cp's OEIS Frontend

A134611 Nonprime numbers such that the root mean cube of their prime factors is an integer (where the root mean cube of c and d is ((c^3+d^3)/2)^(1/3)).

%I A134611 #16 Nov 04 2013 21:50:03
%S A134611 1,4,8,9,16,25,27,32,49,64,81,121,125,128,169,243,256,289,343,361,512,
%T A134611 529,625,729,841,961,1024,1331,1369,1512,1681,1849,2048,2187,2197,
%U A134611 2209,2401,2809,3125,3481,3721,4096,4489,4913,5041,5329,6241,6561,6859,6889
%N A134611 Nonprime numbers such that the root mean cube of their prime factors is an integer (where the root mean cube of c and d is ((c^3+d^3)/2)^(1/3)).
%C A134611 The prime factors are taken with multiplicity.
%C A134611 All perfect prime powers (A025475) are included. First term not included in A025475 is a(30) = 1512 = A134613(2) = A134613(1).
%C A134611 Most terms have a last digit of 1 or 9 (i.e., 8326 out of 9000 terms). Mainly, this comes from the fact that all squares of primes are included. Since each prime > 10 has a last digit of 1, 3, 7 or 9, its square has a last digit of 1 or 9. In addition, m-th powers of primes have a last digit of 1, if m == 0 (mod 4), and have a last digit of 1 or 9 if m == 2 (mod 4), and have a 50% chance, roughly, for a last digit of 1 or 9, if m == 1 (mod 4) or m == 3 (mod 4). Since the number of terms <= N which are squares of primes is PrimePi(sqrt(N)) = A000720(sqrt(N)), it follows that the number of terms <= N which have a last digit of 1 or 9 is greater than PrimePi(sqrt(N)). This can be estimated as 2*N^(1/2)/log(N), approximately.
%H A134611 Hieronymus Fischer, <a href="/A134611/b134611.txt">Table of n, a(n) for n = 1..9000</a>
%e A134611 a(6) = 25, since 25 = 5*5 and ((5^3+5^3)/2)^(1/3) = 5.
%e A134611 a(30) = 1512, since 1512 = 2*2*2*3*3*3*7 and ((3*2^3+3*3^3+7^3)/7)^(1/3) = 64^(1/3) = 4.
%o A134611 (PARI) lista(m) = {for (i=2, m, if (! isprime(i), f = factor(i); s = sum (j=1, length(f~), f[j,1]^3*f[j,2]); s /= bigomega(i); if (type(s) == "t_INT" && ispower(s, 3), print1(i, ", "));););}  \\ _Michel Marcus_, Apr 14 2013
%Y A134611 Cf. A001597, A025475, A134333, A134344, A134376.
%Y A134611 Cf. A134600, A134602, A134605, A134608, A134617, A134619, A134621.
%K A134611 nonn
%O A134611 1,2
%A A134611 _Hieronymus Fischer_, Nov 11 2007
%E A134611 Edited by _Hieronymus Fischer_, May 30 2013