A077498 -id:A077498 - cp's OEIS frontend

A005109 Class 1- (or Pierpont) primes: primes of the form 2^t*3^u + 1.

2, 3, 5, 7, 13, 17, 19, 37, 73, 97, 109, 163, 193, 257, 433, 487, 577, 769, 1153, 1297, 1459, 2593, 2917, 3457, 3889, 10369, 12289, 17497, 18433, 39367, 52489, 65537, 139969, 147457, 209953, 331777, 472393, 629857, 746497, 786433, 839809, 995329, 1179649, 1492993, 1769473, 1990657

Offset: 1

N. J. A. Sloane, Simon Plouffe

The definition is given by Guy: a prime p is in class 1- if the only prime divisors of p - 1 are 2 or 3; and p is in class r- if every prime factor of p - 1 is in some class <= r- - 1, with equality for at least one prime factor. - N. J. A. Sloane, Sep 22 2012
See A005105 for the definition of class r+ primes.
Gleason, p. 191: a regular polygon of n sides can be constructed by ruler, compass and angle-trisector iff n = 2^r * 3^s * p_1 * p_2 * ... * p_k, where p_1, p_2, ..., p_k are distinct elements of this sequence and > 3.
Sequence gives primes solutions to p == +1 (mod phi(p-1)). - Benoit Cloitre, Feb 22 2002
These are the primes p for which p-1 is 3-smooth.  Primes for which either p+1 or p-1 have many small factors are more easily proved prime, so most of the largest primes found have this property. - Michael B. Porter, Feb 19 2013
For terms p > 3, omega(p-1) = 3 - p mod 3. Consider terms > 3. Clearly, t > 0. If p == 1 mod 3, u > 0: hence omega(p-1) = 2 because p-1 has two prime factors. If p == 2 mod 3, u = 0: hence omega(p-1) = 1 because p-1 is a power of 2. The latter case corresponds to terms that are Fermat primes > 3. Similar arguments demonstrate the converse, that for p > 3, if omega(p-1) = 3 - p mod 3, p is a term. - Chris Boyd, Mar 22 2014
The subset of A055600 which are prime. - Robert G. Wilson v, Jul 19 2014
Named after the American mathematician James Pierpont (1866-1938). - Amiram Eldar, Jun 09 2021

			97 = 2^5*3 + 1 is a term.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, section A18, p. 66.
George E. Martin, Geometric Constructions, Springer, 1998. ISBN 0-387-98276-0.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robert G. Wilson v, Table of n, a(n) for n = 1..8396 (terms 1..795 from T. D. Noe, terms 796..1602 from Joerg Arndt)
Claudi Alsina and Roger B. Nelson, A Panoply of Polygons, Dolciani Math. Expeditions Vol. 58, AMS/MAA (2023), see page 112.
Chris K. Caldwell, The Prime Pages.
David A. Cox and Jerry Shurman, Geometry and number theory on clovers, Amer. Math. Monthly, Vol. 112, No. 8 (2005), pp. 682-704.
Andrew M. Gleason, Angle Trisection, the Heptagon and the Triskaidecagon, American Mathematical Monthly, Vol. 95, No. 3 (1988), pp. 185-194.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Joel C. Langer and David A. Singer, Subdividing the trefoil by origami, Geometry, Vol. 2013 (Hindawi Publishing Company, 2013), Article ID 897320. - From _N. J. A. Sloane_, Feb 08 2013
James Pierpont, On an Undemonstrated Theorem of the Disquisitiones Arithmeticae, American Mathematical Society Bulletin, Vol. 2, No. 3 (1895-1896), pp. 77-83.
Eric Weisstein's World of Mathematics, Pierpont Prime.
Index entries for sequences related to the Erdos-Selfridge classification

Cf. A048135, A048136, A056637, A077497, A077498, A077500, A081426, A122259, A019434, A000668, A000040, A003586, A374577, A374578.

K:=10^7;; # to get all terms <= K.
A:=Filtered([1..K],IsPrime);;
B:=List(A,i->Factors(i-1));;
C:=[];;  for i in B do if Elements(i)=[2] or Elements(i)=[2,3]  then Add(C,Position(B,i)); fi; od;
A005109:=Concatenation([2],List(C,i->A[i])); # Muniru A Asiru, Sep 10 2017

[p: p in PrimesUpTo(10^8) | forall{d: d in PrimeDivisors(p-1) | d le 3}]; // Bruno Berselli, Sep 24 2012

