A293194 -id:A293194 - cp's OEIS frontend

A293199 Primes of the form 2^q * 3^r * 7^s - 1.

2, 3, 5, 7, 11, 13, 17, 23, 31, 41, 47, 53, 71, 83, 97, 107, 127, 167, 191, 223, 251, 293, 383, 431, 503, 587, 647, 863, 881, 971, 1151, 1511, 1567, 2267, 2351, 2591, 2687, 3023, 3527, 3583, 4373, 4703, 4801, 6047, 6143

Offset: 1

Muniru A Asiru, Oct 02 2017

nonn

Mersenne primes A000668 occur when (q, r, s) = (q, 0 ,0) with q > 0.
a(2) = 3 is a Mersenne prime but a(3) = 5 is not.
For n > 2, all terms = {1, 5} mod 6.

			3 is a member because it is a prime number and 2^2 * 3^0 * 7^0 - 1 = 3.
503 is a member because it is a prime number and 2^3 * 3^2 * 7^1 - 1 = 503.
list of (q, r, s): (0, 1 ,0), (2, 0, 0), (1, 1, 0), (3, 0, 0), (2, 1, 0), (1, 0, 1), (1, 2, 0), (3, 1, 0),(5, 0, 0), (1, 1, 1), (4, 1, 0), (1, 3, 0), (3, 2, 0), (2, 1, 1), ...

Robert Israel, Table of n, a(n) for n = 1..4000

Cf. A000668, A005105, A293194.

K:=10^5+1;; # to get all terms <=K
A:=Filtered([1..K],IsPrime);; I:=[3,7];;
B:=List(A,i->Elements(Factors(i+1)));;
C:=List([0..Length(I)],j->List(Combinations(I,j),i->Concatenation([2],i)));
A293199:=Concatenation([2],List(Set(Flat(List([1..Length(C)],i->List([1..Length(C[i])],j->Positions(B,C[i][j]))))),i->A[i]));

N:= 10^4: # for terms <= N
S:= {1}:
for p in {2,3,7} do S:= map(proc(s) local i; seq(s*p^i,i=0..floor(log[p](N/s))) end proc, S) od:
sort(convert(select(isprime,map(`-`,S,1)),list)); # Robert Israel, Dec 17 2020

A293425 Primes of the form 2^a * 3^b * 5^c - 1 for positive a, b, c.

Original entry on oeis.org

29, 59, 89, 149, 179, 239, 269, 359, 449, 479, 599, 719, 809, 1439, 1499, 1619, 2399, 2699, 2879, 2999, 4049, 4799, 5399, 7499, 8999, 9719, 10799, 11519, 12149, 12959, 13499, 15359, 18749, 20249, 21599, 23039, 25919, 33749, 35999, 40499, 51839, 56249, 59999, 65609, 67499, 69119, 71999

Offset: 1

Muniru A Asiru, Oct 09 2017

nonn

a(n) is congruent to 29 (mod 30).

			a(1) = 29 = 2^1 * 3^1 * 5^1 - 1.
a(2) = 59 = 2^2 * 3^1 * 5^1 - 1.
a(3) = 89 = 2^1 * 3^2 * 5^1 - 1.
a(4) = 149 = 2^1 * 3^1 * 5^2 - 1.
a(5) = 179 = 2^2 * 3^2 * 5^1 - 1.
list of (a, b, c): (1, 1, 1), (2, 1, 1), (1, 2, 1), (1, 1, 2), (2, 2, 1), (4, 1, 1), (1, 3, 1), (3, 2, 1), (1, 2, 2), (5, 1, 1), (3, 1, 2), (4, 2, 1), (1, 4, 1), (5, 2, 1), (2, 1, 3), (2, 4, 1), ...

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

Cf. A000040, A005105, A030433, A132236, A293194.

K:=10^5+1;; # to get all terms <= K.
A:=Filtered([1..K],IsPrime);;
A293425:=List(Positions(List(A,i->Elements(Factors(i+1))),[2,3,5]),i->A[i]);

N:= 10^6: # to get all terms < N
R:= {}:
for c from 1 to floor(log[5]((N+1)/6)) do
for b from 1 to floor(log[3]((N+1)/2/5^c)) do
     R:= R union select(isprime, {seq(2^a*3^b*5^c-1,
             a=1..ilog2((N+1)/(3^b*5^c)))})
od od:
sort(convert(R,list)); # Robert Israel, Oct 15 2017

