A195352 -id:A195352 - cp's OEIS frontend

A103153 a(n) is the smallest odd prime p such that 2n+1 = 2p + A000040(k) for some k>1, or 0 if no such prime exists.

0, 0, 0, 3, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 7, 5, 3, 3, 5, 5, 3, 7, 3, 3, 5, 3, 7, 5, 3, 7, 5, 3, 3, 5, 5, 3, 7, 3, 3, 5, 5, 3, 7, 3, 19, 5, 3, 7, 5, 11, 3, 11, 3, 3, 5, 3, 3, 5, 3, 7, 5, 11, 7, 11, 11, 3, 11, 3, 13, 5, 3, 3, 5, 5, 7, 7, 3, 3, 5, 5, 3, 7, 5, 3, 7, 3, 13, 5, 3, 7, 5, 3, 3, 5, 5, 7, 7, 3

Offset: 1

Lei Zhou, Feb 09 2005

nonn

			For n < 4 there are no such primes, thus a(1)-a(3)=0. For n=4, 2*4+1 = 9 = 2*3+3, thus a(4)=3. For n=7, 2*7+1 = 15 = 2*5+5, thus a(7)=7.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

a(n)=0 if A103507(n)=0, otherwise A000040(A103507(n)).
Cf. A103151, A103152, A002373.
Cf. A195352 (similar definition, but p=2 is allowed).

Do[m = 3; While[ ! (PrimeQ[m] && ((n - 2*m) > 2) && PrimeQ[n - 2*m]), m = m + 2]; Print[m], {n, 9, 299, 2}]

(define (A103153 n) (let ((ind (A103507 n))) (if (zero? ind) 0 (A000040 ind))))

Edited and Scheme code added by Antti Karttunen, Jun 19 2007

Definition corrected by Hugo Pfoertner, Sep 16 2011

A219252 Smallest prime q such that 2n+1 = p + 4q for some odd prime p, otherwise 0 if no such q exists.

Original entry on oeis.org

0, 0, 0, 0, 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 5, 3, 2, 2, 3, 3, 2, 7, 2, 2, 3, 2, 5, 3, 2, 5, 3, 2, 2, 3, 3, 2, 0, 2, 2, 3, 3, 2, 7, 2, 5, 3, 2, 5, 3, 5, 2, 7, 2, 2, 3, 2, 2, 3, 2, 5, 3, 5, 5, 7, 5, 2, 7, 2, 7, 3, 2, 2, 3, 3, 11, 7, 2, 2, 3, 3, 2, 7, 3, 2, 19, 2, 5, 3, 2

Offset: 1

Michel Lagneau, Apr 11 2013

nonn

a(38) = 0.

Conjecture: except m = 77, all odd number > 9 are of the form m = p + 4*q where p and q are prime numbers.

			3 + 4*2 = 11 => a(5) = 2;
5 + 4*2 = 13 => a(6) = 2;
7 + 4*2 = 15 => a(7) = 2;
5 + 4*3 = 17 => a(8) = 3.

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

Cf. A195352, A002091, A219254.

for n from 11 by 2 to 200 do:jj:=0:for j from 1 to 1000 while (jj=0) do:q:=ithprime(j):p:=n-4*q:if p> 0 and type(p,prime)=true  then jj:=1:printf(`%d, `,q):else fi:od:if jj=0 then printf(`%d, `,0):else fi:od:

A195353 Odd numbers n>5 in the representation n=2*p+q, p, q prime, q maximal, at which a larger p than for any smaller n is needed. A195354 gives values of p.

Original entry on oeis.org

7, 13, 31, 91, 451, 539, 1397, 1417, 1777, 3139, 14419, 39751, 77507, 96157, 158173, 214979, 263113, 496723, 1046179, 3415297, 3546371, 4306549, 9456677, 18338311, 45521269, 243377803, 766707661, 1023263789, 8032822531, 92635306249, 151318414531, 352799777983

Offset: 1

Hugo Pfoertner, Sep 16 2011

nonn

2*(Positions of records in A195352) + 1.

Checked up to n = 10^13.

			a(3)=31 because it is the first number for which the representation n=2*p+q needs a larger value of p than for all smaller odd numbers. 31=2*7+17, whereas all smaller odd n can be expressed using p=2 or p=3.

See A002091.

Hugo Pfoertner, Table of n, a(n) for n = 1..41

Cf. A195352, A195354 (records of p), A194828 (similar, but looking for records of q with p maximized)

a(36)-a(41) from Hugo Pfoertner, Sep 26 2011

A195354 Records of primes p in the representation of odd n>5 by n=2*p+q, p, q prime, q maximal. A195353 gives the values of n at which an increase of p is required.

Original entry on oeis.org

2, 3, 7, 19, 31, 41, 47, 67, 79, 139, 181, 229, 233, 277, 307, 383, 421, 463, 619, 643, 659, 691, 743, 967, 1231, 1483, 1609, 1931, 2389, 2719, 2791, 2953, 2971, 3079, 3121, 3217, 3301, 3319, 3617, 3719, 3767

Offset: 1

Hugo Pfoertner, Sep 16 2011

nonn

Values of records in A195352. Related to Lemoine's conjecture. Due to the large number of possible representations (A046927) it is both possible to represent n=2*p+q with q<

A194829) and with p<

Checked up to n=10^13.

See A002091.

Cf. A195352, A195353, A194829.

a(36)-a(41) from Hugo Pfoertner, Sep 26 2011

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cp's OEIS Frontend

A103153 a(n) is the smallest odd prime p such that 2n+1 = 2p + A000040(k) for some k>1, or 0 if no such prime exists.

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A219252 Smallest prime q such that 2n+1 = p + 4q for some odd prime p, otherwise 0 if no such q exists.

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A195353 Odd numbers n>5 in the representation n=2*p+q, p, q prime, q maximal, at which a larger p than for any smaller n is needed. A195354 gives values of p.

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A195354 Records of primes p in the representation of odd n>5 by n=2*p+q, p, q prime, q maximal. A195353 gives the values of n at which an increase of p is required.

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cp's OEIS Frontend

A103153 a(n) is the smallest odd prime p such that 2*n+1 = 2*p + A000040(k) for some k>1, or 0 if no such prime exists.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Scheme

Extensions

A219252 Smallest prime q such that 2*n+1 = p + 4*q for some odd prime p, otherwise 0 if no such q exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

A195353 Odd numbers n>5 in the representation n=2*p+q, p, q prime, q maximal, at which a larger p than for any smaller n is needed. A195354 gives values of p.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Extensions

A195354 Records of primes p in the representation of odd n>5 by n=2*p+q, p, q prime, q maximal. A195353 gives the values of n at which an increase of p is required.

Views

Author

Keywords

Comments

References

Crossrefs

Extensions

A103153 a(n) is the smallest odd prime p such that 2n+1 = 2p + A000040(k) for some k>1, or 0 if no such prime exists.

A219252 Smallest prime q such that 2n+1 = p + 4q for some odd prime p, otherwise 0 if no such q exists.