PrimeFactors[n_Integer] := Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[n]]; f[n_Integer] := Block[{m = n}, If[m == 0, m = 1, While[ IntegerQ[m/2], m /= 2]; While[ IntegerQ[m/3], m /= 3]]; Apply[Times, PrimeFactors[m] - 1]]; ClassMinusNbr[n_] := Length[NestWhileList[f, n, UnsameQ, All]] - 3; Prime[ Select[ Range[3, 6300], ClassMinusNbr[ Prime[ # ]] == 1 &]]
Select[Prime /@ Range[10^5], Max @@ First /@ FactorInteger[ # - 1] < 5 &] (* Ray Chandler, Nov 01 2005 *)
mx = 2*10^6; Select[Sort@ Flatten@ Table[2^i*3^j + 1, {i, 0, Log[2, mx]}, {j, 0, Log[3, mx/2^i]}], PrimeQ] (* Robert G. Wilson v, Jul 16 2014, edited by Michael De Vlieger, Aug 23 2017 *)

N=10^8; default(primelimit,N);
pq(p)={p-=1; (p/(2^valuation(p,2)*3^valuation(p,3)))==1;}
forprime(p=2,N,if(pq(p),print1(p,", ")));
/* Joerg Arndt, Sep 22 2012 */

/* much more efficient: */
A005109_upto(lim=1e10)={my(L=List(), k2=1);
until ( lim <= k2 *= 2, my(k23 = k2);
    until ( lim <= k23 *= 3, isprime(k23+1) && listput(L, k23+1));
); Set(L) } /* Joerg Arndt, Sep 22 2012, edited by M. F. Hasler, Mar 17 2024 */

N=10^8; default(primelimit, N);
print1("2, 3, ");forprime(p=5,N,if(omega(p-1)==3-p%3,print1(p", "))) \\ Chris Boyd, Mar 22 2014

from itertools import islice
from sympy import nextprime
def A005109_gen(): # generator of terms
    p = 2
    while True:
        q = p-1
        q >>= (~q&q-1).bit_length()
        a, b = divmod(q,3)
        while not b:
            a, b = divmod(q:=a,3)
        if q==1:
            yield p
        p = nextprime(p)
A005109_list = list(islice(A005109_gen(),30)) # Chai Wah Wu, Mar 17 2023

A122257(a(n)) = 1; A122258(n) = number of Pierpont primes <= n; A122260 gives numbers having only Pierpont primes as factors. - Reinhard Zumkeller, Aug 29 2006
{primes p: A126805(PrimePi(p)) = 1}. - R. J. Mathar, Sep 24 2012
a(n) = 2^A374577(n) * 3^A374578(n) + 1. - Amiram Eldar, Sep 02 2024

Comments and additional references from Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr)
More terms from David W. Wilson
More terms from Benoit Cloitre, Feb 22 2002
More terms from Robert G. Wilson v, Mar 20 2003

A077497 Primes of the form 2^r*5^s + 1.

Original entry on oeis.org

2, 3, 5, 11, 17, 41, 101, 251, 257, 401, 641, 1601, 4001, 16001, 25601, 40961, 62501, 65537, 160001, 163841, 16384001, 26214401, 40960001, 62500001, 104857601, 167772161, 256000001, 409600001, 655360001, 2441406251, 2500000001, 4194304001, 10485760001

Offset: 1

Amarnath Murthy, Nov 07 2002

nonn

These are also the prime numbers p for which there is an integer solution x to the equation p*x = p*10^p + x, or equivalently, the prime numbers p for which (p*10^p)/(p-1) is an integer. - Vicente Izquierdo Gomez, Feb 20 2013

For n > 2, all terms are congruent to 5 (mod 6). - Muniru A Asiru, Sep 03 2017

			101 is in the sequence, since 101 = 2^2*5^2 + 1 and 101 is prime.

Ray Chandler, Table of n, a(n) for n = 1..3150

Cf. A005109, A077497, A077498, A077500, A003592, A077313, A019434.

K:=10^7;; # to get all terms <= K.
A:=Filtered(Filtered([1..K],i-> i mod 6=5),IsPrime);;
B:=List(A,i->Factors(i-1));;
C:=[];;  for i in B do if Elements(i)=[2] or Elements(i)=[2,5]  then Add(C,Position(B,i)); fi; od;
A077497:=Concatenation([2,3],List(C,i->A[i])); # Muniru A Asiru, Sep 03 2017

Do[p=Prime[k];s=FindInstance[p x == p 10^p+x,x,Integers];If[s!={},Print[p]],{k,10000}] (* Vicente Izquierdo Gomez, Feb 20 2013 *)

list(lim)=my(v=List(),t);for(r=0,log(lim)\log(5),t=5^r;while(t<=lim,if(isprime(t+1),listput(v,t+1)); t<<=1)); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jan 29 2013

Corrected and extended by Reinhard Zumkeller, Nov 19 2002

More terms from Ray Chandler, Aug 02 2003

A077500 Primes of the form 2^r*p^s + 1, where p is an odd prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 41, 47, 53, 59, 73, 83, 89, 97, 101, 107, 109, 113, 137, 149, 163, 167, 173, 179, 193, 197, 227, 233, 251, 257, 263, 269, 293, 317, 347, 353, 359, 383, 389, 401, 433, 449, 467, 479, 487, 503, 509, 557, 563, 569, 577, 587

Offset: 1

Amarnath Murthy, Nov 07 2002

nonn

Primes p such that p-1 has at most one odd prime divisor.

