A058383 -id:A058383 - cp's OEIS frontend

A060211 Larger term of a pair of twin primes such that the prime factors of their average are only 2 and 3. Proper subset of A058383.

7, 13, 19, 73, 109, 193, 433, 1153, 2593, 139969, 472393, 786433, 995329, 57395629, 63700993, 169869313, 4076863489, 10871635969, 2348273369089, 56358560858113, 79164837199873, 84537841287169, 150289495621633, 578415690713089, 1141260857376769, 57711166318706689

Offset: 1

Labos Elemer, Mar 20 2001

nonn

Larger of twin primes p such that p-1 = (2^u)*(3^w), u,w >= 1.

			a(4) = 73, {71,73} are twin primes and (71 + 73)/2 = 72 = 2*2*2*3*3.

Ray Chandler, Table of n, a(n) for n = 1..61 (terms < 10^1000)
Harsh Aggarwal, Table of n, a(n) for n = 62..91 (terms from 10^1000 to 10^20000)

Cf. A001454, A002822, A027856, A033845, A058383, A059960.

Take[Select[Sort[Flatten[Table[2^a 3^b,{a,250},{b,250}]]],AllTrue[#+{1,-1},PrimeQ]&]+1,23] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 17 2019 *)

isok(p) = isprime(p) && isprime(p-2) && (vecmax(factor(p-1)[,1]) == 3); \\ Michel Marcus, Sep 05 2017

a(n) = A027856(n+1) + 1. - Amiram Eldar, Mar 17 2025

Name corrected by Sean A. Irvine, Oct 31 2022

A027856 Dan numbers: numbers m of the form 2^j * 3^k such that m +- 1 are twin primes.

Original entry on oeis.org

4, 6, 12, 18, 72, 108, 192, 432, 1152, 2592, 139968, 472392, 786432, 995328, 57395628, 63700992, 169869312, 4076863488, 10871635968, 2348273369088, 56358560858112, 79164837199872, 84537841287168, 150289495621632, 578415690713088, 1141260857376768

Offset: 1

Richard C. Schroeppel

nonn

Special twin prime averages (A014574).

Intersection of A014574 and A003586. - Jeppe Stig Nielsen, Sep 05 2017

			a(14) = 243*4096 = 995328 and {995327, 995329} are twin primes.

Ray Chandler, Table of n, a(n) for n = 1..62 (terms < 10^1000, first 55 terms from Donovan Johnson)

Cf. A014574, A060211, A078883, A078884.

Cf. also A002822, A033845, A058383, A003586.

Select[#, Total@ Boole@ Map[PrimeQ, # + {-1, 1}] == 2 &] &@ Select[Range[10^7], PowerMod[6, #, #] == 0 &] (* Michael De Vlieger, Dec 31 2016 *)
seq[max_] := Select[Sort[Flatten[Table[2^i*3^j, {i, 1, Floor[Log2[max]]}, {j, 0, Floor[Log[3, max/2^i]]}]]], And @@ PrimeQ[# + {-1, 1}] &]; seq[2*10^15] (* Amiram Eldar, Aug 27 2024 *)

a(n) = A078883(n) + 1 = A078884(n) - 1. - Amiram Eldar, Aug 27 2024

Offset corrected by Donovan Johnson, Dec 02 2011

Entry revised by N. J. A. Sloane, Jan 01 2017

A035095 Smallest prime congruent to 1 (mod prime(n)).

Original entry on oeis.org

3, 7, 11, 29, 23, 53, 103, 191, 47, 59, 311, 149, 83, 173, 283, 107, 709, 367, 269, 569, 293, 317, 167, 179, 389, 607, 619, 643, 1091, 227, 509, 263, 823, 557, 1193, 907, 1571, 653, 2339, 347, 359, 1087, 383, 773, 3547, 797, 2111, 2677, 5449, 2749, 467

Offset: 1

Labos Elemer

nonn

This is a version of the "least prime in special arithmetic progressions" problem.
Smallest numbers m such that largest prime factor of Phi(m) = prime(n), the n-th prime, also seems to be prime and identical to n-th term of A035095. See A068211, A068212, A065966: Min[x : A068211(x)=prime(n)] = A035095(n); e.g., Phi(a(7)) = Phi(103) = 2*3*17, of which 17 = p(7) is the largest prime factor, arising first here.
It appears that A035095, A066674, A125878 are probably all the same, but see the comments in A066674. - N. J. A. Sloane, Jan 05 2013
Minimum of the smallest prime factors of F(n,i) = (i^prime(n)-1)/(i-1), when i runs through all integers in [2, prime(n)]. Every prime factor of F(n,i) is congruent to 1 modulo prime(n). - Vladimir Shevelev, Nov 26 2014
Conjecture: a(n) is the smallest prime p such that gpf(p-1) = prime(n). See A023503. - Thomas Ordowski, Aug 06 2017
For n>1, a(n) is the smallest prime congruent to 1 mod (2*prime(n)). - Chai Wah Wu, Apr 28 2025

			a(8) = 191 because in the prime(8)k+1 = 19k+1 sequence, 191 is the smallest prime.

