A016014 -id:A016014 - cp's OEIS frontend

A239746 Records in A016014.

1, 2, 3, 5, 6, 7, 10, 12, 15, 17, 24, 30, 42, 54, 57, 60, 63, 72, 74, 83, 84, 91, 96, 104, 106, 120, 123, 129, 138, 139, 144, 147, 149, 153, 155, 203, 219, 240, 245, 259

Offset: 1

Zak Seidov, Mar 26 2014

Cf. A016014, A239727.

km = 0; Reap[Do[k = 1;  While[ ! PrimeQ[2*n*k + 1], k++]; If[k > km, km = k; Sow[k]], {n,  20000}]][[2, 1]]
lk[n_]:=Module[{k=1},While[!PrimeQ[2n k+1],k++];k];DeleteDuplicates[Array[ lk,10^7],GreaterEqual] (* Harvey P. Dale, Jun 20 2023 *)

A016014(n)=my(k); while(!isprime(2*n*k+++1), ); k
r=0; for(n=1, 1e8, t=A016014(n); if(t>r, r=t; print1(r", "))) \\ Jianing Song, Mar 09 2021, based on the program of A239727

a(37)-a(40) from Giovanni Resta, Mar 26 2014

A239800 First appearance of n in A016014, or 0 if n never occurs.

Original entry on oeis.org

1, 4, 12, 42, 19, 59, 92, 196, 184, 159, 334, 227, 317, 415, 256, 521, 514, 796, 734, 1861, 1691, 1997, 2053, 706, 5006, 5731, 3814, 2348, 5641, 1466, 19016, 5542, 26815, 8762, 18637, 5794, 31667, 5227, 17054, 35246, 51148, 5207, 59537, 75862, 54737, 117899, 58603, 81313, 30332, 100042, 205471, 113018, 102307, 21209, 63971, 321994, 62809, 648512

Offset: 1

Zak Seidov, Mar 27 2014

nonn

Conjecture: a(n) > 0 for all n. - Jianing Song, Mar 09 2021

Zak Seidov, Table of n, a(n) for n = 1..100

Cf. A016014, A239746, A239727.

isA239800 := proc(n, m) local k;
for k from 1 to n-1 while not isprime(2*m*k + 1) do od;
if k < n then false else isprime(2*m*n + 1) fi end:
A239800 := proc(n, SearchLimit) local k;
for k from 1 to SearchLimit do if isA239800(n, k) then return k fi od;
error "Search limit reached!" end:
A239800List := (ListLength, SearchLimit) ->
map(n -> A239800(n, SearchLimit), [$1..ListLength]):
A239800List(32, 65537); # Peter Luschny, Oct 05 2021

A016014(a(n)) = n.

Escape clause added by Jianing Song, Mar 09 2021

A067760 a(n) is the least positive k such that (2n+1) + 2^k is prime, or 0 if no such k exists.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 3, 2, 1, 1, 4, 2, 1, 2, 1, 1, 2, 1, 5, 2, 1, 3, 2, 1, 1, 8, 2, 1, 2, 1, 1, 4, 2, 1, 2, 1, 7, 2, 1, 3, 4, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 7, 4, 5, 3, 4, 2, 1, 2, 1, 3, 2, 1, 1, 10, 3, 3, 2, 1, 1, 4, 2, 1, 4, 2, 1, 2, 1, 5, 2, 1, 3, 2, 1, 1, 4, 3, 3, 2, 1, 1, 2, 1, 1, 6, 5, 3, 6

Offset: 0

Don Reble, Feb 05 2002

nonn

From Phil Moore (moorep(AT)lanecc.edu), Dec 14 2009: (Start)
It is known that a(39278) = 0, since no such prime exists for the Sierpiński number 78557 (cf. A076336).
It has recently been discovered that 2131+2^4583176 and 41693+2^5146295 are probable primes, so a(1065) is probably 4583176 and a(20846) is probably 5146295.
At present, the only odd value less than 78557 for which no prime or strong probable prime of the form t+2^k is known is t = 40291, so a(20145) is completely unknown. In addition, for 25 values of t < 78557, only strong probable primes are known. (End)
The last case was resolved in 2011 when the probable prime 40291+2^9092392 was found as a part of a distributed project "Five or Bust". See links. - Jeppe Stig Nielsen, Mar 29 2019

			a(15)=4 because (2*15+1)+2^k is composite for k=1,2,3 and prime for k=4.

