A005109 -id:A005109 - cp's OEIS frontend

A175257 a(n) is the smallest prime p such that 2^(p-1) == 1 (mod a(1)...a(n-1)*p).

3, 5, 13, 37, 73, 109, 181, 541, 1621, 4861, 9721, 19441, 58321, 87481, 379081, 408241, 2041201, 2449441, 7348321, 14696641, 22044961, 95528161, 382112641, 2292675841, 8024365441, 40121827201, 481461926401, 722192889601, 2888771558401, 7944121785601, 55608852499201, 111217704998401, 889741639987201, 1779483279974401

Offset: 1

Manuel Valdivia, Mar 15 2010

nonn

Conjecture: a(n) is the smallest integer k > 1 such that 2^(k-1) == 1 (mod a(0)*...*a(n-1)*k), with a(0) = 1. - Thomas Ordowski, Mar 13 2019

Either a(n) > a(n-1), or a(n) = a(n-1) is a Wieferich prime (A001220). - Max Alekseyev, Sep 29 2024

Max Alekseyev, Table of n, a(n) for n = 1..1000

Cf. A007663, A096060, A002200, A005109, A306826.

i=1;Do[p=Prime[n];If[Mod[2^(p-1)-1,p*i]==0,Print[p];i=p*i],{n,2,78498}]

findprime(prd) = {forprime(p=2, , if (Mod(2, p*prd)^(p-1) == 1, return (p)););}
lista(nn) = {my(prd = 1, na); for (n=1, nn, na = findprime(prd); print1(na, ", "); prd *= na;);} \\ Michel Marcus, Mar 14 2019

{ a175257_first_terms(N=1000) = my(P,L,t); P=[3]; L=2; for(n=#P,N, print(n," ",P[n]); forstep(p=P[n],oo,Mod(1,L), if(p==P[n], if(Mod(2,p^2)^(p-1)==1, error("Wieferich prime!"), next)); if(ispseudoprime(p), P=concat(P,[p]); t=Mod(2,p)^L; fordiv((p-1)\L,d, if(t^d==1, L*=d; break)); break))); P; } \\ Max Alekseyev, Sep 29 2024

a(17)-a(26) from Amiram Eldar, Feb 03 2019
Name corrected by Thomas Ordowski, Mar 13 2019
a(27) from Hans Havermann, Mar 29 2019
Eliminated a(0)=1 in the definition (empty products equal 1). - R. J. Mathar, Jun 19 2021
Terms a(28) onward from Max Alekseyev, Sep 29 2024

A284037 Primes p such that p-1 and p+1 have two distinct prime factors.

Original entry on oeis.org

11, 13, 19, 23, 37, 47, 53, 73, 97, 107, 163, 193, 383, 487, 577, 863, 1153, 2593, 2917, 4373, 8747, 995327, 1492993, 1990657, 5308417, 28311553, 86093443, 6879707137, 1761205026817, 2348273369087, 5566277615617, 7421703487487, 21422803359743, 79164837199873

Offset: 1

Giuseppe Coppoletta, Mar 28 2017

nonn

Either p-1 or p+1 must be of the form 2^i * 3^j, since among three consecutive numbers exactly one is a multiple of 3. - Giovanni Resta, Mar 29 2017

Subsequence of A219528. See the previous comment. - Jason Yuen, Mar 08 2025

			7 is not a term because n + 1 = 8 has only one prime factor.
23 is a term because it is prime and n - 1 = 22 has two distinct prime factors (2, 11) and n + 1 = 24 has two distinct prime factors (2, 3).
43 is not a term because n - 1 = 42 has three distinct prime factors (2, 3, 7).

Robert Israel, Table of n, a(n) for n = 1..51 (all terms < 10^75)

Cf. A000668, A001221, A005105, A005109, A033845, A058383, A067386, A092506, A215504, A219528, A275598.

N:= 10^20: # To get all terms <= N
Res:= {}:
for i from 1 to ilog2(N) do
  for j from 1 to floor(log[3](N/2^i)) do
    q:= 2^i*3^j;
    if isprime(q-1) and nops(numtheory:-factorset((q-2)/2^padic:-ordp(q-2,2)))=1 then Res:= Res union {q-1} fi;
    if isprime(q+1) and nops(numtheory:-factorset((q+2)/2^padic:-ordp(q+2,2)))=1 then Res:= Res union {q+1} fi
od od:
sort(convert(Res,list)); # Robert Israel, Apr 16 2017