With[{n = 10^5}, Sort@ Select[Flatten@ Table[2^a*3^b*5^c - 1, {a, Log2@ n}, {b, Log[3, n/(2^a)]}, {c, Log[5, n/(2^a*3^b)]}], PrimeQ]] (* Michael De Vlieger, Oct 11 2017 *)

lista(nn) = {forprime(p=2,nn, f = factor(p+1); if ((vecmax(f[,1]) <= 5) && (#f~==3), print1(p, ", ")););} \\ Michel Marcus, Oct 09 2017

from itertools import count, islice
from sympy import integer_log, isprime
def A293425_gen(): # generator of terms
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c = x
        for i in range(1,integer_log(x,5)[0]+1):
            for j in range(1,integer_log(m:=x//5**i,3)[0]+1):
                c -= (m//3**j).bit_length()-1
        return c
    yield from filter(isprime,(bisection(lambda k:n+f(k),n,n)-1 for n in count(1)))
A293425_list = list(islice(A293425_gen(),30)) # Chai Wah Wu, Mar 31 2025

A347977 Primes of the form 2^p * 3^q * 5^r * 7^s - 1.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 53, 59, 71, 79, 83, 89, 97, 107, 127, 139, 149, 167, 179, 191, 199, 223, 239, 251, 269, 293, 349, 359, 383, 419, 431, 449, 479, 499, 503, 587, 599, 647, 719, 809, 839, 863, 881, 971, 1049, 1151, 1249, 1259, 1279, 1399, 1439, 1499, 1511, 1567, 1619, 1889

Offset: 1

Flávio V. Fernandes, Sep 21 2021

nonn

Restricting to r = s = 0 gives A005105; s = 0 gives A293194.

Primes of the form A002473(k) - 1.

			251 = 2^2 * 3^2 * 5^0 * 7^1 - 1 and 839 = 2^3 * 3^1 * 5^1 * 7^1 - 1 are terms.

Flávio V. Fernandes, Table of n, a(n) for n = 1..10000

Cf. A002473, A005105, A293194.

With[{n = 2000}, Sort@ Select[Flatten@ Table[2^p * 3^q * 5^r * 7^s - 1, {p, 0, Log[2, n]}, {q, 0, Log[3, n/(2^p)]}, {r, 0, Log[5, n/(2^p * 3^q)]}, {s, 0, Log[7, n/(2^p * 3^q * 5^r)]}], PrimeQ]] (* Amiram Eldar, Sep 25 2021 after Michael De Vlieger at A293194 *)

isok(p) = isprime(p) && (vecmax(factor(p+1)[,1]) < 11); \\ Michel Marcus, Nov 10 2021

upto(limit)={my(P=[2,3,5,7]); local(L=List()); my(recurse(k,t) = if(t<=limit+1, if(k>#P, if(isprime(t-1), listput(L,t-1)), my(b=P[k]); for(e=0, logint(limit+1,b), self()(k+1, t*b^e))))); recurse(1, 1); vecsort(Vec(L))} \\ Andrew Howroyd, Nov 20 2021

from itertools import count, islice
from sympy import integer_log, isprime
def A347977_gen(): # generator of terms
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c = x
        for i in range(integer_log(x,7)[0]+1):
            for j in range(integer_log(m:=x//7**i,5)[0]+1):
                for k in range(integer_log(r:=m//5**j,3)[0]+1):
                    c -= (r//3**k).bit_length()
        return c
    yield from filter(isprime,(bisection(lambda k:n+f(k),n,n)-1 for n in count(1)))
A347977_list = list(islice(A347977_gen(),30)) # Chai Wah Wu, Mar 31 2025

A373464 Largest of a quadruple of primes p[1..4] such that (p[k]+1, k=1..4) is in geometric progression.

Original entry on oeis.org

23, 47, 107, 191, 499, 647, 719, 809, 863, 1249, 1439, 1999, 2591, 2879, 3023, 3779, 4079, 5323, 6911, 7039, 7127, 7559, 8231, 8231, 8747, 9839, 10289, 10289, 10499, 10499, 10529, 10691, 11279, 11519, 12959, 13229, 13309, 13999, 15551, 15551, 15971, 18143, 19207

Offset: 1

M. F. Hasler, Jul 12 2024

nonn

a(10) = 1249 is the first term not in A299171, a(15) = 3023 is the first term not in A293194, a(17) = 4079 is the first term not in A347977 and also the first term not in A374482, and a(21) = 7127 is the first term not in A184856.

			The terms of the sequence are column "p[4]" in the following table which lists the sequences of primes, and ratios of the geometric progression (p[k]+1):
   n  | p[1], p[2], p[3], p[4]  |  r = (p[k+1]+1) / (p[k]+1)
------+-------------------------+---------------------------
   1  |    2,    5,   11,   23  |  2 = 6/3 = 12/6 = 24/12
   2  |    5,   11,   23,   47  |  2 = 12/6 = 24/12 = 48/24
   3  |   31,   47,   71,  107  |  3/2 = 48/32 = 72/48 = 108/72
   4  |    2,   11,   47,  191  |  4 = 12/3 = 48/12 = 192/48
   5  |   31,   79,  199,  499  |  5/2 = 80/32 = 200/80 = 500/200
   6  |    2,   17,  107,  647  |  6 = 18/3 = 108/18 = 648/108
   7  |   89,  179,  359,  719  |  2 = 180/90 = ...
   8  |   29,   89,  269,  809  |  3 = 90/30 = ...
   9  |  499,  599,  719,  863  |  6/5 = 600/500 = ...
  10  |   79,  199,  499, 1249  |  5/2 = 200/80 = ...
  11  |  179,  359,  719, 1439  |  2 = 360/180 = ...
  12  |   53,  179,  599, 1999  |  10/3 = 180/54 = ...

