A002473 -id:A002473 - cp's OEIS frontend

A068185 Number of ways writing n^n as a product of decimal digits of some other number which has no digits equal to 1.

0, 2, 3, 81, 1, 102136, 1, 1389537, 4181, 4972825, 0, 12718670252691776, 0, 4506838380, 11472991008, 53560898629395777, 0, 514875062240230100091396, 0, 164997736300578242823300, 241098942106440, 0, 0, 3203410440031870942324022423896806853153460, 1, 0, 61305790721611591

Offset: 1

Labos Elemer, Feb 19 2002

a(n)= 0 when n has prime-factor larger than 7 [so A067734(n)=0] or when n is in A068191, i.e. not in A002473.

			n=1 has no solution; a(2)=A000073(6)=2 with {4,22} solutions; a(3)=A067734(27)=3=Fibonacci[4]; n=5 and n=7, n^n has single prime factor of which any true multiple have 2 digits so 55555 and 7777777 are the only solutions, so a(5)=a(7)=1; a(4)=A067734(256)=81=A000073(10); a(8)=A067734(2^24)=A000073(26)=1389537; n=9 a(9)=A067734(3^27)=4181.

Cf. A000045, A000073, A000312, A001222, A002473, A067734, A068183-A068187, A068189-A068191.

a(n) = A067734(n^n) = A067734(A000312(n))

Edited By Henry Bottomley, Feb 26 2002.
Edited and extended by Max Alekseyev, Sep 19 2009
a(9) corrected by Sean A. Irvine, Feb 01 2024

A068186 a(n) is the largest number whose product of decimal digits equals n^n.

Original entry on oeis.org

22, 333, 22222222, 55555, 333333222222, 7777777, 222222222222222222222222, 333333333333333333, 55555555552222222222, 0, 333333333333222222222222222222222222, 0, 7777777777777722222222222222

Offset: 2

Labos Elemer, Feb 19 2002

No digit=1 is permitted to avoid infinite number of solutions; a(n)=0 if A067734(n^n)=0.

			n=10, 10^10=10000000000, a(5)=55555555552222222222.

Chai Wah Wu, Table of n, a(n) for n = 2..100

Cf. A000312, A001222, A002473, A067734, A068183-A068187, A068189-A068191.

from sympy import factorint
def A068186(n):
    if n == 1:
        return 1
    pf = factorint(n)
    ps = sorted(pf.keys(),reverse=True)
    if ps[0] > 7:
        return 0
    s = ''
    for p in ps:
        s += str(p)*(n*pf[p])
    return int(s) # Chai Wah Wu, Aug 12 2017

a(n) is obtained as prime factors of n^n concatenated in order of magnitude and with repetitions; a(n)=0 if n has p > 7 prime factors.

a(12) corrected by Chai Wah Wu, Aug 12 2017

A068190 Largest number whose digit product equals n; a(n)=0 if no such number exists, e.g., when n has a prime factor larger than 7; no digit=1 is permitted to avoid an infinite number of solutions.

Original entry on oeis.org

0, 2, 3, 22, 5, 32, 7, 222, 33, 52, 0, 322, 0, 72, 53, 2222, 0, 332, 0, 522, 73, 0, 0, 3222, 55, 0, 333, 722, 0, 532, 0, 22222, 0, 0, 75, 3322, 0, 0, 0, 5222, 0, 732, 0, 0, 533, 0, 0, 32222, 77, 552, 0, 0, 0, 3332, 0, 7222, 0, 0, 0, 5322, 0, 0, 733, 222222, 0, 0, 0, 0, 0

Offset: 1

Labos Elemer, Feb 19 2002

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A001222, A002473, A007954, A067734, A068183-A068187, A068189-A068191.

Array[If[#[[-1, 1]] > 7, 0, FromDigits@ Reverse@ Flatten@ Map[ConstantArray[#1, #2] & @@ # &, #]] &@ FactorInteger@ # &, 69] /. 1 -> 0 (* Michael De Vlieger, Dec 08 2018 *)

a(n) = {my(res = []); for(i=2, 9, v = valuation(n, i); if(v > 0, res = concat(vector(v, j, i), res); n/=i^v)); if(n==1,fromdigits(res), 0)} \\ David A. Corneth, Jul 31 2017

If a solution exists, a(n) is the concatenation of prime factors with repetitions and in order of magnitude, otherwise a(n)=0.

a(36) corrected by David A. Corneth, Jul 31 2017

A069077 Triangular numbers such that the product of digits is also a (positive) triangular number.