E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Bd 1 (reprinted Chelsea 1953).
E. C. Titchmarsh, A divisor problem, Renc. Circ. Math. Palermo, 54 (1930) pp. 414-429.
P. Turan, Über Primzahlen der arithmetischen Progression, Acta Sci. Math. (Szeged), 8 (1936/37) pp. 226-235.

T. D. Noe, Table of n, a(n) for n = 1..10000
P. Erdős, On some application of Brun's method, Acta Sci. Math (Szeged), v. 13, 1949, pp. 57-63.
A. Granville and C. Pomerance, On the least prime in certain arithmetic progressions
A. Granville and C. Pomerance, On the least prime in certain arithmetic progressions J. Lond Math Soc s2-41 (2) (1990), pp. 193-200.
D. R. Heath-Brown, almost-primes in arithmetic progressions in short intervals, Math Proc Cambr. Phil Soc v 83 (1978), pp. 357-375.
D. R. Heath-Brown, Siegel zeros and the least prime in arithmetic progression, Quart. J. of Math 41 (49) (1990), pp. 405-418.
H.-J. Kanold, Uber Primzahlen in arithmetischen Folgen, Math. Ann. v 156 (1964) pp. 393-395.
U. V. Linnik, On the least prime in an arithmetic progression. I. The basic theorem, Rec. Math (N.S.) v 15 (57) (1944), pp 139-178. MR0012111
C. Pomerance, A note on the least prime in an arithmetic progression J. Number Theory 12 (2) (1980), pp. 218-223.
K. Prachar, Uber die kleinste Primzahl in einer arithmetischen Reihe, J Reine Angew Math. 206 (1961) pp. 3-4.
A. Schinzel, Remark on the paper of K. Prachar Uber die kleinste.., J. Reine Angew Math. v 210 (1962) pp. 122-122.
S. S. Wagstaff, Jr., The irregular primes to 125000, Math. Comp., 32 (1978) pp. 583-591.
S. S. Wagstaff, Jr, Greatest of the Least Primes in Arithmetic Progressions Having a Given Modulus, Math. Comp., 33 (147) (1979) pp. 1073-1080.
Index entries for sequences related to primes in arithmetic progressions

Cf. A034694, A032448, A006530, A006093, A035096, A000040, A019434, A058383.
Cf. A068211, A068212, A065966, A000010, A070844-A070858, A061092.
Cf. A000040.

a[n_] := Block[{p = Prime[n]}, r = 1 + p; While[ !PrimeQ[r], r += p]; r]; Array[a, 51] (* Jean-François Alcover, Sep 20 2011, after PARI *)
a[n_]:=If[n<2,3,Block[{p=Prime[n]},r=1+2*p;While[!PrimeQ[r],r+=2*p]];r];Array[a,51] (* Zak Seidov, Dec 14 2013 *)

a(n)=local(p,r);p=prime(n);r=1;while(!isprime(r),r+=p);r

{my(N=66); forprime(p=2, , forprime(q=p+1,10^10, if((q-1)%p==0, print1(q,", "); N-=1; break)); if(N==0,break)); } \\ Joerg Arndt, May 27 2016

from itertools import count
from sympy import prime, isprime
def A035095(n): return 3 if n==1 else next(filter(isprime,count((p:=prime(n)<<1)+1,p))) # Chai Wah Wu, Apr 28 2025

According to a long-standing conjecture (see the 1979 Wagstaff reference), a(n) <= prime(n)^2 + 1. This would be sufficient to imply that a(n) is the smallest prime such that greatest prime divisor of a(n)-1 is prime(n), the n-th prime: A006530(a(n)-1) = A000040(n). This in turn would be sufficient to imply that no value occurs twice in this sequence. - Franklin T. Adams-Watters, Jun 18 2010

a(n) = 1 + A035096(n)*A000040(n). - Zak Seidov, Dec 27 2013

Edited by Franklin T. Adams-Watters, Jun 18 2010
Minor edits by N. J. A. Sloane, Jun 27 2010
Edited by N. J. A. Sloane, Jan 05 2013

A059960 Smaller term of a pair of twin primes such that prime factors of their average are only 2 and 3.

