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

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

Author

Muniru A Asiru, Oct 09 2017

Keywords

• nonn

Comments

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

Examples

			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), ...
		

Links

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

Crossrefs

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

Programs

GAP

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]);

Maple

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

Mathematica

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 *)

PARI

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

Python

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