lagneau, Apr 18 2013

nonn

See the conjecture and the comments in A224794, and the corresponding primes p.

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

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

Cf. A224730, A185046, A224794.

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, `,q):else fi:od:if jj=0 then printf(`%d, `,0):else fi:od:

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A185046 Smallest prime p such that 2n+1 = p^3 - 2q for some odd prime q, or 0 if no such prime exists.

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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.

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Maple

A224795 Smallest prime q such that 2n+1 = 2p - q^3 for some odd prime p, or 0 if no such prime exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple