cp's OEIS Frontend

A217419 Numbers factoring 3^r*11^s whose decimal representations are such that each of the digits 0-9 appears a prime number of times.

%I A217419 #48 Apr 07 2020 23:51:55
%S A217419 5586701408957811048315412506328906443,
%T A217419 4756351651546448722293215379849833244291829764770574840668685771747
%N A217419 Numbers factoring 3^r*11^s whose decimal representations are such that each of the digits 0-9 appears a prime number of times.
%C A217419 The 3-adic valuations for the terms are 29 and 83, and the 11-adic valuations are 22 and 26. For the other main sequences of trios corresponding to nonsquare semiprimes other than 10 and through 39, including separate prime valuations in companion sequences, see cross-references.
%C A217419 While heuristics imply this sequence is infinite, finding a(3) is difficult: A program counting digits for numbers of the given form restricted to those having an even number of decimal digits (with heuristics and initial data practically ruling out counts of 2 for any digit) produced no result for this sequence through 61 values having no more than one digit counted a nonprime number of times. The last of these values was 3^14650*11^3032 (10148 digits in length). - _James G. Merickel_, Dec 11 2013
%C A217419 a(3) if it exists has > 10000 digits. - _James G. Merickel_, Dec 11 2013
%e A217419 3^29 * 11^22 has two each of 2's, 7's and 9's; three each of 3's and 6's; and five each of 0's, 1's, 4's, 5's and 8's.  No smaller number with only 3 and 11 as prime factors has a prime number of each digit 0-9, so a(1) is this value.
%o A217419 (PARI)
%o A217419 {
%o A217419 k=33;a=[99];t=1;while(1,
%o A217419 v=vectorsmall(10);m=k;
%o A217419 while(m,d=m%10;m\=10;v[d+1];next());
%o A217419 f=1;for(i=1,10,if(isprime(v[i])==0,f=0;break()));
%o A217419 if(f,F=factor(k);print1(3"^"F[1,2]"*"11"^"F[2,2]"="k"\n"));
%o A217419 if(11^(t+1)<k,t++;a=concat(a,3*11^t));
%o A217419 k*=3;j=1;for(i=1,t,if(a[i]<k,k=a[i];j=i));a[j]*=3;
%o A217419 next())
%o A217419 }
%Y A217419 Cf. A217854, A217404, A217407, A217410, A217413, A217416, A217422, A217425, A217428, A217431.
%K A217419 nonn,base,bref,more,less,hard
%O A217419 1,1
%A A217419 _James G. Merickel_, Oct 05 2012