A103506 -id:A103506 - cp's OEIS frontend

A195352 Smallest prime p such that 2n+1 = 2p + q for some odd prime q.

2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 3, 7, 2, 2, 3, 5, 2, 3, 2, 2, 3, 2, 3, 7, 2, 3, 7, 2, 2, 3, 5, 2, 3, 2, 2, 3, 5, 2, 3, 2, 3, 19, 2, 3, 7, 5, 2, 3, 2, 2, 3, 2, 2, 3, 2, 3, 7, 5, 11, 7, 11, 2, 3, 2, 3, 13, 2, 2, 3, 5, 5, 7, 2, 2, 3, 5, 2, 3, 7, 2, 3, 2, 3, 13

Offset: 3

Hugo Pfoertner, Sep 16 2011

nonn

Related to Lemoine's conjecture, which states that all odd integers > 5 can be represented as 2*p+q, p, q primes.

			a(3)=2 because 2*3+1=7=2*2+3; a(4)=2: 2*3+1=9=2*2+5; a(5)=2: 11=2*2+7; a(6)=3: 13=2*3+7.

See A002091.

Hugo Pfoertner, Table of n, a(n) for n = 3..10000
Wikipedia, Lemoine's conjecture

Cf. A002091, A103506 (smallest q), A046927, A195353 (positions of records), A195354 (records), A103153 (prime p=2 excluded), A185091.

spp[n_]:=Module[{p=2},While[CompositeQ[(2n+1)-2p],p=NextPrime[p]];p]; Array[ spp,90,3] (* Harvey P. Dale, Jun 02 2022 *)

A103509 a(n) is the least j such that 2n+1 = 2*A000040(k) + A000040(j) for some k > 1, or 0 if no such j exists.

Original entry on oeis.org

0, 0, 0, 2, 3, 2, 3, 2, 3, 4, 6, 2, 3, 2, 3, 4, 6, 2, 3, 2, 3, 4, 6, 2, 3, 4, 7, 5, 6, 2, 3, 2, 3, 4, 6, 5, 6, 2, 3, 4, 12, 2, 3, 2, 3, 4, 6, 2, 3, 4, 7, 5, 6, 2, 3, 4, 10, 5, 6, 2, 3, 2, 3, 4, 6, 5, 6, 2, 3, 4, 12, 2, 3, 2, 3, 4, 6, 5, 6, 2, 3, 4, 18, 2, 3, 4, 7, 5, 6, 2, 3, 4, 10, 5, 6, 15, 7, 2, 3, 4, 12, 2, 3, 2, 3

Offset: 1

Lei Zhou, Feb 10 2005

nonn

			For n < 4 there are no such primes, thus a(1)=a(2)=a(3)=0.
For n=4, 2*4+1 = 9 = 2*3+3 and 3=A000040(2), thus a(4)=2.
For n=11, 2*11+1 = 23 = 13+2*5 and 13=A000040(6), thus a(11)=6.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65539

Cf. A049084, A103506.

Can be used to compute A103506 and A103510. Cf. A103507.

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

A103509(n) = if(n<=3,0,my(o=n+n+1); for(i=2,oo, if(isprime((o-prime(i))/2),return(i)))); \\ Antti Karttunen, Mar 30 2021

a(n) = A049084(A103506(n)), for n >= 4.

Edited by Antti Karttunen, Jun 19 2007

A219604 Smallest prime p such that 2n+1 = 4q + p for some odd prime q, or 0 if no such prime exists.

Original entry on oeis.org

0, 0, 0, 0, 3, 5, 3, 5, 7, 13, 3, 5, 7, 17, 3, 5, 7, 17, 11, 13, 23, 17, 3, 5, 7, 41, 3, 5, 7, 17, 11, 13, 23, 17, 3, 5, 7, 0, 3, 5, 7, 17, 11, 13, 23, 17, 3, 5, 7, 73, 11, 13, 31, 17, 19, 37, 23, 41, 3, 5, 7, 73, 3, 5, 7, 17, 11, 13, 23, 17, 19, 29, 23, 73, 3

Offset: 1

Michel Lagneau, Apr 12 2013

nonn

a(38) = 0.