Jinyuan Wang, Table of n, a(n) for n = 0..39278 (terms 0..1064 from T. D. Noe, terms 1065..3000 from Richard N. Smith).
Mersenne Forum, Five or Bust
Carlos Rivera, Puzzle 167. Primes m + 2^j & m - 2^j, The Prime Puzzles and Problems Connection.

Cf. A016014, A050412, A066081, A033919, A094076, A076336, A260350, A263874, A263875 (records).

a(n) = {my(k=1); while (! isprime((2*n+1)+2^k), k++); k;} \\ Michel Marcus, Feb 26 2018

A070846 Smallest prime == 1 (mod 2n).

Original entry on oeis.org

3, 5, 7, 17, 11, 13, 29, 17, 19, 41, 23, 73, 53, 29, 31, 97, 103, 37, 191, 41, 43, 89, 47, 97, 101, 53, 109, 113, 59, 61, 311, 193, 67, 137, 71, 73, 149, 229, 79, 241, 83, 337, 173, 89, 181, 277, 283, 97, 197, 101, 103, 313, 107, 109, 331, 113, 229, 233, 709, 241

Offset: 1

Amarnath Murthy, May 15 2002

nonn

From Jianing Song, Feb 14 2021: (Start)
a(n) is the smallest prime p such that there is a primitive 2n-th root of unity modulo p, i.e., there is an element with order 2n in the multiplicative group of integers modulo p.
For n > 1, a(n) is the smallest prime p such that the 2n-th cyclotomic field Q(exp(2*Pi*i/(2*n))) can be embedded into the p-adic field Q_p. (End)

Dmitry Kamenetsky, Table of n, a(n) for n = 1..50000

Cf. A070847, A070848, A070849, A070850, A070851, A070852, A070853.

Cf. A034694, A016014.

With[{prs=Prime[Range[200]]},Flatten[Table[Select[prs,Mod[#,2n]==1&,1],{n,60}]]] (* Harvey P. Dale, Jan 16 2013 *)

for(n=1,80,s=1; while((isprime(s)*s-1)%(2*n)>0,s++); print1(s,","))

a(n) = 2*n*A016014(n) + 1. - Dmitry Kamenetsky, Oct 26 2016

More terms from Benoit Cloitre, May 18 2002

A117673 a(n) is the least k such that k2prime(n) + 1 is prime.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 5, 1, 1, 5, 2, 1, 2, 3, 1, 6, 3, 2, 4, 2, 2, 1, 1, 2, 3, 3, 3, 5, 1, 2, 1, 3, 2, 4, 3, 5, 2, 7, 1, 1, 3, 1, 2, 9, 2, 5, 6, 12, 6, 1, 1, 3, 1, 3, 3, 4, 3, 2, 1, 3, 1, 2, 3, 3, 13, 3, 5, 3, 5, 7, 1, 3, 2, 6, 6, 12, 3, 4, 2, 1, 5, 1, 2, 5, 1, 4, 15, 3, 6, 3, 4, 2, 1, 2, 3, 1, 16, 5, 9

Offset: 1

Don Reble, Apr 25 2006

nonn

Iff a(n) = 1, prime(n) is a Sophie Germain prime, i.e., in A005384. - A.H.M. Smeets, Feb 01 2018

			a(8)=5 because 2*prime(8)=38 and 5*38 + 1 is prime.

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A016014, A035096, A074884.

