cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 27 results. Next

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

Views

Author

Ray Chandler, May 02 2005

Keywords

Examples

			a(2) = 91 = 7*13.

Links

Crossrefs

Cf. A077326-A077334, A106411-A106419, A106421-A106429.

Programs

  • PARI
    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
  • Python
    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

Extensions

a(18)-a(19) from 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

Views

Author

Ray Chandler, May 02 2005

Keywords

Examples

			a(2) = 9 = 3^2.

Crossrefs

Cf. A077326-A077334, A106411-A106419, A106421-A106429.

Programs

  • PARI
    a(n) = {i = 2^n; while ((digits(i)[1] != 9) || (bigomega(i)!=n), i++); i;} \\ Michel Marcus, Sep 14 2013
  • Python
    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

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

Views

Author

Ray Chandler, May 02 2005

Keywords

Examples

			a(0) = 1, a(5) = 10010 = 2*5*7*11*13.

Links

Crossrefs

Cf. A077326-A077334, A106411-A106419, A106421-A106429.

Programs

  • Python
    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

A106421 Smallest number beginning with 1 and having exactly n prime divisors counted with multiplicity.

Original entry on oeis.org

1, 11, 10, 12, 16, 108, 144, 128, 1296, 1152, 1024, 10368, 10240, 12288, 16384, 110592, 147456, 131072, 1327104, 1179648, 1048576, 10616832, 10485760, 12582912, 16777216, 113246208, 100663296, 134217728, 1006632960, 1207959552
Offset: 0

Views

Author

Ray Chandler, May 02 2005

Keywords

Examples

			a(0) = 1, a(5) = 108 = 2^2*3^3.

Links

Crossrefs

Cf. A077326-A077334, A106411-A106419, A106421-A106429.

Programs

  • Maple
    f:= proc(n) uses priqueue; local pq, t,p,x,i;
    initialize(pq);
    insert([-2^n,2$n],pq);
    do
      t:= extract(pq);
      x:= -t[1];
      if floor(x/10^ilog10(x)) = 1 then return x fi;
      p:= nextprime(t[-1]);
      for i from n+1 to 2 by -1 while t[i] = t[-1] do
        insert([t[1]*(p/t[-1])^(n+2-i), op(t[2..i-1]),p$(n+2-i)],pq)
      od;
    od
end proc:
f(0):= 1:
map(f, [$0..50]); # Robert Israel, Sep 06 2024
  • Python
    from itertools import count
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A106421(n):
    if n <= 1: return 1+10*n
    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

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

Views

Author

Amarnath Murthy, Nov 04 2002

Keywords

Comments

What are the values of n where a(n) differ from A106411(n)? So far only n=4 is such a value. - Chai Wah Wu, May 07 2025

Examples

			a(0) = 1, a(5) = 10010 = 2*5*7*11*13.

Crossrefs

Cf. A077327, A077328, A077329, A077330, A077331, A077332, A077333, A077334.
Cf. A106411-A106419, A106421-A106429.

Extensions

More terms from Sascha Kurz, Jan 04 2003
a(9)-a(10) from Ray Chandler, Apr 17 2005
More terms from Ray Chandler, May 02 2005

A217402 Numbers starting with 9.

Original entry on oeis.org

9, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 900, 901, 902, 903, 904, 905, 906, 907, 908, 909, 910, 911, 912, 913, 914, 915, 916, 917, 918, 919, 920, 921, 922, 923, 924, 925, 926, 927, 928, 929, 930, 931, 932, 933, 934, 935, 936, 937, 938, 939, 940, 941, 942
Offset: 1

Views

Author

Jeremy Gardiner, Oct 02 2012

Keywords

Comments

The lower and upper asymptotic densities of this sequence are 1/81 and 1/9, respectively. - Amiram Eldar, Feb 27 2021

Links

Crossrefs

Cf. A000981, A011539.
Subsequences include: A077685, A045715, A106419, A106429, A077334.
Cf. A131835, A217394, A217395, A217397, A217398, A217399, A217400, A217401.

Programs

  • Mathematica
    Select[Range[1000], IntegerDigits[#][[1]] == 9 &] (* T. D. Noe, Oct 02 2012 *)
  • Python
    def A217402(n): return n+(80*10**(len(str(9*n-8))-1))//9 # Chai Wah Wu, Dec 07 2024

Formula

a(n) = n + (80*10^floor(log_10(9*n-8))-8)/9. - Alan Michael Gómez Calderón, May 17 2023

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

Views

Author

Amarnath Murthy, Nov 04 2002

Keywords

Examples

			a(1) = 2, a(5) = 2310 = 2*3*5*7*11.

Crossrefs

Cf. A077326, A077328, A077329, A077330, A077331, A077332, A077333, A077334.
Cf. A106411-A106419, A106421-A106429.

Programs

  • Python
    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

Extensions

Correct a(2) and a(3), add a(6)-a(11) from Ray Chandler, Apr 17 2005
More terms from Ray Chandler, May 02 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

Views

Author

Amarnath Murthy, Nov 04 2002

Keywords

Examples

			a(1) = 3, a(6) = 30030 = 2*3*5*7*11*13.

Crossrefs

Cf. A077326, A077327, A077329, A077330, A077331, A077332, A077333, A077334.
Cf. A106411-A106419, A106421-A106429.

Extensions

Correct a(3)=30 and add a(5)-a(10) from Ray Chandler, Apr 17 2005
More terms from Ray Chandler, May 02 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

Views

Author

Amarnath Murthy, Nov 04 2002

Keywords

Examples

			a(1) = 4, a(3) = 42.

Crossrefs

Cf. A077326, A077327, A077328, A077330, A077331, A077332, A077333, A077334.
Cf. A106411-A106419, A106421-A106429.

Extensions

a(5)-a(10) from Ray Chandler, Apr 17 2005
More terms from Ray Chandler, May 02 2005

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

Views

Author

Amarnath Murthy, Nov 04 2002

Keywords

Examples

			a(4) = 510 = 2*3*5*17.

Crossrefs

Cf. A077326, A077327, A077328, A077329, A077331, A077332, A077333, A077334.
Cf. A106411-A106419, A106421-A106429.

Extensions

More terms from Ray G. Opao, Aug 04 2004
a(10) from Ray Chandler, Apr 17 2005
More terms from Ray Chandler, May 02 2005
Showing 1-10 of 27 results. Next

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