A034693 -id:A034693 - cp's OEIS frontend

A034782 Numbers n such that A034693(n) = 3: 3n + 1 is prime, but neither n + 1 nor 2n + 1.

24, 32, 34, 64, 76, 80, 92, 94, 104, 110, 122, 124, 132, 144, 152, 154, 182, 202, 206, 214, 220, 236, 242, 244, 246, 252, 274, 286, 294, 302, 322, 332, 340, 344, 356, 362, 364, 374, 376, 390, 412, 416, 434, 474, 482, 484, 494, 496

Offset: 1

Labos Elemer

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A034693, A034694.

Position[#, 3] &@ Table[k = 1; While[! PrimeQ[k n + 1], k++]; k, {n, 500}] // Flatten (* Michael De Vlieger, Mar 02 2017 *)

is(n)=!isprime(n+1)&&!isprime(2*n+1)&&isprime(3*n+1) \\ M. F. Hasler, May 13 2018

Corrected by R. J. Mathar, Jul 26 2015

Name edited by M. F. Hasler, May 13 2018

A034784 Numbers n such that A034693(n) = 2.

Original entry on oeis.org

3, 5, 8, 9, 11, 14, 15, 20, 21, 23, 26, 29, 33, 35, 39, 41, 44, 48, 50, 51, 53, 54, 56, 63, 65, 68, 69, 74, 75, 81, 83, 86, 89, 90, 95, 98, 99, 105, 111, 113, 114, 116, 119, 120, 125, 128, 131, 134, 135, 140, 141, 146, 153, 155, 158, 165, 168, 173, 174, 176

Offset: 1

Labos Elemer

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A034693, A034694.

CompositeNext[n_]:=Module[{k=n+1},While[PrimeQ[k],k++ ];k]; lst={};Do[p=n+CompositeNext[n];If[PrimeQ[p],AppendTo[lst,n]],{n,2,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jun 15 2009 *)
Position[#, 2] &@ Table[k = 1; While[! PrimeQ[k n + 1], k++]; k, {n, 180}] // Flatten (* Michael De Vlieger, Mar 02 2017 *)

Edited by N. J. A. Sloane, Oct 27 2012

A034780 Numbers k such that A034693(k) = 4.

Original entry on oeis.org

7, 13, 25, 27, 37, 43, 45, 49, 57, 67, 73, 79, 84, 87, 93, 97, 115, 127, 139, 142, 160, 163, 169, 175, 177, 193, 199, 205, 207, 213, 234, 235, 253, 255, 258, 265, 267, 277, 279, 295, 298

Offset: 1

Labos Elemer

nonn

R. J. Mathar, Table of n, a(n) for n = 1..656

Cf. A034693, A034694.

Position[#, 4] &@ Table[k = 1; While[! PrimeQ[k n + 1], k++]; k, {n, 300}] // Flatten (* Michael De Vlieger, Mar 02 2017 *)

A034783 Numbers k such that A034693(k) = 6.

Original entry on oeis.org

17, 47, 55, 61, 77, 91, 101, 103, 107, 118, 121, 123, 137, 143, 147, 151, 161, 170, 181, 187, 195, 208, 217, 237, 241, 247, 248, 257, 263, 266, 271, 283, 287, 291, 297, 305, 311, 313, 331, 333, 335, 342, 347, 355, 367, 381, 385, 391, 395, 397, 406

Offset: 1

Labos Elemer

nonn

Cf. A034693, A034694.

A194945 Position of records in A034693.

Original entry on oeis.org

1, 3, 7, 17, 19, 59, 159, 227, 317, 361, 521, 1637, 1691, 1997, 2053, 3833, 5207, 21209, 62809, 152351, 170167, 424783, 860831, 1415551, 2679809, 4722079, 6238447, 7050811, 17886697, 27507569, 30581429, 57868997, 127813579, 154641337, 1101795593, 11907340427

Offset: 1

Charles R Greathouse IV, Sep 06 2011

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..40

Cf. A034693, A194944.