Table[k := 1; While[ ! PrimeQ[2*k*Prime[n] + 1], k++ ]; k, {n, 1, 120}] (* Stefan Steinerberger, May 01 2006 *)

a(n) = {my(p=prime(n), k=1); while (!isprime(2*k*p+1), k++); k;} \\ Michel Marcus, Feb 12 2018

A239727 Numbers n such that the least prime of the form 2nk + 1 has a value of k that is larger than the k values for all smaller n.

Original entry on oeis.org

1, 4, 12, 19, 59, 92, 159, 227, 256, 514, 706, 1466, 5207, 21209, 62809, 86914, 152351, 170167, 321472, 424783, 491702, 860831, 1415551, 1581442, 2679809, 4691576, 6238447, 6630812, 17886697, 27507569, 30581429, 57868997, 108830332, 116156102, 127813579, 154641337, 1072567492, 1101795593, 3546087418, 10371779744

Offset: 1

Charles R Greathouse IV, Mar 25 2014

nonn

RECORDS transform of A016014.

Sequence is infinite; a terrible upper bound can be derived from Linnik's theorem together with the Chinese Remainder Theorem, giving a(n) << exp(a(n-1)^6).

			2*4*2+1 = 17 is prime with k = 2, but 1 through 3 have k = 1.
2*12*3+1 = 73 is prime with k = 3, but k = 2 for 4, 7, 10 and k = 1 for the other n < 12.

Cf. A016014, A239746.

A016014(n)=my(k); while(!isprime(2*n*k+++1),);k
r=0;for(n=1,1e8,t=A016014(n);if(t>r,r=t;print1(n", ")))

A016014(a(n)) = A239746(n). - Zak Seidov, Mar 27 2014

a(37)-a(40) from Giovanni Resta, Mar 26 2014

A037035 Least k such that 2^n+1+k is a prime.

Original entry on oeis.org

0, 0, 0, 2, 0, 4, 2, 2, 0, 8, 6, 4, 2, 16, 26, 2, 0, 28, 2, 20, 6, 16, 14, 8, 42, 34, 14, 28, 2, 10, 2, 10, 14, 16, 24, 52, 30, 8, 6, 22, 14, 26, 14, 28, 6, 58, 14, 4, 20, 68, 54, 20, 20, 4, 158, 2, 80, 8, 68, 130, 32, 14, 134, 28, 12, 130, 8, 2, 32, 28, 24, 10, 14, 28, 36, 32, 14

Offset: 0

Felice Russo

nonn

			a(5)=4 because 2^5+1+4=37 that is a prime.

Cf. A016014.

A013597(n) - 1.

a(n) = k = 0; while (! isprime(2^n+k+1), k++); k; \\ Michel Marcus, Sep 27 2013

More terms from Erich Friedman

A103961 Least k such that 2nk - 1 is a prime.

Original entry on oeis.org

2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 4, 3, 1, 1, 2, 2, 1, 2, 1, 1, 3, 1, 3, 2, 1, 3, 3, 1, 1, 2, 2, 1, 2, 1, 1, 2, 3, 1, 2, 1, 3, 3, 1, 4, 3, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 3, 3, 2, 4, 5, 2, 1, 3, 1, 3, 2, 1, 1, 2, 3, 7, 3, 1, 1, 2, 2, 1, 3, 4, 1, 2, 1, 3, 5, 1, 7, 8, 1, 1, 2, 3, 3, 2, 1, 1, 3, 1, 1, 5, 5, 3, 5, 2

Offset: 1

Lei Zhou, Feb 23 2005

Question: Is the sequence unbounded (like A016014)? - Dmitry Kamenetsky, Oct 26 2016

Answer: Yes. Essentially the same argument works. To get n such that a(n) > K, take distinct odd primes p_k, k=1..K with p_k not dividing k, and take n such that n == (2*k)^(-1) mod p_k and 2*k*n-1 > p_k for k=1..K. - Robert Israel, Oct 27 2016

			2*1*2-1 = 3, so a(1) = 2;
2*5*2-1 = 19, so a(5) = 2.

