A106419
Smallest number beginning with 9 that is the product of exactly n distinct primes.
Original entry on oeis.org
97, 91, 902, 910, 9030, 91770, 903210, 9699690, 900029130, 9146807670, 902340208770, 9426343036110, 900781858106130, 90004386781078770, 914836017997511610, 90100977291211496610, 9000008798605567472730, 900002983747159323401370, 9146570985683589524055990
Offset: 1
-
a(n) = {i = prod(i=1, n, prime(i)); while ((digits(i)[1] != 9) || (omega(i) != n) || (bigomega(i) != n), i++); i;} \\ Michel Marcus, Sep 14 2013
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from itertools import count
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi, primorial
def A106419(n):
if n == 1: return 97
def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) 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(primorial(n)))-1):
kmin, kmax = 9*10**l-1, 10**(l+1)-1
mmin, mmax = f(kmin), f(kmax)
if mmax>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, Aug 29 2024
A106429
Smallest number beginning with 9 and having exactly n prime divisors counted with multiplicity.
Original entry on oeis.org
97, 9, 92, 90, 918, 96, 972, 960, 9072, 9600, 90624, 9216, 93312, 90112, 903168, 98304, 995328, 917504, 9043968, 9175040, 90243072, 9437184, 95551488, 92274688, 924844032, 922746880, 9042919424, 905969664, 9172942848, 9059696640
Offset: 1
-
a(n) = {i = 2^n; while ((digits(i)[1] != 9) || (bigomega(i)!=n), i++); i;} \\ Michel Marcus, Sep 14 2013
-
from itertools import count
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A106429(n):
if n == 1: return 97
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, Aug 29 2024
A077334
Smallest number beginning with 9 and having exactly n distinct prime divisors.
Original entry on oeis.org
9, 91, 90, 910, 9030, 90090, 903210, 9699690, 900029130, 9146807670, 901741380540, 9426343036110, 900781858106130, 90004386781078770, 914836017997511610, 90100977291211496610, 9000008798605567472730, 900002983747159323401370
Offset: 1
-
a(n) = {i = prod(i=1, n, prime(i)); while ((digits(i)[1] != 9) || (omega(i)!= n), i++); i;} \\ Michel Marcus, Sep 14 2013
A106411
Smallest number beginning with 1 that is the product of exactly n distinct primes.
Original entry on oeis.org
1, 11, 10, 102, 1110, 10010, 101010, 1009470, 11741730, 1001110110, 10407767370, 1000287585570, 10293281928930, 1001230315195110, 13082761331670030, 1004819888620217670, 100015003602410826930, 1922760350154212639070
Offset: 0
a(0) = 1, a(5) = 10010 = 2*5*7*11*13.
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from itertools import count
from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi, primorial
def A106411(n):
if n <= 1: return 1+10*n
def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) 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(primorial(n)))-1):
kmin, kmax = 10**l-1, 2*10**l-1
mmin, mmax = f(kmin), f(kmax)
if mmax>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
A077326
Smallest number beginning with 1 and having exactly n distinct prime divisors.
Original entry on oeis.org
1, 11, 10, 102, 1020, 10010, 101010, 1009470, 11741730, 1001110110, 10407767370, 1000287585570, 10293281928930, 1001230315195110, 13082761331670030, 1004819888620217670, 100015003602410826930, 1922760350154212639070
Offset: 0
a(0) = 1, a(5) = 10010 = 2*5*7*11*13.
A077327
Smallest number beginning with 2 and having exactly n distinct prime divisors.
Original entry on oeis.org
2, 20, 204, 210, 2310, 200970, 2012010, 20030010, 223092870, 20090100030, 200560490130, 20055767721990, 2000029432190790, 20384767656323070, 2000848249650860610, 200001648981983238390, 2183473617971732996910
Offset: 1
a(1) = 2, a(5) = 2310 = 2*3*5*7*11.
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from sympy import primorial, factorint
def a(n, begins_with=2): # use begins_with 1-9 for A077326-A077334
m, start_digit = primorial(n), str(begins_with)
while len(factorint(m)) != n or str(m)[0] != start_digit:
m += 1
s = str(m)
if s[0] == start_digit: continue
elif s[0] < start_digit: m = int(start_digit+'0'*(len(s)-1))
else: m = int(start_digit+'0'*len(s))
return m
print([a(n) for n in range(1, 10)]) # Michael S. Branicky, Feb 20 2021
Correct a(2) and a(3), add a(6)-a(11) from
Ray Chandler, Apr 17 2005
A077328
Smallest number beginning with 3 and having exactly n distinct prime divisors.
Original entry on oeis.org
3, 33, 30, 330, 3570, 30030, 3008460, 30120090, 300690390, 30043474230, 304075581810, 30035662366710, 304250263527210, 30078810535603830, 3001252188252588270, 32589158477190044730, 3003056284355533696290
Offset: 1
a(1) = 3, a(6) = 30030 = 2*3*5*7*11*13.
Correct a(3)=30 and add a(5)-a(10) from
Ray Chandler, Apr 17 2005
A077329
Smallest number beginning with 4 and having exactly n distinct prime divisors.
Original entry on oeis.org
4, 40, 42, 420, 4290, 43890, 4001970, 40029990, 406816410, 40026056070, 401120980260, 40013061952710, 405332750552730, 40111962162442170, 4000228915204892370, 40909794684132183810, 4000669166940700163910
Offset: 1
A077330
Smallest number beginning with 5 and having exactly n distinct prime divisors.
Original entry on oeis.org
5, 50, 504, 510, 5460, 51870, 510510, 50169210, 504894390, 50007124860, 503520607590, 50000602191540, 501601785815130, 50073188107872930, 5000089945706645790, 50617203592231346070, 5000858931483646541310
Offset: 1
A077331
Smallest number beginning with 6 and having exactly n distinct prime divisors.
Original entry on oeis.org
61, 6, 60, 630, 6006, 60060, 690690, 60090030, 601380780, 6469693230, 600319429710, 60007743265470, 600277546959090, 60039293728424010, 614889782588491410, 60865792091025932010, 6000526229622444289770, 600025752738409899231330
Offset: 1
Corrected and extended by Sam Handler (sam_5_5_5_0(AT)yahoo.com), Jul 21 2004
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