mx = 10^30; ok[t_] := PrimeQ[t] && PrimeNu[t-1]==2==PrimeNu[t+1]; Sort@ Reap[Do[ w = 2^i 3^j; Sow /@ Select[ w+ {1,-1}, ok], {i, Log2@ mx}, {j, 1, Log[3, mx/2^i]}]][[2, 1]] (* terms up to mx, Giovanni Resta, Mar 29 2017 *)

isok(n) = isprime(n) && (omega(n-1)==2) && (omega(n+1)==2); \\ Michel Marcus, Apr 17 2017

omega=sloane.A001221; [n for n in prime_range(10^6) if 2==omega(n-1)==omega(n+1)]

sorted([2^i*3^j+k for i in (1..40) for j in (1..20) for k in (-1,1) if is_prime(2^i*3^j+k) and sloane.A001221(2^i*3^j+2*k)==2])

A001221(a(n)) = 1 and A001221(a(n) - 1) = A001221(a(n) + 1) = 2.

a(33)-a(34) from Giovanni Resta, Mar 29 2017

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

A375906 Positions of primes in the sequence of numbers of the form 2^t * 3^u + 1 (A055600).

Original entry on oeis.org

1, 2, 4, 5, 8, 9, 10, 14, 18, 20, 21, 24, 25, 28, 32, 33, 35, 38, 42, 43, 44, 51, 52, 54, 55, 68, 70, 75, 76, 87, 91, 95, 107, 108, 114, 122, 128, 134, 137, 138, 139, 142, 146, 150, 154, 156, 162, 176, 177, 187, 193, 198, 206, 214, 232, 234, 237, 242, 246, 248, 250

Offset: 1

Amiram Eldar, Sep 02 2024

nonn

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

Cf. A003586, A005109, A055600, A174099.

With[{lim = 10^10}, Position[Sort@ Flatten@ Table[2^i*3^j + 1, {i, 0, Log2[lim]}, {j, 0, Log[3, lim/2^i]}], _?PrimeQ] // Flatten]

lista(lim) = {my(s = List()); for(i = 0, logint(lim, 2), for(j = 0, logint(lim >> i, 3), listput(s, 2^i * 3^j + 1))); s = Set(s); for(i = 1, #s, if(isprime(s[i]), print1(i, ", ")));}

A055600(a(n)) = A003586(a(n)) + 1 = A005109(n).

A086983 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, 31, 37, 43, 47, 53, 61, 67, 71, 73, 79, 97, 103, 107, 127, 151, 157, 163, 191, 193, 199, 211, 223, 241, 271, 277, 283, 313, 331, 337, 367, 383, 397, 421, 431, 457, 463, 487, 499, 523, 541, 547, 577, 607, 613, 631, 647, 661, 673

Offset: 1

Ray Chandler, Aug 02 2003

nonn

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

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

Cf. A005105, A077497, A077498, A077499, A077500.

Cf. also A005109, A077313, A077314, A077315.

N:= 1000: # to get all terms <= N
Primes:= select(isprime, [$3..(N+1)/2]):
sort(convert(select(isprime, {2,seq(seq(seq(2^r*p^s-1, r = 1 .. ilog2((N+1)/p^s)),s=0..floor(log[p]((N+1)/2))),p=Primes)}),list)); # Robert Israel, Jun 13 2018

A113412 Number of Pierpont primes less than 10^(2^n).

Original entry on oeis.org

4, 10, 25, 58, 125, 250, 505, 1020, 2075, 4227, 8597, 17213

Offset: 0

Eric W. Weisstein, Oct 29 2005

nonn

Computed by Chris Caldwell.

Eric Weisstein's World of Mathematics, Pierpont Prime

Cf. A005109, A113420.

a(10) from Eric W. Weisstein, Nov 01 2005

a(11) from Donovan Johnson, Feb 13 2010

A114991 Primes of the form 2^a * 3^b * 5^c + 1 for positive a, b, c.

Original entry on oeis.org

31, 61, 151, 181, 241, 271, 541, 601, 751, 811, 1201, 1621, 1801, 2161, 2251, 3001, 4051, 4801, 4861, 6481, 7681, 8101, 8641, 9001, 9601, 9721, 11251, 14401, 15361, 19441, 21601, 21871, 22501, 23041, 24001, 32401, 33751, 36451, 37501, 43201, 54001, 57601, 58321

Offset: 1

Jonathan Vos Post, Feb 22 2006

			a(1) = 31 = 2^1 * 3^1 * 5^1 + 1.
a(2) = 61 = 2^2 * 3^1 * 5^1 + 1.
a(3) = 151 = 2^1 * 3^1 * 5^2 + 1.
a(4) = 181 = 2^2 * 3^2 * 5^1 + 1.
Values include 30000001 = 2^7 * 3^1 * 5^7 + 1, 90000000001 = 2^9 * 3^2 * 5^9 + 1.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A000040, A005109, A030430.