Original entry on oeis.org

1, 3, 6, 66, 153, 231, 351, 465, 741, 1326, 2556, 5671, 6786, 14535, 21115, 24531, 25651, 33411, 43956, 57291, 58311, 71253, 92665, 95266, 123753, 153181, 167331, 278631, 325221, 341551, 351541, 372816, 459361, 491536, 516636, 521731, 567645, 572985, 613278

Offset: 1

Amarnath Murthy, Apr 05 2002

Amiram Eldar, Table of n, a(n) for n = 1..457

Cf. A000217, A002473, A007954, A061380.

[m:m in [1..600000]|not 0 in Intseq(m) and IsSquare(8*m+1) and IsSquare(8*(&*Intseq(m))+1)]; // Marius A. Burtea, Aug 12 2019

aQ[n_] := (p = Times @@ IntegerDigits[n]) > 0 && IntegerQ @ Sqrt[8p + 1]; t[n_] := n(n+1)/2; Select[t[Range[10^3]], aQ] (* Amiram Eldar, Aug 12 2019 *)
Select[Accumulate[Range[1500]],FreeQ[IntegerDigits[#],0]&&OddQ[Sqrt[8 Times@@IntegerDigits[ #]+1]]&] (* Harvey P. Dale, May 01 2023 *)

More terms from Jason Earls and Lior Manor, May 15 2002

A080195 11-smooth numbers which are not 7-smooth.

Original entry on oeis.org

11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132, 154, 165, 176, 198, 220, 231, 242, 264, 275, 297, 308, 330, 352, 363, 385, 396, 440, 462, 484, 495, 528, 539, 550, 594, 605, 616, 660, 693, 704, 726, 770, 792, 825, 847, 880, 891, 924, 968, 990, 1056, 1078, 1089

Offset: 1

Klaus Brockhaus, Feb 10 2003

Numbers of the form 2^r*3^s*5^t*7^u*11^v with r, s, t, u >= 0, v > 0.

			33 = 3*11 is a term but 35 = 5*7 is not.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A051038, A002473.

N:= 10^6; # to get all terms <= N
A:= NULL;
for v from 1 to floor(log[11](N)) do
   V:= 11^v;
   for u from 0 to floor(log[7](N/V)) do
      U:= 7^u*V;
      for t from 0 to floor(log[5](N/U)) do
        T:= 5^t*U;
        for s from 0 to floor(log[3](N/T)) do
          S:= 3^s*T;
          for r from 0 to floor(log[2](N/S)) do
             A:= A, 2^r*S
          od
         od
       od
    od
od:
{A}; # Robert Israel, May 28 2014

Select[Range[1000], FactorInteger[#][[-1, 1]] == 11 &] (* Amiram Eldar, Nov 10 2020 *)

{m=1100; z=[]; for(r=0,floor(log(m)/log(2)),a=2^r; for(s=0,floor(log(m/a)/log(3)),b=a*3^s; for(t=0, floor(log(m/b)/log(5)),c=b*5^t; for(u=0,floor(log(m/c)/log(7)),d=c*7^u; for(v=1,floor(log(m/d)/log(11)), z=concat(z,d*11^v)))))); z=vecsort(z); for(i=1,length(z),print1(z[i],","))}

from sympy import integer_log, prevprime
def A080195(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(x,m): return sum((x//3**i).bit_length() for i in range(integer_log(x,3)[0]+1)) if m==3 else sum(g(x//(m**i),prevprime(m))for i in range(integer_log(x,m)[0]+1))
    def f(x): return n+x-g(x,11)
    return 11*bisection(f,n,n) # Chai Wah Wu, Oct 22 2024

a(n) = 11 * A051038(n). - David A. Corneth, May 27 2017

Sum_{n>=1} 1/a(n) = 7/16. - Amiram Eldar, Nov 10 2020

A080786 Triangle T(n,k) = number of k-smooth numbers <= n, read by rows.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 3, 4, 4, 1, 3, 4, 4, 5, 1, 3, 5, 5, 6, 6, 1, 3, 5, 5, 6, 6, 7, 1, 4, 6, 6, 7, 7, 8, 8, 1, 4, 7, 7, 8, 8, 9, 9, 9, 1, 4, 7, 7, 9, 9, 10, 10, 10, 10, 1, 4, 7, 7, 9, 9, 10, 10, 10, 10, 11, 1, 4, 8, 8, 10, 10, 11, 11, 11, 11, 12, 12, 1, 4, 8, 8, 10, 10, 11, 11, 11, 11, 12, 12, 13, 1, 4

Offset: 1

Reinhard Zumkeller, Mar 12 2003

T(n,n-1) = A014684(n) for n>1;
T(n,2) = A029837(n) for n>1; T(n,3) = A071521(n) for n>2; T(n,5) = A071520(n) for n>4.
A036234(n) = number of distinct terms in n-th row. - Reinhard Zumkeller, Sep 17 2013

			Triangle begins:
.................. 1
................ 1...2
.............. 1...2...3
............ 1...3...4...4
.......... 1...3...4...4...5
........ 1...3...5...5...6...6
...... 1...3...5...5...6...6...7
.... 1...4...6...6...7...7...8...8
.. 1...4...7...7...8...8...9...9...9.