Dmitry Kamenetsky, Table of n, a(n) for n = 1..10000

Cf. A016014.

Do[k = 1; cp = n*k - 1; While[ ! PrimeQ[cp], k++; cp = n*k - 1]; Print[k], {n, 2, 400, 2}]
lkp[n_]:=Module[{k=1},While[!PrimeQ[2n*k-1],k++];k]; Array[lkp,120] (* Harvey P. Dale, Nov 13 2020 *)

a(n) = {my(k=1); while (!isprime(2*n*k-1), k++); k;} \\ Michel Marcus, Oct 27 2016

A342364 The primes associated with A239727.

Original entry on oeis.org

3, 17, 73, 191, 709, 1289, 3181, 5449, 7681, 17477, 33889, 87961, 437389, 2290573, 7160227, 10429681, 19196227, 24504049, 47577857, 70513979, 82605937, 156671243, 271785793, 328939937, 568119509, 1125978241, 1534657963, 1710749497, 4936728373, 7647104183

Offset: 1

Jianing Song, Mar 09 2021

nonn

			The smallest prime of the form 2*A239727(3)*k + 1 = 24*k+1 is 73, hence a(3) = 73.
The smallest prime of the form 2*A239727(4)*k + 1 = 38*k+1 is 191, hence a(4) = 191.

Jianing Song, Table of n, a(n) for n = 1..40

Cf. A239727, A239746, A016014, A070846, A342365.

A016014(n)=my(k); while(!isprime(2*n*k+++1), ); k
r=0; for(n=1, 1e8, t=A016014(n); if(t>r, r=t; print1(2*n*r+1", "))) \\ based on the program of A239727

a(n) = 2*A239727(n)*A239746(n) + 1.

A342365 The primes associated with A239800 (1 if A239800(n) = 0).

Original entry on oeis.org

3, 17, 73, 337, 191, 709, 1289, 3137, 3313, 3181, 7349, 5449, 8243, 11621, 7681, 16673, 17477, 28657, 27893, 74441, 71023, 87869, 94439, 33889, 250301, 298013, 205957, 131489, 327179, 87961, 1178993, 354689, 1769791, 595817, 1304591, 417169, 2343359

Offset: 1

Jianing Song, Mar 09 2021

nonn

			The smallest m such that 2*m*i + 1 is not prime until i = 3 is m = 12, and the corresponding prime is 2*12*3 + 1 = 73 = a(3).
The smallest m such that 2*m*i + 1 is not prime until i = 4 is m = 42, and the corresponding prime is 2*42*4 + 1 = 337 = a(4).
The smallest m such that 2*m*i + 1 is not prime until i = 5 is m = 19, and the corresponding prime is 2*19*5 + 1 = 191 = a(5).

Jianing Song, Table of n, a(n) for n = 1..100

Cf. A239800, A016014, A070846, A342364.

isok(n, m) = for(i=1, n-1, if(isprime(2*m*i+1), return(0))); if(isprime(2*m*n+1), 1, 0)
a(n) = for(m=1, oo, if(isok(n, m), return(2*n*m+1))) \\ based on the conjecture that all numbers occur in A016014

a(n) = 2*n*A239800(n)+1.

cp's OEIS Frontend

A239746 Records in A016014.

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A239800 First appearance of n in A016014, or 0 if n never occurs.

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A067760 a(n) is the least positive k such that (2n+1) + 2^k is prime, or 0 if no such k exists.

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A070846 Smallest prime == 1 (mod 2n).

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A117673 a(n) is the least k such that k*2*prime(n) + 1 is prime.

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A239727 Numbers n such that the least prime of the form 2nk + 1 has a value of k that is larger than the k values for all smaller n.

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PARI

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A037035 Least k such that 2^n+1+k is a prime.

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PARI

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A103961 Least k such that 2*n*k - 1 is a prime.

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A117673 a(n) is the least k such that k2prime(n) + 1 is prime.

A103961 Least k such that 2nk - 1 is a prime.