A077497 -id:A077497 - 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

A077313 Primes of the form 2^r*5^s - 1.

Original entry on oeis.org

3, 7, 19, 31, 79, 127, 199, 499, 1249, 1279, 1999, 4999, 5119, 8191, 12799, 20479, 31249, 49999, 51199, 79999, 81919, 131071, 199999, 524287, 799999, 1249999, 1310719, 3124999, 3276799, 4999999, 7812499, 12499999, 19999999, 20479999

Offset: 1

Amarnath Murthy, Nov 04 2002

nonn

Primes p such that 10^p is divisible by p+1. Primes p whose fractions p/(p+1) are terminating decimals, i.e., primes p such that A158911(p)=0. Primes p such that the prime divisors of p+1 are also prime divisors of the numbers m obtained by the concatenation of p and p+1. For example, for p=19, m = 1920, the prime divisors of 20 are {2, 5} and the prime divisors of 1920 are {2, 3, 5}. - Jaroslav Krizek, Feb 25 2013

For n > 1, all terms are congruent to 1 (mod 6). - Muniru A Asiru, Sep 29 2017

			1250000 = 2*2*2*2*5*5*5*5*5*5*5 and 1250000 - 1 = A000040(96469), therefore 1249999 is a term.
List of (r, s): (2, 0), (3, 0), (2, 1), (5, 0), (4, 1), (7, 0), (3, 2), (2, 3), (1, 4), (8, 1), (4, 3), (3, 4), (10, 1), ...  - _Muniru A Asiru_, Sep 29 2017

Ray Chandler, Table of n, a(n) for n = 1..4222

Cf. A003592, A023509, A077497.

A:=Filtered([1..10^7],IsPrime);;    I:=[5];;
B:=List(A,i->Elements(Factors(i+1)));;
C:=List([0..Length(I)],j->List(Combinations(I,j),i->Concatenation([2],i)));;
A077313:=List(Set(Flat(List([1..Length(C)],i->List([1..Length(C[i])],j->Positions(B,C[i][j]))))),i->A[i]); # Muniru A Asiru, Sep 29 2017

With[{n = 10^8}, Union@ Select[Flatten@ Table[2^p*5^q - 1, {p, 0, Log[2, n/(1)]}, {q, 0, Log[5, n/(2^p)]}], PrimeQ]] (* Michael De Vlieger, Sep 30 2017 *)

More terms from Reinhard Zumkeller, Nov 15 2002

More terms from Vladeta Jovovic, May 08 2003

A077498 Primes of the form 2^r*7^s + 1.

Original entry on oeis.org

2, 3, 5, 17, 29, 113, 197, 257, 449, 1373, 3137, 50177, 65537, 114689, 268913, 470597, 614657, 1075649, 3294173, 7340033, 9834497, 210827009, 275365889, 359661569, 469762049, 1129900997, 1438646273, 1927561217, 7909306973, 52613349377

Offset: 1

Amarnath Murthy, Nov 07 2002

nonn

			197 = 2^2*7^2 + 1 is a member.

Cf. A005109, A077497, A077499, A077500.

Corrected and extended by 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

A002200 Primes of the form 2^q3^r5^s + 1.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 31, 37, 41, 61, 73, 97, 101, 109, 151, 163, 181, 193, 241, 251, 257, 271, 401, 433, 487, 541, 577, 601, 641, 751, 769, 811, 1153, 1201, 1297, 1459, 1601, 1621, 1801, 2161, 2251, 2593, 2917, 3001, 3457, 3889, 4001, 4051, 4801, 4861

Offset: 1

N. J. A. Sloane

nonn

M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 53.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 200 terms from Harry J. Smith)

Cf. A174144, A005109, A077497.

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]  or Elements(i)=[2,5] or Elements(i)=[2,3,5]  then Add(C,Position(B,i)); fi; od;
A002200:=Concatenation([2],List(C,i->A[i])); # Muniru A Asiru, Sep 10 2017

[p: p in PrimesUpTo(5000) | forall{d: d in PrimeDivisors(p-1) | d le 5}]; // Bruno Berselli, Sep 24 2012

up=10^6; a=1; Sort[Reap[While[ aGiovanni Resta, Jul 18 2017 *)

{ default(primelimit, 16600000); n=0; forprime (p=2, 16600000, m=p-1; p2=p3=p5=0; s=m; r=0; while(r==0, q=s\2; r=s-2*q; s=q; if(r==0, p2++)); s=m; r=0; while(r==0, q=s\3; r=s-3*q; s=q; if(r==0, p3++)); s=m; r=0; while(r==0, q=s\5; r=s-5*q; s=q; if(r==0, p5++)); if (m == 2^p2*3^p3*5^p5, n++; write("b002200.txt", n, " ", p)); if (n >= 200, break); ); } \\ Harry J. Smith, May 25 2009

{ n=5000; cache=10^5; v=vector(cache); x2=2; x3=3; x5=5; i=j=k=1; v[1]=1; for(m=2,cache,v[m]=t=min(x2,min(x3,x5)); if(x2==t,x2=2*v[i++]); if(x3==t,x3=3*v[j++]); if(x5==t,x5=5*v[k++]);); i=0; c=0; while(cJean-Marie Madiot, Jul 17 2017

Better description and more terms from Vladeta Jovovic, May 08 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

A178070 Primes dividing repunits R(10^n) for some n.

