A068191 -id:A068191 - cp's OEIS frontend

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

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

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

A198375 Smallest n-digit number whose product of digits is n or 0 if no number exists.

Original entry on oeis.org

1, 12, 113, 1114, 11115, 111116, 1111117, 11111118, 111111119, 1111111125, 0, 111111111126, 0, 11111111111127, 111111111111135, 1111111111111128, 0, 111111111111111129, 0, 11111111111111111145, 111111111111111111137, 0, 0, 111111111111111111111138

Offset: 1

Jaroslav Krizek, Oct 23 2011

			113, 131, and 311 are the 3-digit numbers whose product of digits is 3; 113 is the smallest.

Michael S. Branicky, Table of n, a(n) for n = 1..1000

Cf. A198376 (largest), A002473, A068191.

Table[If[FactorInteger[n][[-1, 1]] > 9, 0, i = (10^n - 1)/9; While[i < 10^n && Times @@ IntegerDigits[i] != n, i++]; If[i == 10^n, 0, i]], {n, 30}] (* T. D. Noe, Oct 24 2011 *)

def A198375(n): return int(str(A198376(n))[::-1])
print([A198375(n) for n in range(1, 25)]) # Michael S. Branicky, Jan 21 2021

a(A068191(n)) = 0 for n >=1.

a(n) <> 0 iff n in { A002473 }. - Michael S. Branicky, Jan 21 2021

A191599 Numbers k that do not divide Ramanujan's tau(k).

Original entry on oeis.org

11, 13, 17, 19, 22, 23, 26, 29, 31, 33, 34, 37, 38, 39, 41, 43, 44, 46, 47, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 68, 69, 71, 73, 74, 76, 77, 78, 79, 82, 83, 85, 86, 87, 89, 93, 94, 95, 97, 99, 101, 102, 103, 104, 106, 107, 109, 110, 111, 113, 114

Offset: 1

Luis H. Gallardo, Jun 08 2011

nonn

This sequence has its first 45 terms in common with A068191.
Subsequence of A068191.
Complement of A063938. - Omar E. Pol, Aug 28 2011

			For n=1, a(1)=11 since 11 does not divide tau(11) = 534612.

T. D. Noe, Table of n, a(n) for n = 1..1000
B. Ramakrishnan and Brundaban Sahu, Rankin-Cohen brackets and Van der Pol-type identities for the Ramanujan's tau function, arXiv:0711.3512 [math.NT], 2007, pp. 14. See Corollary 2.11

Cf. A000594, A068191.

with(numtheory): tn := proc(n) modp(-840*sum(k^4*sigma(k)*sigma(n-k),k=1..n-1),n); end; ser := proc(a,b) local n,lis; lis := []; for n from a to b do if tn(n) <> 0 then lis := [op(lis),n]; fi; od; lis; end;

Select[Range[120], !Divisible[RamanujanTau[#], #]&] (* Jean-François Alcover, Nov 29 2017 *)

isok(k) = ramanujantau(k) % k; \\ Michel Marcus, Aug 14 2021

A198376 Largest n-digit number whose product of digits is n or 0 if no such number exists.

Original entry on oeis.org

1, 21, 311, 4111, 51111, 611111, 7111111, 81111111, 911111111, 5211111111, 0, 621111111111, 0, 72111111111111, 531111111111111, 8211111111111111, 0, 921111111111111111, 0, 54111111111111111111, 731111111111111111111, 0, 0, 831111111111111111111111

Offset: 1

Jaroslav Krizek, Oct 23 2011

			113, 131, and 311 are the 3-digit numbers whose product of digits is 3; 311 is the largest.

Michael S. Branicky, Table of n, a(n) for n = 1..1000

Cf. A198375 (smallest), A002473, A068191.

Table[If[FactorInteger[n][[-1, 1]] > 9, 0, i = (10^n - 1)/9; While[i < 10^n && Times @@ (d = IntegerDigits[i]) != n, i++]; If[i == 10^n, 0, FromDigits[Reverse[d]]]], {n, 30}] (* T. D. Noe, Oct 24 2011 *)

def A198376(n):
  ncopy, p, an = n, 1, ""
  for d in range(9, 1, -1):
    while ncopy%d == 0: ncopy//=d; p *= d; an += str(d)
  if p == n and len(an) <= n: return int(an+'1'*(n-len(an)))
  return 0
print([A198376(n) for n in range(1, 25)]) # Michael S. Branicky, Jan 21 2021

a(A068191(n)) = 0 for n >=1.

a(n) <> 0 iff n in { A002473 }. - Michael S. Branicky, Jan 21 2021

A373641 The number of positive n-digit integers whose digit product is n.

