A103153 -id:A103153 - cp's OEIS frontend

A103514 a(n) is the smallest m such that primorial(n)/2 - 2^m is prime.

0, 1, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 25, 2, 1, 6, 6, 19, 1, 13, 3, 3, 11, 29, 2, 1, 6, 3, 4, 2, 6, 4, 15, 6, 4, 20, 4, 1, 7, 16, 4, 7, 22, 3, 12, 13, 9, 35, 2, 3, 3, 52, 35, 3, 32, 15, 13, 10, 53, 56, 9, 16, 36, 5, 8, 5, 22, 3, 14, 2, 64, 37, 8, 22, 42, 11, 22, 22, 12, 11, 26, 1, 54, 187, 20, 9

Offset: 2

Lei Zhou, Feb 15 2005

nonn

			P(2)/2-2^0=2 is prime, so a(2)=0;
P(10)/2-2^3=3234846607 is Prime, so a(10)=3.

R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2018.

Cf. A002110, A005234, A014545, A018239, A006794, A057704, A057705, A103153, A067026, A067027, A139439, A139440, A139441, A139442, A139443, A139444, A139445, A139446, A139447, A139448, A139449, A139450, A139451, A139452, A139453, A139454, A139455, A139456, A139457, A103514.

nmax = 2^8192; npd = 1; n = 2; npd = npd*Prime[n]; While[npd < nmax, tn = 1; tt = 2; cp = npd - tt; While[(cp > 1) && (! (PrimeQ[cp])), tn = tn + 1; tt = tt*2; cp = npd - tt]; If[cp < 2, Print["*"], Print[tn]]; n = n + 1; npd = npd*Prime[n]]
(* Second program: *)
k = 1; a = {}; Do[k = k*Prime[n]; m = 1; While[ ! PrimeQ[k - 2^m], m++ ]; Print[m]; AppendTo[a, m], {n, 2, 200}]; a (* Artur Jasinski, Apr 21 2008 *)

a(n)=my(t=prod(i=2,n,prime(i)),m); while(!isprime(t-2^m),m++); m \\ Charles R Greathouse IV, Apr 28 2015

Edited by N. J. A. Sloane, May 16 2008 at the suggestion of R. J. Mathar

A103506 Smallest prime p such that 2n+1 = 2q + p for some odd prime q, or 0 if no such prime exists.

Original entry on oeis.org

0, 0, 0, 3, 5, 3, 5, 3, 5, 7, 13, 3, 5, 3, 5, 7, 13, 3, 5, 3, 5, 7, 13, 3, 5, 7, 17, 11, 13, 3, 5, 3, 5, 7, 13, 11, 13, 3, 5, 7, 37, 3, 5, 3, 5, 7, 13, 3, 5, 7, 17, 11, 13, 3, 5, 7, 29, 11, 13, 3, 5, 3, 5, 7, 13, 11, 13, 3, 5, 7, 37, 3, 5, 3, 5, 7, 13, 11

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=11, 2*11+1 = 23 = 2*5+13, thus a(11)=13.

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

a(n)=0 if A103509(n)=0, otherwise A000040(A103509(n)).

Cf. A103151, A103152, A103153.

Join[{0,0,0}, Table[m=3; While[! (PrimeQ[m] && (((n-m)/2) > 2) && PrimeQ[(n-m)/2]), m=m+2]; m, {n, 9, 299, 2}]]

(define (A103506 n) (let ((ind (A103509 n))) (if (zero? ind) 0 (A000040 ind)))) ;; Antti Karttunen, Jun 19 2007

Edited by Antti Karttunen, Jun 19 2007

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

Original entry on oeis.org

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 *)

A103507 a(n) = Least i > 1, such that 2n+1 = 2*A000040(i)+A000040(k) for some k>1, 0 if no such i exists.

Original entry on oeis.org

0, 0, 0, 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 4, 3, 2, 2, 3, 3, 2, 4, 2, 2, 3, 2, 4, 3, 2, 4, 3, 2, 2, 3, 3, 2, 4, 2, 2, 3, 3, 2, 4, 2, 8, 3, 2, 4, 3, 5, 2, 5, 2, 2, 3, 2, 2, 3, 2, 4, 3, 5, 4, 5, 5, 2, 5, 2, 6, 3, 2, 2, 3, 3, 4, 4, 2, 2, 3, 3, 2, 4, 3, 2, 4, 2, 6, 3, 2, 4, 3, 2, 2, 3, 3, 4, 4, 2, 2, 3, 2, 2, 3, 3, 4, 4, 5, 2

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 and 3=A000040(2), thus a(4)=2. For n=7, 2*7+1 = 15 = 2*5+5 and 5=A000040(3), thus a(7)=3.

a(n) = A049084(A103153(n)), for n >= 4. Can be used to compute A103153 and A103508. Cf. A103509.

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

Edited, Scheme-code added and starting offset changed from 0 to 1 by Antti Karttunen, Jun 19 2007

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

A103515 Primes of the form primorial P(k)*2^n-1 with minimal n, n>=0, k>=2.

Original entry on oeis.org

5, 29, 419, 2309, 30029, 1021019, 19399379, 892371479, 51757545839, 821495767572479, 14841476269619, 304250263527209, 54873078184468933509119, 2459559130353965639, 521426535635040715679, 15751252788463309939261439

Offset: 1

Lei Zhou, Feb 15 2005

nonn

Conjecture: sequence is defined for all k>=2

			P(2)*2^0-1=3*2-1=5 is prime, so a(2)=5;
P(4)*2^1-1=7*5*3*2*2-1=419 is prime, so a(4)=419;

Cf. A002110, A005234, A014545, A018239, A006794, A057704, A057705, A103153, A103154.

nmax = 2^2048; npd = 2; n = 2; npd = npd*Prime[n]; While[npd < nmax, tt = 1; cp = npd*tt - 1; While[ ! (PrimeQ[cp]), tt = tt*2; cp = npd*tt - 1]; Print[cp]; n = n + 1; npd = npd*Prime[n]]

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A103514 a(n) is the smallest m such that primorial(n)/2 - 2^m is prime.

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A103506 Smallest prime p such that 2n+1 = 2q + p for some odd prime q, or 0 if no such prime exists.

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

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A103507 a(n) = Least i > 1, such that 2n+1 = 2*A000040(i)+A000040(k) for some k>1, 0 if no such i 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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Extensions

A103515 Primes of the form primorial P(k)*2^n-1 with minimal n, n>=0, k>=2.

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Mathematica

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