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

Author

M. F. Hasler, Sep 17 2012

Keywords

• nonn
• base

Comments

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.

Links

• 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.

Crossrefs

Subsequence of A215876.

Programs

PARI

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)

PARI

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

Formula

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