A034693(n)=my(k=n+1);while(!isprime(k),k+=n);(k-1)/n
r=0;forstep(n=1,1e9,2,t=A034693(n);if(t>r,r=t;print1(n", ")))

A194944(n) = A034693(a(n)). - R. J. Mathar, May 05 2023

Name corrected by Peter Munn, May 04 2023

A194944 Records in A034693.

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 20, 24, 26, 30, 32, 38, 42, 44, 46, 60, 84, 108, 114, 126, 144, 166, 182, 192, 212, 234, 246, 258, 276, 278, 288, 294, 310, 406, 480, 494, 504, 534, 618, 634

Offset: 1

Charles R Greathouse IV, Sep 06 2011

Cf. A034693, A194945 (positions).

A034693(n)=my(k=n+1);while(!isprime(k),k+=n);(k-1)/n
r=0;forstep(n=1,1e9,2,t=A034693(n);if(t>r,r=t;print1(t", ")))

a(n) = A034693(A194945(n)). - R. J. Mathar, May 05 2023

a(36)-a(40) from Charles R Greathouse IV, Sep 18 2011

NAME corrected by R. J. Mathar, May 05 2023

A072920 a(n) = Sum_{k=1..n} A034693(k).

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 12, 14, 16, 17, 19, 20, 24, 26, 28, 29, 35, 36, 46, 48, 50, 51, 53, 56, 60, 62, 66, 67, 69, 70, 80, 83, 85, 88, 90, 91, 95, 100, 102, 103, 105, 106, 110, 112, 116, 117, 123, 125, 129, 131, 133, 134, 136, 138, 144, 146, 150, 151, 163, 164, 170, 175

Offset: 1

Benoit Cloitre, Aug 11 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A013661, A034693.

f[n_] := Module[{k = 1}, While[!PrimeQ[k*n + 1], k++]; k]; Accumulate[Table[f[n], {n, 1, 100}]] (* Amiram Eldar, May 05 2022 *)

f(n) = if(n<0, 0, s=1; while(isprime(s*n+1)==0, s++); s); \\ A034693
a(n) = sum(k=1, n, f(k)); \\ Michel Marcus, May 05 2022

a(n) appears to be asymptotic to (zeta(2)-1)*n*log(n) where zeta(2)-1 = Pi^2/6-1 = 0.6449... . Example: a(10^5)/10^5/log(10^5) = 0.6449(1)... .

A006093 a(n) = prime(n) - 1.

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 52, 58, 60, 66, 70, 72, 78, 82, 88, 96, 100, 102, 106, 108, 112, 126, 130, 136, 138, 148, 150, 156, 162, 166, 172, 178, 180, 190, 192, 196, 198, 210, 222, 226, 228, 232, 238, 240, 250, 256, 262, 268, 270