Cf. A143207.

A114992 Primes of the form 2^a * 5^b * 7^c + 1 for positive a, b, c.

Original entry on oeis.org

71, 281, 491, 701, 2801, 4481, 7001, 7841, 12251, 13721, 17921, 28001, 34301, 54881, 70001, 78401, 85751, 122501, 125441, 137201, 168071, 240101, 280001, 286721, 437501, 490001

Offset: 1

Jonathan Vos Post, Feb 22 2006

nonn

Since the factors of 2 and 5 are the same as a factor of 10, a subset of A030430 "primes of form 10n+1." There are subsequences such as 71, 701, 7001, 70001, 700001, 700000001, 7000000001; 281, 2801, 280001, 2800001; 491, 490001, 4900001, 490000001, 49000000001, 490000000001.

			a(1) = 71 = 2^1 * 5^1 * 7^1 + 1.
a(2) = 281 = 2^3 * 5^1 * 7^1 + 1.
a(3) = 491 = 2^1 * 5^1 * 7^2 + 1.
a(4) = 701 = 2^2 * 5^2 * 7^1 + 1.
a(5) = 2801 = 2^4 * 5^2 * 7^1 + 1.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000040, A005109, A030430.

With[{nn=30},Take[Select[Union[2^#[[1]]*5^#[[2]]*7^#[[3]]+1&/@Tuples[ Range[nn],3]],PrimeQ],nn]] (* Harvey P. Dale, Aug 24 2012 *)

find(lim)=my(v=List(), t); for(b=1,log(lim\14)\log(5), for(c=1,log(lim\2\5^b)/log(7), t=2*5^b*7^c; while(tCharles R Greathouse IV, Feb 17 2011

Corrected by T. D. Noe, Nov 15 2006

A122262 Number of numbers having only factors that are Pierpont primes.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 20, 20, 21, 22, 23, 24, 25, 25, 26, 26, 27, 27, 28, 29, 30, 31, 32, 33, 34, 34, 35, 35, 35, 36, 36, 36, 37, 38, 39, 40, 41, 41, 42, 42, 43, 44, 44, 44, 45, 45, 45, 46, 47, 48, 48, 48, 49, 49, 50, 50, 51

Offset: 1

Reinhard Zumkeller, Aug 29 2006

nonn

Eric Weisstein's World of Mathematics, Pierpont Prime.

Partial sums of A122261.

Cf. A005109.

smooth3Q[n_] := Times @@ ({2, 3}^IntegerExponent[n, {2, 3}]) == n; s[n_] := Boole@ AllTrue[FactorInteger[n][[;;, 1]]-1, smooth3Q]; s[1] = 1; Accumulate[Table[s[n], {n, 1, 100}]] (* Amiram Eldar, May 14 2025 *)

A137990 Least prime p of the form c*3^n+1 with c not divisible by 3.

Original entry on oeis.org

2, 7, 19, 109, 163, 487, 1459, 17497, 52489, 39367, 472393, 4960117, 5314411, 102036673, 19131877, 57395629, 86093443, 258280327, 3874204891, 23245229341, 90656394427, 585779779369, 251048476873, 9790890598009, 4518872583697

Offset: 0

Andrew V. Sutherland, May 01 2008

a(n) is also the least prime such that 3^(n+1), but not 3^(n+2), divides 2^(a(n)-1)-1.

			a(8)=52489 because 52489=8*3^8+1 is prime and no smaller prime p has p-1 divisible by 3^8 but not 3^9.

Carlos Rivera, Puzzle 1129 OEIS A137990, The Prime Puzzles & Problems Connection.

Cf. A005109, A057775.

cp's OEIS Frontend

A175257 a(n) is the smallest prime p such that 2^(p-1) == 1 (mod a(1)*...*a(n-1)*p).

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A284037 Primes p such that p-1 and p+1 have two distinct prime factors.

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

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A375906 Positions of primes in the sequence of numbers of the form 2^t * 3^u + 1 (A055600).

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Mathematica

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

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Maple

A113412 Number of Pierpont primes less than 10^(2^n).

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A114991 Primes of the form 2^a * 3^b * 5^c + 1 for positive a, b, c.

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A114992 Primes of the form 2^a * 5^b * 7^c + 1 for positive a, b, c.

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A175257 a(n) is the smallest prime p such that 2^(p-1) == 1 (mod a(1)...a(n-1)*p).