Original entry on oeis.org

5, 11, 17, 71, 107, 191, 431, 1151, 2591, 139967, 472391, 786431, 995327, 57395627, 63700991, 169869311, 4076863487, 10871635967, 2348273369087, 56358560858111, 79164837199871, 84537841287167, 150289495621631, 578415690713087, 1141260857376767

Offset: 1

Labos Elemer, Mar 02 2001

nonn

Lesser of twin primes p such that p+1 = (2^u)*(3^w), u,w >= 1.

Primes p(k) such that the number of distinct prime divisors of all composite numbers between p(k) and p(k+1) is 2. - Amarnath Murthy, Sep 26 2002

			a(11)+1 = 2*2*2*3*3*3*3*3*3*3*3*3*3 = 472392.

Ray Chandler, Table of n, a(n) for n = 1..61 (terms < 10^1000, first 49 terms from T. D. Noe)

Cf. A014574, A002822, A033845, A058383, A027856.
Cf. A052297, A075581, A075580, A075583, A075584, A075585, A075586, A075587, A075588, A075589.
Apart from initial terms, same as A078883.

nn=10^15; Sort[Reap[Do[n=2^i 3^j; If[n<=nn && PrimeQ[n-1] && PrimeQ[n+1], Sow[n-1]], {i, Log[2, nn]}, {j, Log[3, nn]}]][[2, 1]]]
Select[Select[Partition[Prime[Range[38*10^5]],2,1],#[[2]]-#[[1]]==2&][[All,1]],FactorInteger[#+1][[All,1]]=={2,3}&] (* The program generates the first 15 terms of the sequence. *)
seq[max_] := Select[Sort[Flatten[Table[2^i*3^j - 1, {i, 1, Floor[Log2[max]]}, {j, 1, Floor[Log[3, max/2^i]]}]]], And @@ PrimeQ[# + {0, 2}] &]; seq[2*10^15] (* Amiram Eldar, Aug 27 2024 *)

a(n) = A027856(n+1) - 1. - Amiram Eldar, Mar 17 2025

A060212 Primes q such that 6q-1 and 6q+1 are twin primes. Proper subset of A002822.

Original entry on oeis.org

2, 3, 5, 7, 17, 23, 47, 103, 107, 137, 283, 313, 347, 373, 397, 443, 467, 577, 593, 653, 773, 787, 907, 1033, 1117, 1423, 1433, 1613, 1823, 2027, 2063, 2137, 2153, 2203, 2287, 2293, 2333, 2347, 2677, 2903, 3257, 3307, 3407, 3413, 3593, 3623, 3673, 3923

Offset: 1

Labos Elemer, Mar 20 2001

nonn

Primes in A182521. Also all primes p for which A182481(p)=1. - Vladimir Shevelev, May 03 2012

Conjecture: a(n) ~ n*log(n)*log(n*log(n))*log(log(n)). - Carl R. White, Nov 16 2023

David Radcliffe, Table of n, a(n) for n = 1..10000

Cf. A001359, A002822, A014574, A027856, A058383, A059960, A182481, A182521, A294731.

lst={}; Do[p=Prime[n]; If[PrimeQ[6*p-1] && PrimeQ[6*p+1], AppendTo[lst,p]], {n,100}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 16 2009 *)

forprime(p=2, 9999, if(isprime(6*p+1) & isprime(6*p-1), print(p))) \\ David Radcliffe, Apr 02 2016

from sympy import *; print([p for p in primerange(2,9999) if isprime(6*p-1) and isprime(6*p+1)]) # David Radcliffe, Apr 02 2016

A125866 Odd numbers k such that cos(2*Pi/k) is an algebraic number of a 3-smooth degree, but not 2-smooth.

Original entry on oeis.org

7, 9, 13, 19, 21, 27, 35, 37, 39, 45, 57, 63, 65, 73, 81, 91, 95, 97, 105, 109, 111, 117, 119, 133, 135, 153, 163, 171, 185, 189, 193, 195, 219, 221, 243, 247, 259, 273, 285, 291, 315, 323, 327, 333, 351, 357, 365, 399, 405, 433, 455, 459, 481, 485, 487, 489

Offset: 1

Artur Jasinski, Dec 13 2006

nonn

Odd terms of A051913.
This sequence is infinite (unlike A004729), because it contains any A058383(n) times any power of 3.
A regular polygon of a(n) sides can be constructed if one also has an angle trisector.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A004729, A051913, A058383, A125867-A125878.

filter:= proc(n) local r,a,b;
  r:= numtheory:-phi(n);
  a:= padic:-ordp(r,2);
  b:= padic:-ordp(r,3);
  if b = 0 then return false fi;
  r = 2^a*3^b;
end proc:
select(filter, [seq(i,i=3..1000,2)]); # Robert Israel, May 11 2020

Do[If[Take[FactorInteger[EulerPhi[2n+1]][[ -1]], 1]=={3},Print[2n+1]],{n,1,10000}]

Edited by Don Reble, Apr 24 2007

A216148 Primes of the form 2*k^k + 1 = A216147(k).