Reinhard Zumkeller, Rows n=1..120 of triangle, flattened
Eric Weisstein's World of Mathematics, Smooth Number.

Cf. A000079, A002473, A003586, A006530, A014684, A029837, A036234, A051037, A051038, A071520, A071521, A080197, A080681, A080682, A080683.

a080786 n k = a080786_tabl !! (n-1) !! (k-1)
a080786_row n = a080786_tabl !! (n-1)
a080786_tabl = map reverse $ iterate f [1] where
   f xs@(x:_) = (x + 1) :
                (zipWith (+) xs (map (fromEnum . (lpf <=)) [x, x-1 ..]))
        where lpf = fromInteger $ a006530 $ fromIntegral (x + 1)
-- Reinhard Zumkeller, Sep 17 2013

A080786 := proc(x,y)
    local a,n ;
    a := 0 ;
    for n from 1 to x do
        if A006530(n) <= y then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Aug 31 2013

P[n_] := FactorInteger[n][[-1, 1]]; P[1]=1; T[n_, k_] := (For[j=0; m=1, m <= n, m++, If[P[m] <= k, j++]]; j); Table[T[n, k], {n, 1, 15}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 22 2015 *)

from itertools import count, islice
from sympy import prevprime, integer_log
def A080786_T(n,k):
    if k==1: return 1
    def g(x,m): return x.bit_length() if m==2 else sum(g(x//(m**i),prevprime(m))for i in range(integer_log(x,m)[0]+1))
    return g(n,prevprime(k+1))
def A080786_gen(): # generator of terms
    return (A080786_T(n,k) for n in count(1) for k in range(1,n+1))
A080786_list = list(islice(A080786_gen(),100)) # Chai Wah Wu, Oct 22 2024

A084891 Multiples of 2, 3, 5, or 7, but not 7-smooth.

Original entry on oeis.org

22, 26, 33, 34, 38, 39, 44, 46, 51, 52, 55, 57, 58, 62, 65, 66, 68, 69, 74, 76, 77, 78, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 99, 102, 104, 106, 110, 111, 114, 115, 116, 117, 118, 119, 122, 123, 124, 129, 130, 132, 133, 134, 136, 138, 141, 142, 145, 146

Offset: 1

Reinhard Zumkeller, Jul 13 2003

nonn

Intersection of A068191 with (A005843, A008585, A008587 and A008589); union of (A005843, A008585, A008587 and A008589) without A002473.

A020639(a(n)) <= 7, A006530(a(n)) > 7.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Diagram showing numbers k in this sequence instead as k mod 210, in black, else white if k is coprime to 210, purple if k = 1, red if k | 210, and gold if rad(k) | 210, magnification 5X.
Eric Weisstein's World of Mathematics, Smooth Number.

Cf. A002473, A005843, A006530, A008585, A008587, A008589, A020639, A068191, A080672.

okQ[n_] := AnyTrue[{2, 3, 5, 7}, Divisible[n, #]&] && FactorInteger[n][[-1, 1]] > 7;
Select[Range[1000], okQ] (* Jean-François Alcover, Oct 15 2021 *)

mult2357(m,n) = \\ mult 2,3,5,7 not 7 smooth
{
local(x,a,j,f,ln);
for(x=m,n,
f=0;
if(gcd(x,210)>1,
a = ifactor(x);
for(j=1,length(a),
if(a[j]>7,f=1;break);
);
if(f,print1(x","));
);
);
}
ifactor(n) = \\ The vector of the prime factors of n with multiplicity.
{
local(f,j,k,flist);
flist=[];
f=Vec(factor(n));
for(j=1,length(f[1]),
for(k = 1,f[2][j],flist = concat(flist,f[1][j])
);
);
return(flist)
}
\\ Cino Hilliard, Jul 03 2009

from sympy import primefactors
def ok(n):
    pf = set(primefactors(n))
    return pf & {2, 3, 5, 7} and pf - {2, 3, 5, 7}
print(list(filter(ok, range(147)))) # Michael S. Branicky, Oct 15 2021