Chai Wah Wu, Table of n, a(n) for n = 1..2372
Doddy Kastanya, Fun Math #241, Number Theory group on LinkedIn.com, Jul 04 2024

Subsequence of A089199 (primes p such that p+1 is divisible by a cube).

A373464_upto(N, show=0, D = 1, LIM=N\2) = { my(L=List()); forprime(p=1, LIM, my(denom = p+D); for(numer=denom+1, sqrtnint((N+D) * denom^2, 3), my(r=numer/denom); for(k=1,3, (type(denom * r^k)=="t_INT" && isprime(denom * r^k - D)) || next(2)); listput(L, denom * r^3 - D); show && printf(" | %4d, %4d, %4d, %4d | %s\n",denom-D, denom*r-D, denom*r^2-D, denom*r^3-D, numer/denom))); vecsort(L)}

from itertools import islice
from fractions import Fraction
from sympy import nextprime
def A373464_gen(): # generator of terms
    p, plist, pset = 1, [], set()
    while True:
        p = nextprime(p)
        for q in plist:
            r = Fraction(q+1,p+1)
            q2 = r*(q+1)-1
            if q2 < 2:
                break
            if q2.denominator == 1:
                q2 = int(q2)
                if q2 in pset:
                    q3 = r*(q2+1)-1
                    if q3 < 2:
                        break
                    if q3.denominator == 1 and int(q3) in pset:
                        yield p
        plist = [p]+plist
        pset.add(p)
A373464_list = list(islice(A373464_gen(),20)) # Chai Wah Wu, Jul 16 2024

a(26)-a(43) from Chai Wah Wu, Jul 16 2024

A293074 Primes of the form 2^q * 3^r * 11^s - 1.

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 23, 31, 43, 47, 53, 71, 107, 127, 131, 191, 197, 241, 263, 383, 431, 593, 647, 863, 967, 971, 1151, 1187, 1451, 1583, 2111, 2591, 2903, 3167, 4373, 4751, 5323, 5807, 6143, 6911, 7127, 8191, 8447, 8747, 10691, 12671, 13121, 15551, 15971, 21383, 23327

Offset: 1

Muniru A Asiru, Oct 01 2017

nonn

Mersenne primes A000668 occur when (q, r, s) = (q, 0, 0) with q > 0.
a(2) = 3 is a Mersenne prime but a(3) = 5 is not a Mersenne prime.
For n > 2, all terms = {1, 5} mod 6.

			3 = a(2) = 2^2 * 3^0 * 11^0 - 1.
131 = a(15) = 2^2 * 3^1 * 11^1 - 1.
list of (q, r, s): (0, 1, 0), (2, 0, 0), (1, 1, 0), (3, 0, 0), (2, 1, 0), (1, 2, 0), (3, 1, 0), (5, 0, 0), (2, 0, 1), (4, 1, 0), (1, 3, 0), ...

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

Cf. A000668, A005105, Primes of the form 2^q * 3^r * b^s - 1: A293194 (b = 5), A293199 (b = 7).

K:=10^5+1;; # to get all terms <= K.
A:=Filtered([1..K],IsPrime);;    I:=[3,11];;
B:=List(A,i->Elements(Factors(i+1)));;
C:=List([0..Length(I)],j->List(Combinations(I,j),i->Concatenation([2],i)));;
A293074:=Concatenation([2],List(Set(Flat(List([1..Length(C)],i->List([1..Length(C[i])],j->Positions(B,C[i][j]))))),i->A[i]));

N:= 10^5: # to get all terms < N
S:=select(isprime, {seq(seq(seq(2^q*3^r*11^s-1, q=0..ilog2(floor(N/3^r/11^s))),r=0..floor(log[3](N/11^s))),s=0..floor(log[11](N)))}):
sort(convert(S,list)); # Robert Israel, Oct 03 2017

With[{nn=20},Take[Select[Union[Flatten[Table[2^q 3^r 11^s-1,{q,0,nn},{r,0,nn},{s,0,nn}]]],PrimeQ],60]] (* Harvey P. Dale, May 12 2019 *)

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A293199 Primes of the form 2^q * 3^r * 7^s - 1.

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A293425 Primes of the form 2^a * 3^b * 5^c - 1 for positive a, b, c.

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A347977 Primes of the form 2^p * 3^q * 5^r * 7^s - 1.

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A373464 Largest of a quadruple of primes p[1..4] such that (p[k]+1, k=1..4) is in geometric progression.

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A293074 Primes of the form 2^q * 3^r * 11^s - 1.

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