Offset: 1

N. J. A. Sloane

These are also the numbers that cannot be written as i*j + i + j (i,j >= 1). - Rainer Rosenthal, Jun 24 2001; Henry Bottomley, Jul 06 2002
The values of k for which Sum_{j=0..n} (-1)^j*binomial(k, j)*binomial(k-1-j, n-j)/(j+1) produces an integer for all n such that n < k. Setting k=10 yields [0, 1, 4, 11, 19, 23, 19, 11, 4, 1, 0] for n = [-1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], so 10 is in the sequence. Setting k=3 yields [0, 1, 1/2, 1/2] for n = [-1, 0, 1, 2], so 3 is not in the sequence. - Dug Eichelberger (dug(AT)mit.edu), May 14 2001
n such that x^n + x^(n-1) + x^(n-2) + ... + x + 1 is irreducible. - Robert G. Wilson v, Jun 22 2002
Records for Euler totient function phi.
Together with 0, n such that (n+1) divides (n!+1). - Benoit Cloitre, Aug 20 2002; corrected by Charles R Greathouse IV, Apr 20 2010
n such that phi(n^2) = phi(n^2 + n). - Jon Perry, Feb 19 2004
Numbers having only the trivial perfect partition consisting of a(n) 1's. - Lekraj Beedassy, Jul 23 2006
Numbers n such that the sequence {binomial coefficient C(k,n), k >= n } contains exactly one prime. - Artur Jasinski, Dec 02 2007
Record values of A143201: a(n) = A143201(A001747(n+1)) for n > 1. - Reinhard Zumkeller, Aug 12 2008
From Reinhard Zumkeller, Jul 10 2009: (Start)
The first N terms can be generated by the following sieving process:
  start with {1, 2, 3, 4, ..., N - 1, N};
  for i := 1 until SQRT(N) do
  (if (i is not striked out) then
  (for j := 2 * i + 1 step i + 1 until N do
  (strike j from the list)));
  remaining numbers = {a(n): a(n) <= N}. (End)
a(n) = partial sums of A075526(n-1) = Sum_{1..n} A075526(n-1) = Sum_{1..n} (A008578(n+1) - A008578(n)) = Sum_{1..n} (A158611(n+2) - A158611(n+1)) for n >= 1. - Jaroslav Krizek, Aug 04 2009
A006093 U A072668 = A000027. - Juri-Stepan Gerasimov, Oct 22 2009
A171400(a(n)) = 1 for n <> 2: subsequence of A171401, except for a(2) = 2. - Reinhard Zumkeller, Dec 08 2009
Numerator of (1 - 1/prime(n)). - Juri-Stepan Gerasimov, Jun 05 2010
Numbers n such that A002322(n+1) = n. This statement is stronger than repeating the property of the entries in A002322, because it also says in reciprocity that this sequence here contains no numbers beyond the Carmichael numbers with that property. - Michel Lagneau, Dec 12 2010
a(n) = A192134(A095874(A000040(n))); subsequence of A192133. - Reinhard Zumkeller, Jun 26 2011
prime(a(n)) + prime(k) < prime(a(k) + k) for at least one k <= a(n): A212210(a(n),k) < 0. - Reinhard Zumkeller, May 05 2012
Except for the first term, numbers n such that the sum of first n natural numbers does not divide the product of first n natural numbers; that is, n*(n + 1)/2 does not divide n!. - Jayanta Basu, Apr 24 2013
BigOmega(a(n)) equals BigOmega(a(n)*(a(n) + 1)/2), where BigOmega = A001222. Rationale: BigOmega of the product on the right hand side factorizes as BigOmega(a/2) + Bigomega(a+1) = BigOmega(a/2) + 1 because a/2 and a + 1 are coprime, because BigOmega is additive, and because a + 1 is prime. Furthermore Bigomega(a/2) = Bigomega(a) - 1 because essentially all 'a' are even. - Irina Gerasimova, Jun 06 2013
Record values of A060681. - Omar E. Pol, Oct 26 2013
Deficiency of n-th prime. - Omar E. Pol, Jan 30 2014
Conjecture: All the sums Sum_{k=s..t} 1/a(k) with 1 <= s <= t are pairwise distinct. In general, for any integers d >= -1 and m > 0, if Sum_{k=i..j} 1/(prime(k)+d)^m = Sum_{k=s..t} 1/(prime(k)+d)^m with 0 < i <= j and 0 < s <= t then we must have (i,j) = (s,t), unless d = m = 1 and {(i,j),(s,t)} = {(4,4),(8,10)} or {(4,7),(5,10)}. (Note that 1/(prime(8)+1)+1/(prime(9)+1)+1/(prime(10)+1) = 1/(prime(4)+1) and Sum_{k=5..10} 1/(prime(k)+1) = 1/(prime(4)+1) + Sum_{k=5..7} 1/(prime(k)+1).) - Zhi-Wei Sun, Sep 09 2015
Numbers n such that (prime(i)^n + n) is divisible by (n+1), for all i >= 1, except when prime(i) = n+1. - Richard R. Forberg, Aug 11 2016
a(n) is the period of Fubini numbers (A000670) over the n-th prime. - Federico Provvedi, Nov 28 2020