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

Cf. A005109, A077497, A077498, A077499.

Select[Prime[Range[110]],Length[Select[FactorInteger[#-1] [[All, 1]], OddQ]]<2&] (* Harvey P. Dale, Oct 09 2017 *)

Corrected and extended by Sascha Kurz, Jan 04 2003

A077499 Primes of the form 2^r*11^s + 1.

Original entry on oeis.org

2, 3, 5, 17, 23, 89, 257, 353, 1409, 2663, 30977, 65537, 170369, 495617, 5767169, 23068673, 59969537, 82458113, 453519617, 3429742097, 4715895383, 15352201217, 39909726209, 1857616347137, 45732811767809, 96757023244289

Offset: 1

Amarnath Murthy, Nov 07 2002

nonn

Primes p such that p-1 has at most one odd prime divisor 11.

Cf. A005109, A077497, A077498, A077500.

More terms from Ray Chandler, Aug 02 2003

A291049 Primes of the form 2^r * 17^s + 1.

Original entry on oeis.org

2, 3, 5, 17, 137, 257, 65537, 157217, 295937, 557057, 1336337, 96550277, 1212153857, 2281701377, 5473632257, 395469930497, 1401249857537, 2637646790657, 4964982194177, 28572702478337, 1271035441709057, 38280596832649217, 1872540629620228097, 6634884445436379137

Offset: 1

Muniru A Asiru, Sep 15 2017

nonn

Primes of the forms a^r * b^s + 1 where (a, b) = (2,1), (2,3), (2,5), (2,7), (2,11) and (2,13) are A092506, A005109, A077497, A077498, A077499 and A173236.
Fermat prime exponents r are 0, 1, 2, 4, 8, 16.
For n > 2, all terms are congruent to 5 (mod 6).
Also, these are prime numbers p for which (p*34^p)/(p-1) is an integer.

			With n = 1, a(1) = 2^0 * 17^0 + 1 = 2.
With n = 5, a(5) = 2^3 * 17^1 + 1 = 137.
list of (r,s): (0,0), (1,0), (2,0), (4,0), (3,1), (8,0), (16,0), (5,3), (10,2), (15,1), (4,4), (2,6).

Robert Israel, Table of n, a(n) for n = 1..625

Cf. Sequences of primes of form 2^n * q^u + 1: A092506 (q=1), A005109 (q=3), A077497 (q=5), A077498 (q=7), A077499 (q=11), A173236 (q=13).

K:=26*10^7+1;; # to get all terms <= K.
A:=Filtered(Filtered([1,3..K],i-> i mod 6=5),IsPrime);;    I:=[17];;
B:=List(A,i->Elements(Factors(i-1)));;
C:=List([0..Length(I)],j->List(Combinations(I,j),i->Concatenation([2],i)));;
A291049:=Concatenation([2,3],List(Set(Flat(List([1..Length(C)],i->List([1..Length(C[i])],j->Positions(B,C[i][j]))))),i->A[i]));

N:= 10^20: # to get all terms <= N+1
S:= NULL:
for r from 0 to ilog2(N) do
  for s from 0 to floor(log[17](N/2^r)) do
    p:= 2^r*17^s +1;
    if isprime(p) then
     S:= S, p
    fi
od od:
sort([S]); # Robert Israel, Sep 26 2017

With[{nn = 10^19, q = 17}, Select[Sort@ Flatten@ Table[2^i*q^j + 1, {i, 0, Log[2, nn]}, {j, 0, Log[q, nn/2^i]}], PrimeQ]] (* Michael De Vlieger, Sep 18 2017, after Robert G. Wilson v at A005109 *)

lista(nn) = my(t, v=List([])); for(r=0, logint(nn, 2), t=2^r; for(s=0, logint(nn\t, 17), if(isprime(t+1), listput(v, t+1)); t*=17)); Vec(vecsort(v)) \\ Jinyuan Wang, Jun 26 2022

cp's OEIS Frontend

A005109 Class 1- (or Pierpont) primes: primes of the form 2^t*3^u + 1.

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A077497 Primes of the form 2^r*5^s + 1.

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A077500 Primes of the form 2^r*p^s + 1, where p is an odd prime.

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A077499 Primes of the form 2^r*11^s + 1.

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A291049 Primes of the form 2^r * 17^s + 1.

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A086983 Primes of the form 2^r*p^s - 1, where p is an odd prime.

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