Original entry on oeis.org

3, 17832200896513, 78692816150593075150849

Offset: 1

M. F. Hasler, Sep 02 2012

The sequence should be extended through A110932, which lists the corresponding values of k: The next term, 2*251^251 + 1 = A216147(A110932(4)) ~ 4.16*10^602, is too large to include here.

M. F. Hasler, Table of n, a(n) for n = 1..4
C. Caldwell, G.L. Honaker (Eds), Prime Curios!: 78692816150593075150849.
"Jim", Topic: The Lost Proof of Fermat, on mathforum.org, Jan 31 2004

Cf. A110932.

A subsequence of A133663, with b=a and c=1.

Select[Table[2n^n+1,{n,20}],PrimeQ] (* Harvey P. Dale, Mar 27 2016 *)

for(n=1,999, ispseudoprime(p=n^n*2+1) & print1(p","))

a(2) = A216147(12) = A005109(95) = A070855(12) = A058383(89) = A133663(18).

a(3) = A216147(18) = A005109(183)= A070855(18) = A058383(177)= A133663(36).

A060213 Lesser of twin primes whose average is 6 times a prime.

Original entry on oeis.org

11, 17, 29, 41, 101, 137, 281, 617, 641, 821, 1697, 1877, 2081, 2237, 2381, 2657, 2801, 3461, 3557, 3917, 4637, 4721, 5441, 6197, 6701, 8537, 8597, 9677, 10937, 12161, 12377, 12821, 12917, 13217, 13721, 13757, 13997, 14081, 16061, 17417

Offset: 1

Labos Elemer, Mar 20 2001

nonn

Lowest factor-density among all positive consecutive integer triples; for p > 41, last digit of p can be only 1 or 7 (see Alexandrov link, p. 15). - Lubomir Alexandrov, Nov 25 2001

			102197 is here because 102198 = 17033*6 and 17033 is prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Lubomir Alexandrov, On the nonasymptotic prime number distribution, arXiv:math/9811096 [math.NT], 1998, see p. 15.

Cf. A002822, A058383, A059960, A027856, A001454, A001359, A060212.

map(t -> 6*t-1, select(p -> isprime(p) and isprime(6*p-1) and isprime(6*p+1), [2,seq(i,i=3..10000,2)]));

Transpose[Select[Partition[Prime[Range[2500]],2,1],#[[2]]-#[[1]] == 2 && PrimeQ[Mean[#]/6]&]][[1]] (* Harvey P. Dale, May 04 2014 *)

isok(n) = isprime(n) && isprime(n+2) && !((n+1) % 6) && isprime((n+1)/6); \\ Michel Marcus, Dec 14 2013

a(n) = 6 * A060212(n) - 1. - Sean A. Irvine, Oct 31 2022

Offset changed to 1 by Michel Marcus, Dec 14 2013

A336773 a(n) is the least prime of the form 2^j*3^k + 1, j > 0, k > 0, j + k = n. a(n) = 0 if no such prime exists.

Original entry on oeis.org

7, 13, 37, 73, 97, 193, 577, 769, 3457, 10369, 0, 12289, 629857, 839809, 147457, 995329, 1990657, 786433, 5308417, 120932353, 14155777, 28311553, 0, 113246209, 29386561537, 3439853569, 6879707137, 1811939329, 18345885697, 3221225473, 1253826625537, 0, 85691213438977

Offset: 2

Hugo Pfoertner, Aug 28 2020

nonn

Robert Israel, Table of n, a(n) for n = 2..2193

Cf. A033845, A058383, A336772 (positions of 0).

f:= proc(n) local k,p;
   for k from 1 to n-1 do
     p:= 2^(n-k)*3^k+1;
     if isprime(p) then return p fi
   od;
   0
end proc:
map(f, [$2..40]); # Robert Israel, Aug 30 2020

for(n=2,34, my(pm=oo); for(j=1,n-1, my(k=n-j,p=2^j*3^k+1);if(isprime(p),pm=min(p,pm))); print1(if(pm==oo,0,pm),", "))

A051913 Numbers k such that phi(k)/phi(phi(k)) = 3.

