A106422
Smallest number beginning with 2 and having exactly n prime divisors counted with multiplicity.
Original entry on oeis.org
2, 21, 20, 24, 200, 216, 288, 256, 2592, 2304, 2048, 20736, 20480, 24576, 204800, 221184, 294912, 262144, 2654208, 2359296, 2097152, 21233664, 20971520, 25165824, 209715200, 226492416, 201326592, 268435456, 2013265920, 2415919104
Offset: 1
a(1) = 2, a(5) = 200 = 2^3*5^2.
-
f:= proc(n) uses priqueue; local pq, t, p, x, i;
initialize(pq);
insert([-2^n, 2$n], pq);
do
t:= extract(pq);
x:= -t[1];
if floor(x/10^ilog10(x)) = 2 then return x fi;
p:= nextprime(t[-1]);
for i from n+1 to 2 by -1 while t[i] = t[-1] do
insert([t[1]*(p/t[-1])^(n+2-i), op(t[2..i-1]), p$(n+2-i)], pq)
od;
od
end proc:
map(f, [$1..40]); # Robert Israel, Apr 15 2025
-
from itertools import count
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A106422(n):
if n == 1: return 2
def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
def f(x): return int(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,n)))
for l in count(len(str(1<mmin:
while kmax-kmin > 1:
kmid = kmax+kmin>>1
mmid = f(kmid)
if mmid > mmin:
kmax, mmax = kmid, mmid
else:
kmin, mmin = kmid, mmid
return kmax # Chai Wah Wu, Sep 12 2024
A106423
Smallest number beginning with 3 and having exactly n prime divisors counted with multiplicity.
Original entry on oeis.org
3, 33, 30, 36, 32, 324, 320, 384, 3200, 3456, 3072, 31104, 30720, 36864, 32768, 331776, 327680, 393216, 3276800, 3538944, 3145728, 31850496, 31457280, 37748736, 33554432, 339738624, 301989888, 3057647616, 3019898880, 3623878656
Offset: 1
a(1) = 3, a(6) = 324 = 2^2*3^4.
-
f:= proc(n) uses priqueue; local pq, t,p,x,i;
initialize(pq);
insert([-2^n,2$n],pq);
do
t:= extract(pq);
x:= -t[1];
if floor(x/10^ilog10(x)) = 3 then return x fi;
p:= nextprime(t[-1]);
for i from n+1 to 2 by -1 while t[i] = t[-1] do
insert([t[1]*(p/t[-1])^(n+2-i), op(t[2..i-1]),p$(n+2-i)],pq)
od;
od
end proc:
map(f, [$1..50]); # Robert Israel, Sep 06 2024
-
from itertools import count
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A106423(n):
if n == 1: return 3
def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
def f(x): return int(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,n)))
for l in count(len(str(1<mmin:
while kmax-kmin > 1:
kmid = kmax+kmin>>1
mmid = f(kmid)
if mmid > mmin:
kmax, mmax = kmid, mmid
else:
kmin, mmin = kmid, mmid
return kmax # Chai Wah Wu, Sep 12 2024
A106424
Smallest number beginning with 4 and having exactly n prime divisors counted with multiplicity.
Original entry on oeis.org
41, 4, 42, 40, 48, 400, 432, 4000, 4032, 40000, 4608, 4096, 41472, 40960, 49152, 409600, 442368, 4063232, 4128768, 40310784, 4718592, 4194304, 42467328, 41943040, 411041792, 419430400, 452984832, 402653184, 4076863488, 4026531840
Offset: 1
a(1) = 41, a(3) = 42 = 2*3*7.
-
from itertools import count
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A106424(n):
if n == 1: return 41
def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
def f(x): return int(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,n)))
for l in count(len(str(1<mmin:
while kmax-kmin > 1:
kmid = kmax+kmin>>1
mmid = f(kmid)
if mmid > mmin:
kmax, mmax = kmid, mmid
else:
kmin, mmin = kmid, mmid
return kmax # Chai Wah Wu, Sep 12 2024
A106425
Smallest number beginning with 5 and having exactly n prime divisors counted with multiplicity.
Original entry on oeis.org
5, 51, 50, 54, 500, 504, 5000, 576, 512, 5184, 5120, 50176, 51200, 55296, 507904, 516096, 5038848, 589824, 524288, 5308416, 5242880, 51380224, 52428800, 56623104, 50331648, 509607936, 503316480, 5096079360, 536870912, 5435817984
Offset: 1
-
from itertools import count
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A106425(n):
if n == 1: return 5
def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
def f(x): return int(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,n)))
for l in count(len(str(1<mmin:
while kmax-kmin > 1:
kmid = kmax+kmin>>1
mmid = f(kmid)
if mmid > mmin:
kmax, mmax = kmid, mmid
else:
kmin, mmin = kmid, mmid
return kmax # Chai Wah Wu, Sep 12 2024
A106426
Smallest number beginning with 6 and having exactly n prime divisors counted with multiplicity.
Original entry on oeis.org
61, 6, 63, 60, 612, 64, 648, 640, 6048, 6400, 6912, 6144, 62208, 61440, 602112, 65536, 663552, 655360, 6029312, 6553600, 60162048, 6291456, 63700992, 62914560, 616562688, 67108864, 679477248, 603979776, 6115295232, 6039797760
Offset: 1
-
from itertools import count
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A106426(n):
if n == 1: return 61
def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
def f(x): return int(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,n)))
for l in count(len(str(1<mmin:
while kmax-kmin > 1:
kmid = kmax+kmin>>1
mmid = f(kmid)
if mmid > mmin:
kmax, mmax = kmid, mmid
else:
kmin, mmin = kmid, mmid
return kmax # Chai Wah Wu, Sep 12 2024
A106427
Smallest number beginning with 7 and having exactly n prime divisors counted with multiplicity.
Original entry on oeis.org
7, 74, 70, 708, 72, 729, 704, 7056, 768, 7776, 7168, 70656, 71680, 702464, 73728, 746496, 720896, 7225344, 786432, 7962624, 7077888, 71663616, 70778880, 700710912, 75497472, 764411904, 704643072, 7113539584, 7046430720
Offset: 1
-
from itertools import count
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A106427(n):
if n == 1: return 7
def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
def f(x): return int(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,n)))
for l in count(len(str(1<mmin:
while kmax-kmin > 1:
kmid = kmax+kmin>>1
mmid = f(kmid)
if mmid > mmin:
kmax, mmax = kmid, mmid
else:
kmin, mmin = kmid, mmid
return kmax # Chai Wah Wu, Sep 12 2024
A106428
Smallest number beginning with 8 and having exactly n prime divisors counted with multiplicity.
Original entry on oeis.org
83, 82, 8, 81, 80, 810, 800, 864, 8000, 8064, 80000, 80640, 8192, 82944, 81920, 802816, 819200, 884736, 8126464, 8257536, 80621568, 80216064, 8388608, 84934656, 83886080, 822083584, 838860800, 8120172544, 805306368, 8153726976
Offset: 1
-
from itertools import count
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A106428(n):
if n == 1: return 83
def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
def f(x): return int(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,n)))
for l in count(len(str(1<mmin:
while kmax-kmin > 1:
kmid = kmax+kmin>>1
mmid = f(kmid)
if mmid > mmin:
kmax, mmax = kmid, mmid
else:
kmin, mmin = kmid, mmid
return kmax # Chai Wah Wu, Sep 12 2024