A216855 -id:A216855 - cp's OEIS frontend

A216854 3-smooth numbers (i.e., of the form 2^a*3^b) such that each digit 0-9 occurs a prime number of times.

38294359833110460235776, 17428188652935605013970944, 20655630996071828164706304, 2414716733802996890553286656, 3721583168194563820184970816, 4829433467605993781106573312, 10455147226971833612810452992, 27880392605258222967494541312, 30526789326102084147241549824

Offset: 1

M. F. Hasler, Sep 17 2012

Obviously such numbers need to have at least 10 x 2 = 20 digits, but the smallest solution a(1)=2^56*3^12 has actually 5 digits "3", and 2 of each of the other digits.

Charles R Greathouse IV, Table of n, a(n) for n = 1..40
J. Merickel, Conjecture, Sept. 2012
James Merickel and others, Conjecture, digest of 30 messages in primenumbers Yahoo group, Sep 16, 2012 - Jul 7, 2014.

Subsequence of A215876.

is_A216854(n)=my(t=divrem(n,10),c=vector(10)); until(!t=divrem(t[1],10), c[t[2]+1]++); vecmin(isprime(c)) && return(c)

prDigits(n)=my(d=digits(n), v=vector(10)); for(i=1, #d, v[d[i]+1]++); for(i=1, 10, if(!isprime(v[i]), return(0))); 1
list(lim)=my(v=List(), t); for(a=0, log(lim+.5)\log(3), t=3^a; while(t<=lim, if(prDigits(t), listput(v, t)); t<<=1)); vecsort(Vec(v)) \\ Charles R Greathouse IV, Sep 19 2013

A216854(n) = 2^A216855(n) * 3^A216856(n).

A217051 Smallest k such that the number k^n in its decimal representation has a prime number of copies of the digit d for each d from 0 through 9.

Original entry on oeis.org

3164252736, 258479, 69636, 15165, 3123, 1019, 1315, 815, 307, 205, 475, 347, 143, 151, 272, 1388, 618, 245, 12080, 48, 8635, 23, 287467, 17, 23118, 8440, 48387, 127009, 65457, 70662, 13181, 42911, 4965, 162192, 14460, 226994, 12, 55853, 4104749, 2674855

Offset: 2

James G. Merickel, Sep 25 2012

			23^23 = 20880467999847912034355032910567 has a prime number of copies of each digit (two 1's and two 6's; three 2's, 3's, 4's, 5's, 7's and 8's; and five each of 9's and 0's), and no k < 23 is such that k^23 has this property.

Cf. A216855.

cp's OEIS Frontend

A216854 3-smooth numbers (i.e., of the form 2^a*3^b) such that each digit 0-9 occurs a prime number of times.

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