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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. 10
5586701408957811048315412506328906443, 4756351651546448722293215379849833244291829764770574840668685771747 (list; graph; refs; listen; history; text; internal format)



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.

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

a(3) if it exists has > 10000 digits. - James G. Merickel, Dec 11 2013


Table of n, a(n) for n=1..2.


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.




k=33; a=[99]; t=1; while(1,

v=vectorsmall(10); m=k;

while(m, d=m%10; m\=10; v[d+1]; next());

f=1; for(i=1, 10, if(isprime(v[i])==0, f=0; break()));

if(f, F=factor(k); print1(3"^"F[1, 2]"*"11"^"F[2, 2]"="k"\n"));

if(11^(t+1)<k, t++; a=concat(a, 3*11^t));

k*=3; j=1; for(i=1, t, if(a[i]<k, k=a[i]; j=i)); a[j]*=3;




Cf. A217854, A217404, A217407, A217410, A217413, A217416, A217422, A217425, A217428, A217431.

Sequence in context: A003847 A095466 A115555 * A246289 A095468 A115556

Adjacent sequences:  A217416 A217417 A217418 * A217420 A217421 A217422




James G. Merickel, Oct 05 2012



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