Archimedeans Problems Drive, Eureka, 40 (1979), 28.
Harvey Dubner, Generalized Fermat primes, J. Recreational Math., 18 (1985): 279-280.
M. Gardner, The Colossal Book of Mathematics, pp. 31, W. W. Norton & Co., NY, 2001.
M. Gardner, Mathematical Circus, pp. 251-2, Alfred A. Knopf, NY, 1979.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
Thomas F. Bloom, Unit fractions with shifted prime denominators, arXiv:2305.02689 [math.NT], 2023.
R. P. Boas & N. J. A. Sloane, Correspondence, 1974
Harvey Dubner, Generalized Fermat primes, J. Recreational Math. 18.4 (1985-1986), 279. (Annotated scanned copy)
Armel Mercier, Problem E 3065, American Mathematical Monthly, 1984, p. 649.
Armel Mercier, S. K. Rangarajan, J. C. Binz and Dan Marcus, Problem E 3065, American Mathematical Monthly, No. 4, 1987, pp. 378.
Poo-Sung Park, Additive uniqueness of PRIMES-1 for multiplicative functions, arXiv:1708.03037 [math.NT], 2017.
J. R. Rickard and J. J. Hitchcock, Problem Drive 4, Archimedeans Problems Drive, Eureka, 40 (1979), 28-29, 40. (Annotated scanned copy)
Index entries for sequences generated by sieves

a(n) = K(n, 1) and A034693(K(n, 1)) = 1 for all n. The subscript n refers to this sequence and K(n, 1) is the index in A034693. - Labos Elemer
Cf. A000040, A034694. Different from A075728.
Complement of A072668 (composite numbers minus 1), A072670(a(n))=0.
Essentially the same as A039915.
Cf. A084920, A006093, A050997, A008864, A060800, A131991, A131992, A131993.
Cf. A101301 (partial sums), A005867 (partial products).
Column 1 of the following arrays/triangles: A087738, A249741, A352707, A378979, A379010.
The last diagonal of A162619, and of A174996, the first diagonal in A131424.
Row lengths of irregular triangles A086145, A124223, A212157.

Filtered([1..280],IsPrime)-1; # Muniru A Asiru, Nov 25 2018

a006093 = (subtract 1) . a000040  -- Reinhard Zumkeller, Mar 06 2012

[NthPrime(n)-1: n in [1..100]]; // Vincenzo Librandi, Nov 17 2015

for n from 2 to 271 do if (n! mod n^2 = n*(n-1) and (n<>4) then print(n-1) fi od; # Gary Detlefs, Sep 10 2010
# alternative
A006093 := proc(n)
    ithprime(n)-1 ;
end proc:
seq(A006093(n),n=1..100) ; # R. J. Mathar, Feb 06 2019

Table[Prime[n] - 1, {n, 1, 30}] (* Vladimir Joseph Stephan Orlovsky, Apr 27 2008 *)
a[ n_] := If[ n < 1, 0, -1 + Prime @ n] (* Michael Somos, Jul 17 2011 *)
Prime[Range[60]] - 1 (* Alonso del Arte, Oct 26 2013 *)

isA006093(n) = isprime(n+1) \\ Michael B. Porter, Apr 09 2010

A006093(n) = prime(n)-1 \\ Michael B. Porter, Apr 09 2010

\\ Sieve as described in Rainer Rosenthal's comment.
m=270;s=vector(m);for(i=1,m,for(j=i,m,k=i*j+i+j;if(k<=m,s[k]=1)));for(k=1,m,if(s[k]==0,print1(k,", "))); \\ Hugo Pfoertner, May 14 2019