Original entry on oeis.org

1, 2, 3, 10, 5, 36, 7, 120, 45, 90, 0, 924, 0, 182, 210, 3860, 0, 3060, 0, 3800, 420, 0, 0, 61824, 300, 0, 3627, 10584, 0, 25230, 0, 375968, 0, 0, 1190, 441000, 0, 0, 0, 426400, 0, 70602, 0, 0, 44550, 0, 0, 11936496, 1176, 58800, 0, 0, 0, 1491102, 0, 1638560

Offset: 1

Graham Holmes, Jun 12 2024

Trivially, for the four single-digit primes p, a(p)=p.

It's not possible by definition to have a digit product equal to a prime number greater than 10, so a(p)=0 for prime p > 10.

			a(4) = 10: 1114, 1122, 1141, 1212, 1221, 1411, 2112, 2121, 2211, 4111.

Graham Holmes, Table of n, a(n) for n = 1..10000

Cf. A002033, A002473, A068191, A074206, A163767.

b:= proc(n, t, i) option remember; `if`(n=1, 1/t!, `if`(i<2, 0,
      add(b(n/i^j, t-j, i-1)/j!, j=0..padic[ordp](n, i))))
    end:
a:= n-> n!*b(n$2, 9):
seq(a(n), n=1..56);  # Alois P. Heinz, Jun 12 2024

a(n) = 0 <=> n in { A068191 }.
a(n) > 0 <=> n in { A002473 }.
a(n) = A163767(n) for n <= 9.

A376740 Numbers that have at least one two-digit prime factor.

Original entry on oeis.org

11, 13, 17, 19, 22, 23, 26, 29, 31, 33, 34, 37, 38, 39, 41, 43, 44, 46, 47, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 68, 69, 71, 73, 74, 76, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 97, 99, 102, 104, 106, 110, 111, 114, 115, 116, 117

Offset: 1

Kishin Ikemoto, Oct 03 2024

Subsequence of A068191, first differing at A068191(55) = 101 which is not a term here.
Numbers k such that gcd(k,10978895066407230594062391177770267) > 1. - Chai Wah Wu, Nov 18 2024 [The big number is A109819(10) - Alois P. Heinz, Nov 18 2024]
The asymptotic density of this sequence is A051953(A109819(10))/A109819(10) = 1329644281346285477858013527/2807455661493975149742813527 = 0.473611... . - Amiram Eldar, Nov 19 2024

			201 = 3*67 is in this sequence because it has one two-digit prime factor.
202 = 2*101 is not, because neither of them is two-digit.

Kishin Ikemoto, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, order 10978895066407230594062391177770267.

Cf. A051953, A068191, A084891, A109819.

q:= convert(select(isprime, [seq(i,i=11 .. 99, 2)]),`*`):
filter:= n -> igcd(n,q) > 1:
select(filter, [$1..200]); # Robert Israel, Nov 18 2024

A376740Q[k_] := GCD[k, 10978895066407230594062391177770267] > 1;
Select[Range[200], A376740Q] (* Paolo Xausa, Jun 24 2025 *)

is(k) = {forprime(p = 11, 97, if(!(k % p), return(1))); 0;} \\ Amiram Eldar, Nov 19 2024

def ok(n): return any(n%p == 0 for p in [11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97])
print([k for k in range(1, 118) if ok(k)]) # Michael S. Branicky, Oct 15 2024

a(n + A051953(A109819(10))) = a(n) + A109819(10). - Amiram Eldar, Nov 19 2024

cp's OEIS Frontend

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

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A198375 Smallest n-digit number whose product of digits is n or 0 if no number exists.

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A191599 Numbers k that do not divide Ramanujan's tau(k).

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A198376 Largest n-digit number whose product of digits is n or 0 if no such number exists.

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Python

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A373641 The number of positive n-digit integers whose digit product is n.

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Maple

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