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

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

Cf. A219252, A219254, A103506.

for n from 11 by 2 to 200 do:jj:=0:for j from 1 to 1000 while (jj=0) do:p:=ithprime(j):q:=(n-p)/4: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:

Table[m=3;While[!(PrimeQ[m]&&(((2*n+1-m)/4)>1)&&PrimeQ[(2*n+1-m)/4]),m=m+2];Print[n," ",m],{n,5,200}]

A223174 Smallest prime p such that 2n+1 = p + 8*q for some odd prime q, or 0 if no such prime exists.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 5, 7, 0, 3, 5, 7, 17, 11, 13, 23, 17, 3, 5, 7, 0, 11, 13, 31, 17, 3, 5, 7, 41, 11, 13, 31, 17, 19, 37, 23, 41, 43, 29, 31, 73, 3, 5, 7, 41, 11, 13, 47, 17, 3, 5, 7, 73, 11, 13, 31, 17, 19, 37, 23, 41, 43, 29, 31, 97, 3, 5, 7, 41

Offset: 0

Michel Lagneau, May 09 2013

nonn

For n > 8, a(12) = a(24) = 0.
The corresponding q = 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 0, 3, 3, 3, 2, 3, 3, 2, 3, 5, 5, 5, 0, 5, 5, 3, 5, 7, 7, 7,... are not always the minimum values. The smallest primes q are in A223175.
Conjecture: except m = 25 and 49, all odd numbers > 17 are of the form m = p + 8*q where p and q are prime numbers.

			a(14) = 5 because, for p=5 the corresponding q=3 and 5+8*3 = 29 is prime.

Michel Lagneau, Table of n, a(n) for n = 0..10000

Cf. A219252, A219254, A219604, A219252, A223175, A103506.

for n from 1 by 2 to 200 do:jj:=0:for j from 1 to 1000 while (jj=0) do:p:=ithprime(j):q:=(n-p)/8: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:

A223175 Smallest prime q such that 2n+1 = p + 8*q for some odd prime p, or 0 if no such prime exists.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 0, 2, 2, 3, 2, 2, 3, 2, 3, 3, 2, 2, 0, 5, 2, 3, 2, 2, 3, 2, 3, 3, 2, 3, 7, 2, 2, 7, 5, 2, 3, 2, 2, 3, 5, 2, 3, 2, 5, 3, 2, 3, 7, 5, 2, 7, 2, 2, 3, 2, 2, 3, 2, 3, 3, 7, 3, 7, 5, 2, 7, 2, 5, 3, 2, 2, 7, 7, 3, 3, 2, 2, 7, 5, 2

Offset: 0

Michel Lagneau, May 09 2013

nonn

For n > 8, a(12) = a(24) = 0.
The corresponding p: 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 5, 7, 0, 11, 13, 7, 17, 19, 13, 23, 17, 19, 29, 31, 0, 11,... are not always the minimum values. The smallest primes p are in A223174.
Conjecture: except m = 25 and 49, all odd numbers > 17 are of the form m = p + 8*q where p and q are prime numbers.

			a(14) = 2 because, for q=2 the corresponding p=13 and 13+8*2 = 29 is prime.

Michel Lagneau, Table of n, a(n) for n = 0..10000

Cf. A219252, A219254, A219604, A219252, A223174, A103506.

for n from 1 by 2 to 200 do:jj:=0:for j from 1 to 1000 while (jj=0) do:q:=ithprime(j):p:=n-8*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:

A103508 a(n) = 1 + 2 * least i such that A103507(i)=n+1, 0 if no such i exists.

Original entry on oeis.org

9, 15, 31, 101, 139, 227, 91, 503, 995, 451, 751, 539, 1819, 1397, 2957, 3461, 1831, 1417, 6023, 3769, 1777, 9587, 5411, 9421, 18653, 8089, 4511, 6541, 10529, 16051, 19049, 13163, 3139, 22937, 23929, 43363, 24919, 43571, 97367, 55571, 14419, 75209

Offset: 1

Lei Zhou, Feb 10 2005

Cf. A103510, A103151, A103152, A103153, A103506, A103507, A025017.