from sympy import prime
for n in range(1,100): print(prime(n)-1, end=', ') # Stefano Spezia, Nov 30 2018

a(n) = (p-1)! mod p where p is the n-th prime, by Wilson's theorem. - Jonathan Sondow, Jul 13 2010
a(n) = A000010(prime(n)) = A000010(A006005(n)). - Antti Karttunen, Dec 16 2012
a(n) = A005867(n+1)/A005867(n). - Eric Desbiaux, May 07 2013
a(n) = A000040(n) - 1. - Omar E. Pol, Oct 26 2013
a(n) = A033879(A000040(n)). - Omar E. Pol, Jan 30 2014

Correction for change of offset in A158611 and A008578 in Aug 2009 Jaroslav Krizek, Jan 27 2010
Obfuscating comments removed by Joerg Arndt, Mar 11 2010
Edited by Charles R Greathouse IV, Apr 20 2010

A034694 Smallest prime == 1 (mod n).

Original entry on oeis.org

2, 3, 7, 5, 11, 7, 29, 17, 19, 11, 23, 13, 53, 29, 31, 17, 103, 19, 191, 41, 43, 23, 47, 73, 101, 53, 109, 29, 59, 31, 311, 97, 67, 103, 71, 37, 149, 191, 79, 41, 83, 43, 173, 89, 181, 47, 283, 97, 197, 101, 103, 53, 107, 109, 331, 113, 229, 59, 709, 61, 367, 311

Offset: 1

Labos Elemer, David W. Wilson, Spring 1998

Thangadurai and Vatwani prove that a(n) <= 2^(phi(n)+1)-1. - T. D. Noe, Oct 12 2011
Conjecture: a(n) < n^2 for n > 1. - Thomas Ordowski, Dec 19 2016
Eric Bach and Jonathan Sorenson show that, assuming GRH, a(n) <= (1 + o(1))*(phi(n)*log(n))^2 for n > 1. See the abstract of their paper in the Links section. - Jianing Song, Nov 10 2019
a(n) is the smallest prime p such that the multiplicative group modulo p has a subgroup of order n. - Joerg Arndt, Oct 18 2020

			If n = 7, the smallest prime in the sequence 8, 15, 22, 29, ... is 29, so a(7) = 29.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, section 2.12, pp. 127-130.
P. Ribenboim, The Book of Prime Number Records. Chapter 4,IV.B.: The Smallest Prime In Arithmetic Progressions, 1989, pp. 217-223.

T. D. Noe, Table of n, a(n) for n = 1..10000
Eric Bach and Jonathan Sorenson, Explicit bounds for primes in residue classes, Mathematics of Computation, 65(216) (1996), 1717-1735.
Steven R. Finch, Linnik's Constant
S. Graham, On Linnik's Constant, Acta Arithm. 39, 1981, pp. 163-179.
I. Niven and B. Powell, Primes in Certain Arithmetic Progressions, Amer. Math. Monthly 83(6) (1976), 467-469.
R. Thangadurai and A. Vatwani, The least prime congruent to one modulo n, Amer. Math. Monthly 118(8) (2011), 737-742.

Cf. A034693, A034780, A034782, A034783, A034784, A034785, A034846, A034847, A034848, A034849, A038700, A085420.

Records: A120856, A120857.

a034694 n = until ((== 1) . a010051) (+ n) (n + 1)
-- Reinhard Zumkeller, Dec 17 2013

a[n_] := Block[{k = 1}, If[n == 1, 2, While[Mod[Prime@k, n] != 1, k++ ]; Prime@k]]; Array[a, 64] (* Robert G. Wilson v, Jul 08 2006 *)
With[{prs=Prime[Range[200]]},Flatten[Table[Select[prs,Mod[#-1,n]==0&,1],{n,70}]]] (* Harvey P. Dale, Sep 22 2021 *)

