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.

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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

Views

Author

Amarnath Murthy, Nov 04 2002

Keywords

Examples

			a(3) = 60 = 2^2*3*5.

Crossrefs

Cf. A077326, A077327, A077328, A077329, A077330, A077332, A077333, A077334.
Cf. A106411-A106419, A106421-A106429.

Extensions

Corrected and extended by Sam Handler (sam_5_5_5_0(AT)yahoo.com), Jul 21 2004
a(8)-a(10) from Ray Chandler, Apr 17 2005
More terms from Ray Chandler, May 02 2005

A077332 Smallest number beginning with 7 and having exactly n distinct prime divisors.

Original entry on oeis.org

7, 72, 70, 714, 7140, 71610, 746130, 70136220, 703600590, 70015935990, 700288518930, 7420738134810, 701098433345310, 70007243563797540, 757887406446280110, 70025936403159126390, 7001749954335151685670, 700007496840185797172910
Offset: 1

Views

Author

Amarnath Murthy, Nov 04 2002

Keywords

Examples

			a(3) = 70 = 2*5*7.

Crossrefs

Cf. A077326, A077327, A077328, A077329, A077330, A077331, 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

A077333 Smallest number beginning with 8 and having exactly n distinct prime divisors.

Original entry on oeis.org

8, 80, 84, 840, 8190, 81510, 870870, 80059980, 800509710, 8254436190, 800680310430, 8222980095330, 800160280950030, 80008785365579070, 843685980760953330, 80058789202898516010, 8001338333881400327820, 800009744613910196656290
Offset: 1

Views

Author

Amarnath Murthy, Nov 04 2002

Keywords

Examples

			a(3) = 84 = 2^2*3*7.

Crossrefs

Cf. A077326, A077327, A077328, A077329, A077330, A077331, A077332, A077334.
Cf. A106411-A106419, A106421-A106429.

Extensions

Corrected and extended by Sascha Kurz, Jan 30 2003
a(9)-a(10) from Ray Chandler, Apr 17 2005
More terms from Ray Chandler, May 02 2005

A106412 Smallest number beginning with 2 that is the product of exactly n distinct primes.

Original entry on oeis.org

2, 21, 222, 210, 2310, 201630, 2012010, 20030010, 223092870, 20090100030, 200560490130, 20055767721990, 2000029432190790, 20384767656323070, 2000848249650860610, 200001648981983238390, 2183473617971732996910
Offset: 1

Views

Author

Ray Chandler, May 02 2005

Keywords

Examples

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

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 A106412(n):
    if n == 1: return 2
    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 = 2*10**l-1, 3*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

A106413 Smallest number beginning with 3 that is the product of exactly n distinct primes.

Original entry on oeis.org

3, 33, 30, 330, 3570, 30030, 3015870, 30120090, 300690390, 30043474230, 304075581810, 30035662366710, 304250263527210, 30078810535603830, 3001252188252588270, 32589158477190044730, 3003056284355533696290
Offset: 1

Views

Author

Ray Chandler, May 02 2005

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • Maple
    f:= proc(n) uses priqueue; local pq, t, p, x, i,L,v,Lp;
    initialize(pq);
    L:= [seq(ithprime(i),i=1..n)];
    v:= convert(L,`*`);
    insert([-v, L], pq);
    do
      t:= extract(pq);
      x:= -t[1];
      if floor(x/10^ilog10(x)) = 3 then return x fi;
      L:= t[2];
      p:= nextprime(L[-1]);
      for i from n to 1 by -1 do
        if i < n and L[i] <> prevprime(L[i+1]) then break fi;
        Lp:= [op(L[1..i-1]),op(L[i+1..n]),p];
        insert([-convert(Lp,`*`),Lp], pq)
    od od;
end proc:
map(f, [$1..30]); # Robert Israel, Sep 12 2024
  • Python
    from itertools import count
from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi, primorial
def A106413(n):
    if n == 1: return 3
    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 = 3*10**l-1, 4*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

A106414 Smallest number beginning with 4 that is the product of exactly n distinct primes.

Original entry on oeis.org

41, 46, 42, 462, 4290, 43890, 4001970, 40029990, 406816410, 40026056070, 408036859230, 40013061952710, 405332750552730, 40111962162442170, 4000228915204892370, 40909794684132183810, 4000669166940700163910
Offset: 1

Views

Author

Ray Chandler, May 02 2005

Keywords

Examples

			a(1) = 41, a(3) = 42 = 2*3*7..

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 A106414(n):
    if n == 1: return 41
    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 = 4*10**l-1, 5*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

A106415 Smallest number beginning with 5 that is the product of exactly n distinct primes.

Original entry on oeis.org

5, 51, 506, 510, 5610, 51870, 510510, 50169210, 504894390, 50012172210, 503520607590, 50001975553530, 501601785815130, 50073188107872930, 5000089945706645790, 50617203592231346070, 5000858931483646541310
Offset: 1

Views

Author

Ray Chandler, May 02 2005

Keywords

Examples

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

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 A106415(n):
    if n == 1: return 5
    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 = 5*10**l-1, 6*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

A106416 Smallest number beginning with 6 that is the product of exactly n distinct primes.

Original entry on oeis.org

61, 6, 66, 690, 6006, 62790, 690690, 60138078, 606996390, 6469693230, 600319429710, 60007743265470, 600277546959090, 60039293728424010, 614889782588491410, 60865792091025932010, 6000526229622444289770
Offset: 1

Views

Author

Ray Chandler, May 02 2005

Keywords

Examples

			a(3) = 66 = 2*3*11.

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 A106416(n):
    if n == 1: return 61
    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 = 6*10**l-1, 7*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

A106417 Smallest number beginning with 7 that is the product of exactly n distinct primes.

Original entry on oeis.org

7, 74, 70, 714, 7410, 71610, 746130, 70387590, 703600590, 70015935990, 700288518930, 7420738134810, 701098433345310, 70016268785853390, 757887406446280110, 70025936403159126390, 7001749954335151685670
Offset: 1

Views

Author

Ray Chandler, May 02 2005

Keywords

Examples

			a(3) = 70 = 2*5*7.

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 A106417(n):
    if n == 1: return 7
    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 = 7*10**l-1, 8*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

A106418 Smallest number beginning with 8 that is the product of exactly n distinct primes.

Original entry on oeis.org

83, 82, 805, 858, 8610, 81510, 870870, 80150070, 800509710, 8254436190, 800680310430, 8222980095330, 800160280950030, 80008785365579070, 843685980760953330, 80058789202898516010, 8003887646839494820410
Offset: 1

Views

Author

Ray Chandler, May 02 2005

Keywords

Examples

			a(3) = 805 = 5*7*23.

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 A106418(n):
    if n == 1: return 83
    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)))
    def bisection(f,kmin,kmax,mmin,mmax):
        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
    for l in count(len(str(primorial(n)))-1):
        kmin, kmax = 8*10**l-1, 9*10**l-1
        mmin, mmax = f(kmin), f(kmax)
        if mmax>mmin: return bisection(f,kmin,kmax,mmin,mmax) # Chai Wah Wu, Aug 31 2024
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