Original entry on oeis.org

11, 17, 41, 73, 101, 137, 251, 257, 271, 353, 401, 449, 641, 751, 1201, 1409, 1601, 3541, 4001, 4801, 5051, 9091, 10753, 15361, 16001, 19841, 21001, 21401, 24001, 25601, 27961, 37501, 40961, 43201, 60101, 62501, 65537, 69857, 76001, 76801, 160001, 162251, 163841, 307201, 453377, 524801, 544001, 670001, 952001, 976193, 980801

Offset: 1

Shashank Sharma, May 19 2010, Aug 04 2010

nonn

Repunits are the numbers consisting entirely of 1's. The number represented by R(10^n) contains 10^n digits with all 1's. E.g., R(10^1) = 1111111111.
A prime p > 5 is here if the multiplicative order of 10 (mod p) is of the form 2^i*5^j, with i and j nonnegative.
Includes all terms > 5 of A077497. - Robert Israel, Nov 05 2024

			17 divides R(10^4), so is in the sequence. - _Phil Carmody_, May 26 2011
Note that R(10^n) == 1 mod 3 for all n, so 3 is not a member. - _N. J. A. Sloane_, Jun 18 2014

Robert Israel, Table of n, a(n) for n = 1..110
Dario Alejandro Alpern, Known prime factors of Googolplexplex - 1
Project Euler, Problem 133: Repunit nonfactors
Robert P. Munafo, Notable Properties of Specific Numbers

Cf. A077497, A227246.

filter:= proc(p) local v;
  if not isprime(p) then return false fi;
  v:= numtheory:-order(10,p);
  v = 2^padic:-ordp(v,2) * 5^padic:-ordp(v,5)
end proc:
select(filter, [seq(i, i=7 .. 10^6, 2)]); # Robert Israel, Nov 05 2024

Select[Prime[Range[4, 100000]], Complement[First /@ FactorInteger[MultiplicativeOrder[10, #]], {2, 5}] == {} &] (* T. D. Noe, May 26 2011 *)

g=10^30;forprime(p=7,1000000,z=znorder(Mod(10,p));if(gcd(z,g)==z,print1(p", "))) \\ Phil Carmody, May 26 2011

upTo(lim)=my(v=List(),g=10^(log(lim)\log(2))); forprime(p=7,lim,if(g%znorder(Mod(10,p))==0, listput(v,p))); Vec(v) \\ Charles R Greathouse IV, May 26 2011

Arbitrary limit removed and sequence extended by Phil Carmody, May 26 2011

A072074 Number of integers k such that phi(k) = 10^n.

Original entry on oeis.org

2, 2, 4, 11, 16, 24, 43, 63, 94, 152, 224, 324, 464, 644, 897, 1271, 1790, 2521, 3501, 4814, 6535, 8779, 11739, 15585, 20625, 27166, 35588, 46363, 60065, 77424, 99337, 127020, 161930, 205847, 260929, 329782, 415533, 522173, 654548, 818278, 1020391

Offset: 0

Labos Elemer, Jun 13 2002

nonn

a(n) is the coefficient of x^n*y^n in Product_p Sum_{u, v} x^u*y^v, where the product is taken over all primes p and the sum is taken over such u, v that 2^u*5^v = phi(p^k) for some nonnegative integer k. - Max Alekseyev, Apr 26 2010

Elaborating on above comment, primes p must be in A077497 and k must be 1 for primes other than 2 and 5. - Ray Chandler, Feb 12 2012

			n=3: a(3)=11 because InvPhi(1000) = {1111, 1255, 1375, 1875, 2008, 2222, 2500, 2510, 2750, 3012, 3750}.

Ray Chandler, Table of n, a(n) for n = 0..1000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
Max A. Alekseyev, Computing the Inverses, their Power Sums, and Extrema for Euler's Totient and Other Multiplicative Functions. Journal of Integer Sequences, Vol. 19 (2016), Article 16.5.2.

Cf. A000010, A014197, A014573, A110078, A072075, A072076.

Maple
```
[seq(nops(invphi(10^i)),i=1..8)];
```

a(n) = #invphi(10^n); \\ for invphi see Alekseyev link \\ Michel Marcus, May 14 2020

a(n) = Card{x : A000010(x)=10^n}.

More terms from Max Alekseyev, Apr 26 2010

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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A077313 Primes of the form 2^r*5^s - 1.

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A077498 Primes of the form 2^r*7^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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A002200 Primes of the form 2^q*3^r*5^s + 1.

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

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A178070 Primes dividing repunits R(10^n) for some n.

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Maple

A002200 Primes of the form 2^q3^r5^s + 1.