a(n)=if(n<0,0,s=1; while((prime(s)-1)%n>0,s++); prime(s))

a(n) = min{m: m = k*n + 1 with k > 0 and A010051(m) = 1}. - Reinhard Zumkeller, Dec 17 2013

a(n) = n * A034693(n) + 1. - Joerg Arndt, Oct 18 2020

A035050 a(n) is the smallest k such that k*2^n + 1 is prime.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 3, 2, 1, 15, 12, 6, 3, 5, 4, 2, 1, 6, 3, 11, 7, 11, 25, 20, 10, 5, 7, 15, 12, 6, 3, 35, 18, 9, 12, 6, 3, 15, 10, 5, 6, 3, 9, 9, 15, 35, 19, 27, 15, 14, 7, 14, 7, 20, 10, 5, 27, 29, 54, 27, 31, 36, 18, 9, 12, 6, 3, 9, 31, 23, 39, 39, 40, 20, 10, 5, 58, 29, 15, 36, 18, 9, 13

Offset: 0

Labos Elemer

nonn

From Ulrich Krug (leuchtfeuer37(AT)gmx.de), Jun 05 2010: (Start)
If a(i) = 2 * m then a(i+1) = m.
Proof: (I) a(i) = 2*m, 2*m * 2^i + 1 = m*2^(i+1) + 1 prime, so a(i+1) <= m;
(II) if a(i+1) = m-d for an integer d > 0, (m-d) * 2^(i+1) + 1 = (2*m-2*d) * 2^i + 1 prime;
(2m-2d) < 2m contradiction to a(i) = 2 * m, d = 0.
(End)
Conjecture: for n > 0, a(n) = k < 2^n, so k*2^n + 1 is a Proth prime A080076. - Thomas Ordowski, Apr 13 2019

			a(3)=2 because 1*2^3 + 1 = 9 is composite, 2*2^3 + 1 = 17 is prime.
a(99)=219 because 2^99k + 1 is not prime for k=1,2,...,218. The first term which is not a composite number of this arithmetic progression is 2^99*219 + 1.

T. D. Noe, Table of n, a(n) for n = 0..1000
Gareth A. Jones and Alexander K. Zvonkin, Groups of prime degree and the Bateman-Horn conjecture, 2021.
Index entries for sequences related to primes in arithmetic progressions

Analogous case is A034693. Special subscripts (n's for a(n)=1) are the exponents of known Fermat primes: A000215. See also Fermat numbers A000051.

Cf. A007522, A057778, A080076, A085427, A087522, A126717, A127575, A127576, A127577, A127578, A127580, A127581, A127586.

sol:=[];m:=1; for n in [0..82] do k:=0; while not IsPrime(k*2^n+1) do k:=k+1; end while; sol[m]:=k; m:=m+1; end for; sol; // Marius A. Burtea, Jun 05 2019

a = {}; Do[k = 0; While[ ! PrimeQ[k 2^n + 1], k++ ]; AppendTo[a, k], {n, 0, 100}]; a (* Artur Jasinski *)
Table[Module[{k=1,n2=2^n},While[!PrimeQ[k*n2+1],k++];k],{n,0,90}] (* Harvey P. Dale, May 25 2024 *)

a(n) = {my(k = 1); while (! isprime(2^n*k+1), k++); k;}

a(n) << 19^n by Xylouris' improvement to Linnik's theorem. - Charles R Greathouse IV, Dec 10 2013

cp's OEIS Frontend

A034782 Numbers n such that A034693(n) = 3: 3n + 1 is prime, but neither n + 1 nor 2n + 1.

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A034784 Numbers n such that A034693(n) = 2.

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A034780 Numbers k such that A034693(k) = 4.

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A034783 Numbers k such that A034693(k) = 6.

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A194945 Position of records in A034693.

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A194944 Records in A034693.

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A072920 a(n) = Sum_{k=1..n} A034693(k).

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A006093 a(n) = prime(n) - 1.

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