A085908 Smallest 7-smooth number beginning with n.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 112, 12, 135, 14, 15, 16, 175, 18, 192, 20, 21, 224, 2304, 24, 25, 2625, 27, 28, 294, 30, 315, 32, 336, 343, 35, 36, 375, 384, 392, 40, 4116, 42, 432, 441, 45, 4608, 4704, 48, 49, 50, 512, 525, 5376, 54, 55125, 56, 576, 588, 59049, 60

Offset

1,2

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Example

a(23) = 2304 = 2^8*3^2 is the smallest 7-smooth number beginning with 23. (23, 230, 231, 232, ..., 239, 2301, 2302, 2303 etc. have a divisor > 10.)

Maple

N:= 300: # for a(1) .. a(N)
V:= Vector(N): count:= 0:with(priqueue):
initialize(pq);
insert([-1, 0, 0, 0, 0], pq);
while count < N do
  t:= extract(pq);
  x:= -t[1];
  for d from ilog10(x) to 0 by -1 do
    xd:= floor(x/10^d);
    if xd > N then break fi;
    if V[xd] = 0 then V[xd]:= x; count:= count+1; fi;
  od;
  insert([-7*x, t[2], t[3], t[4], t[5]+1], pq);
  if t[5]=0 then
    insert([-5*x, t[2], t[3], t[4]+1, 0], pq);
    if t[4] = 0 then
      insert([-3*x, t[2], t[3]+1, 0, 0], pq);
      if t[3] = 0 then
        insert([-2*x, t[2]+1, 0, 0, 0], pq);
  fi fi fi;
od:
convert(V, list); # Robert Israel, Sep 18 2024

Mathematica

a[n_] := Module[{d = IntegerDigits[n], k = 1}, While[Max[FactorInteger[k][[;; , 1]]] > 7 || Length[IntegerDigits[k]] < Length[d] || IntegerDigits[k][[1 ;; Length[d]]] != d, k++]; k]; Array[a, 60] (* Amiram Eldar, Apr 30 2022 *)

Prog

(PARI) hc(n) = local(f); f = factor(n); f[matsize(f)[1], 1] < 10;
a(n) = local(d, x); if (hc(n), return(n)); d = 1; while (d, for (i = 1, 10^d - 1, x = n*10^d + i; if (hc(x), return(x))); d++); \\ David Wasserman, Feb 11 2005
(Python)
from itertools import count
from sympy import integer_log
def A085908(n):
    if n<2: return n
    def f(x):
        c = 0
        for i in range(integer_log(x, 7)[0]+1):
            for j in range(integer_log(m:=x//7**i, 5)[0]+1):
                for k in range(integer_log(r:=m//5**j, 3)[0]+1):
                    c += (r//3**k).bit_length()
        return c
    for l in count(0):
        kmin, kmax = n*10**l-1, (n+1)*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 17 2024

Crossrefs

Cf. A002473.

Sequence in context: A258373 A250266 A357936 * A376280 A110304 A043315

Adjacent sequences: A085905 A085906 A085907 * A085909 A085910 A085911

Keyword

base,nonn,look

Author

Amarnath Murthy, Jul 09 2003

Extensions

Corrected and extended by David Wasserman, Feb 11 2005

Name corrected by J. Lowell, Apr 30 2022

Status

approved