Original entry on oeis.org

7, 9, 13, 14, 18, 19, 21, 26, 27, 28, 35, 36, 37, 38, 39, 42, 45, 52, 54, 56, 57, 63, 65, 70, 72, 73, 74, 76, 78, 81, 84, 90, 91, 95, 97, 104, 105, 108, 109, 111, 112, 114, 117, 119, 126, 130, 133, 135, 140, 144, 146, 148, 152, 153, 156, 162, 163, 168, 171, 180, 182

Offset: 1

J. H. Conway and Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), Dec 17 1999

Also numbers k such that phi(k) = 2^a*3^b with a, b > 0.
Also numbers k such that a regular k-gon can be constructed using conics but not with merely a compass and straightedge.
"Constructed using conics" means that one can draw any conic, once its focus, its vertex and a point on its directrix are constructed. Points at intersections are thereby constructed. (Videla's definition is slightly more complicated, but equivalent.) One can use parabolas to take cube roots; hyperbolas yield trisected angles. - Don Reble, Apr 23 2007

			Phi(999) = Phi(3*3*3*37) = 648 = 8*81.

George E. Martin, Geometric Constructions, Springer, 1997, p. 140.

T. D. Noe, Table of n, a(n) for n=1..1000
C. R. Videla, On points constructible from conics, Mathematical Intelligencer, 19, No. 2, pp. 53-57 (1997).

Cf. A000010, A003401, A003586, A058383.

[n: n in [1..200] | EulerPhi(n)/EulerPhi(EulerPhi(n)) eq 3]; // Vincenzo Librandi, Apr 17 2015

lf[x_] := Length[FactorInteger[x]] eu[x_] := EulerPhi[x] Do[s=lf[eu[n]]; If[Equal[s, 2]&&Equal[Mod[eu[n], 6], 0], Print[n]], {n, 1, 1000}] (* Labos Elemer, Dec 28 2001 *)
f[n_] := Block[{m = n}, If[m > 0, While[ IntegerQ[m/2], m /= 2]; While[ IntegerQ[m/3], m /= 3]]; m == 1]; fQ[n_] := Block[{pff = Select[ FactorInteger[n], #[[1]] > 3 &]}, pf = Flatten[{2, Table[ #[[1]], {1}] & /@ pff}]; pfe = Union[ Flatten[{1, Table[ #[[2]], {1}] & /@ pff}]]; If[ Union[f /@ (pf - 1)] == {True} && pfe == {1} && !IntegerQ[ Log[2, EulerPhi[ n]]], True, False]]; Select[ Range[184], fQ[ # ] &] (* Robert G. Wilson v, Apr 05 2005 *)
Select[Range[200],EulerPhi[#]/EulerPhi[EulerPhi[#]]==3&] (* Harvey P. Dale, Jul 11 2025 *)

from itertools import count, islice
from sympy import primefactors, totient
def A051913_gen(): # generator of terms
    yield from filter(lambda n: primefactors(totient(n)) == [2,3], count(1))
A051913_list = list(islice(A051913_gen(),30)) # Chai Wah Wu, Apr 02 2025

Numbers k of the form 2^a*3^b*p*q*r*..., where p, q, r, ... are distinct primes of the form 2^x*3^y + 1 (i.e., belong to A005109) and phi(k) is not a power of 2 [Videla]. - Robert G. Wilson v, Apr 05 2005

Additional comments from Labos Elemer, Dec 28 2001
Additional comments from Benoit Cloitre, Jan 26 2002
Edited by N. J. A. Sloane, Apr 21 2007
Entries checked by Don Reble, Apr 23 2007

cp's OEIS Frontend

A060211 Larger term of a pair of twin primes such that the prime factors of their average are only 2 and 3. Proper subset of A058383.

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A027856 Dan numbers: numbers m of the form 2^j * 3^k such that m +- 1 are twin primes.

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A035095 Smallest prime congruent to 1 (mod prime(n)).

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A059960 Smaller term of a pair of twin primes such that prime factors of their average are only 2 and 3.

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A060212 Primes q such that 6*q-1 and 6*q+1 are twin primes. Proper subset of A002822.

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A125866 Odd numbers k such that cos(2*Pi/k) is an algebraic number of a 3-smooth degree, but not 2-smooth.

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A216148 Primes of the form 2*k^k + 1 = A216147(k).

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A060212 Primes q such that 6q-1 and 6q+1 are twin primes. Proper subset of A002822.