A086247 Differences of successive 7-smooth numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 3, 1, 2, 1, 2, 2, 3, 1, 4, 2, 3, 3, 1, 1, 4, 2, 4, 3, 1, 6, 2, 3, 5, 1, 3, 6, 6, 2, 2, 5, 3, 4, 8, 5, 1, 2, 7, 5, 4, 3, 3, 10, 2, 6, 7, 5, 9, 3, 4, 4, 10, 6, 8, 1, 15, 3, 2, 5, 2, 4, 14, 10, 8, 6, 6, 15, 5, 4, 12, 7, 7, 10, 15, 3, 6, 8, 8, 5, 15, 12

Offset: 1

Reinhard Zumkeller, Jul 13 2003

A002473(n) is a term of A085153 iff a(n)=1.

			a(23) = 3 as A002473(23 + 1) - A002473(23) = 35 - 32 = 3. - _David A. Corneth_, Mar 31 2021

David A. Corneth, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Smooth Number

Cf. A002473, A061987, A085153.

smooth7Q[n_] := n == Times@@({2, 3, 5, 7}^IntegerExponent[n, {2, 3, 5, 7}]);
Select[Range[1000], smooth7Q] // Differences (* Jean-François Alcover, Oct 17 2021 *)

a(n) = A002473(n+1) - A002473(n).

A086291 Maximal exponent in prime factorization of 7-smooth numbers.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 2, 1, 1, 4, 2, 2, 1, 3, 2, 3, 2, 1, 5, 1, 2, 3, 1, 2, 4, 2, 2, 3, 3, 2, 2, 6, 1, 3, 2, 4, 4, 2, 2, 5, 2, 2, 1, 3, 4, 3, 3, 2, 7, 3, 2, 4, 2, 2, 5, 4, 3, 2, 2, 3, 6, 2, 3, 1, 3, 5, 2, 4, 5, 2, 3, 2, 8, 3, 3, 5, 2, 2, 2, 6, 4, 4, 3, 2, 3, 3, 3, 7, 3, 4, 4, 2, 4, 2, 6, 2

Offset: 1

Reinhard Zumkeller, Jul 15 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002473, A051903, A086289, A086290.

maxExp[1] = 0; maxExp[n_] := Max @@ Last /@ FactorInteger[n]; maxExp/@Select[Range[500], Max[Transpose[FactorInteger[#]][[1]]] <= 7 &] (* Amiram Eldar, Jan 06 2020 *)

a(n) = A051903(A002473(n)).

A086290(n) <= a(n) <= A086289(n).

A161466 Divisors of 10!.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 35, 36, 40, 42, 45, 48, 50, 54, 56, 60, 63, 64, 70, 72, 75, 80, 81, 84, 90, 96, 100, 105, 108, 112, 120, 126, 128, 135, 140, 144, 150, 160, 162, 168, 175, 180, 189, 192, 200, 210, 216, 224

Offset: 1

Reinhard Zumkeller, Jun 10 2009

Finite subsequence of A002473, the 7-smooth numbers;
10! = A000142(10) = 3628800;
tau(3628800) = A000005(3628800) = 270;
last term: a(270) = 3628800.
A027423(10) = 270; a(n) = A027750(A000142(10),n), n = 1..270. - Reinhard Zumkeller, Aug 27 2013

R. Zumkeller, Table of n, a(n) for n = 1..270 (complete sequence)
Index entries for sequences related to divisors of numbers
Index entries for sequences related to factorial numbers

a161466 n = a161466_list !! (n-1)
a161466_list = a027750_row $ a000142 10
-- Reinhard Zumkeller, Aug 27 2013

Divisors[3628800] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2012 *)

divisors(10!) \\ Joerg Arndt, Apr 07 2013

cp's OEIS Frontend

A068185 Number of ways writing n^n as a product of decimal digits of some other number which has no digits equal to 1.

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A068186 a(n) is the largest number whose product of decimal digits equals n^n.

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A068190 Largest number whose digit product equals n; a(n)=0 if no such number exists, e.g., when n has a prime factor larger than 7; no digit=1 is permitted to avoid an infinite number of solutions.

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A069077 Triangular numbers such that the product of digits is also a (positive) triangular number.

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Magma

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A080195 11-smooth numbers which are not 7-smooth.

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A080786 Triangle T(n,k) = number of k-smooth numbers <= n, read by rows.

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A084891 Multiples of 2, 3, 5, or 7, but not 7-smooth.

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