(define (A103508 n) (+ 1 (* 2 (first-n-where-fun_n-is-i1 A103507 (+ 1 n)))))
(define (first-n-where-fun_n-is-i1 fun i) (let loop ((n 1)) (cond ((= i (fun n)) n) (else (loop (+ n 1))))))

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

A103510 a(n) = 1 + 2 * least i such that A103509(i)=n+1, 0 if no such i exists.

Original entry on oeis.org

9, 11, 21, 57, 23, 55, 245, 241, 115, 833, 83, 523, 437, 193, 447, 733, 167, 689, 1417, 611, 2297, 1081, 2731, 1283, 2755, 5057, 2761, 887, 2719, 9221, 4909, 8179, 4397, 13891, 9557, 2351, 9257, 5869, 10627, 11941, 1487, 2797, 3947, 5899, 11237, 20069

Offset: 1

Lei Zhou, Feb 10 2005

Cf. A103508, A103151, A103152, A103153, A103506, A103509, A025017.

Array[a, 500]; Do[a[n] = 0, {n, 1, 500}]; n = 9; ct = 0; While[ct < 150, m = 3; While[ ! (PrimeQ[m] && (((n - m)/2) > 2) && PrimeQ[(n - m)/2]), m = m + 2]; k = PrimePi[m]; If[a[k] == 0, a[k] = n; ct = ct + 1]; n = n + 2]; Print[a]

(define (A103510 n) (+ 1 (* 2 (first-n-where-fun_n-is-i1 A103509 (+ 1 n)))))
(define (first-n-where-fun_n-is-i1 fun i) (let loop ((n 1)) (cond ((= i (fun n)) n) (else (loop (+ n 1))))))

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

A302564 a(n) is the greatest prime p such that (2*n+1-p)/2 is prime.

Original entry on oeis.org

3, 5, 7, 7, 11, 13, 13, 17, 19, 19, 23, 23, 17, 29, 31, 31, 29, 37, 37, 41, 43, 43, 47, 47, 41, 53, 53, 47, 59, 61, 61, 59, 67, 67, 71, 73, 73, 71, 79, 79, 83, 83, 53, 89, 89, 83, 89, 97, 97, 101, 103, 103, 107, 109, 109, 113, 113, 107, 113, 103, 113, 107, 127, 127, 131, 131, 113, 137, 139, 139

Offset: 3

Robert Israel, Aug 15 2018

nonn

Lemoine's conjecture, also called Levy's conjecture, says a(n) exists for all n >= 3.

Robert Israel, Table of n, a(n) for n = 3..10000
Wikipedia, Lemoine's conjecture

Cf. A046927, A103506.

Lemoine:= proc(x) local s;
s:= x;
while s > 3 do
  s:= prevprime(s);
  if isprime((x-s)/2) then return s fi
od;
end proc:
seq(Lemoine(2*n+1),n=3..100);

a(n) = {my(p = precprime(2*n+1)); while (!isprime((2*n+1-p)/2), p = precprime(p-1)); p;} \\ Michel Marcus, Aug 16 2018

cp's OEIS Frontend

A195352 Smallest prime p such that 2*n+1 = 2*p + q for some odd prime q.

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A103509 a(n) is the least j such that 2n+1 = 2*A000040(k) + A000040(j) for some k > 1, or 0 if no such j exists.

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A219604 Smallest prime p such that 2n+1 = 4q + p for some odd prime q, or 0 if no such prime exists.

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A223174 Smallest prime p such that 2n+1 = p + 8*q for some odd prime q, or 0 if no such prime exists.

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A223175 Smallest prime q such that 2n+1 = p + 8*q for some odd prime p, or 0 if no such prime exists.

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A103508 a(n) = 1 + 2 * least i such that A103507(i)=n+1, 0 if no such i exists.

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A103510 a(n) = 1 + 2 * least i such that A103509(i)=n+1, 0 if no such i exists.

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A302564 a(n) is the greatest prime p such that (2*n+1-p)/2 is prime.

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A195352 Smallest prime p such that 2n+1 = 2p + q for some odd prime q.