A217409 -id:A217409 - cp's OEIS frontend

A217407 Numbers of the form 3^r * 5^s whose decimal representation has a prime number of each digit 0-9.

38171039656829610443115234375, 129892841018736362457275390625, 1766298261467341813095601383375, 83480063729486358039093017578125, 715350795894273434303718560266875, 172661884789704345166683197021484375, 65186341275865666700926353804318984375, 5280093643345119002775034658149837734375

Offset: 1

James G. Merickel, Oct 02 2012

This sequence in particular is motivated by the coincidence that both (2^41)*(3^43) and (3^43)*(5^47) have prime numbers of each digit.

			The first term here is (3^35)*(5^17), corresponding to A217408(1)=35 and A217409(1)=17. Its decimal representation has two each of 0's, 2's, 7's, 8's and 9's; three each of 4's, 5's and 6's; and 5 each of 1's and 3's.

Robert Israel, Table of n, a(n) for n = 1..11

Subsequence of A215876.

Cf. A216854, A217408, A217409, A217404, A217410, A217413, A217416, A217419, A217422, A217425, A217428, A217431.

N:= 10^100: # to get all terms <= N
filter:= proc(n) local L,P,d;
  L:= convert(n,base,10);
  P:= Vector(10);
  for d in L do P[d+1]:= P[d+1]+1 od:
  andmap(isprime,P);
end proc:
sort(select(filter, [seq(seq(3^r*5^s, r=0..floor(log[3](N/5^s))),s=0..floor(log[5](N)))])); # Robert Israel, May 08 2017

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(5), t=5^a; while(t<=lim, if(prDigits(t), listput(v, t)); t*=3)); vecsort(Vec(v)) \\ Charles R Greathouse IV, Sep 19 2013

a(n) = 3^A217408(n) * 5^A217409(n).

More terms from Robert Israel, May 08 2017

A217408 3-adic valuation of A217407.

Original entry on oeis.org

35, 20, 59, 42, 63, 5, 69, 73, 21, 43, 71, 1149

Offset: 1

James G. Merickel, Oct 02 2012

See main sequence for rationale.

			The first number that has only 3 and 5 as prime factors and has prime counts of each digit 0-9 in its decimal representation is (3^35)*(5^17), so, corresponding to that being A217407(1), this sequence's first term is 35.

Cf. A217407, A217409.

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(5), t=5^a; while(t<=lim, if(prDigits(t), listput(v, t)); t*=3)); apply(k -> valuation(k,3), vecsort(Vec(v))) \\ Charles R Greathouse IV, Sep 19 2013

cp's OEIS Frontend

A217407 Numbers of the form 3^r * 5^s whose decimal representation has a prime number of each digit 0-9.

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