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Numbers k such that k and k+1 are both nonsquarefree exponentially odd numbers (A374459).
1

%I #11 Jul 10 2024 09:37:44

%S 135,296,343,351,375,512,999,1160,1375,1431,1592,1624,2079,2295,2375,

%T 2456,2727,2943,3104,3159,3429,3591,3624,3752,3992,4023,4184,4616,

%U 4832,4887,4913,5048,5144,5319,5480,5535,6183,6344,6375,6655,6858,7047,7263,7479,7624

%N Numbers k such that k and k+1 are both nonsquarefree exponentially odd numbers (A374459).

%C The numbers of terms that do not exceed 10^k, for k = 3, 4, ..., are 7, 59, 556, 5539, 55329, 553188, 5531116, 55311354, ... . Apparently, the asymptotic density of this sequence exists and equals 0.005531... .

%H Amiram Eldar, <a href="/A374461/b374461.txt">Table of n, a(n) for n = 1..10000</a>

%e 135 is a term since both 135 = 3^3 * 5 and 136 = 2^3 * 17 are nonsquarefree exponentially odd numbers.

%t q[n_] := q[n] = Module[{e = FactorInteger[n][[;; , 2]]}, AllTrue[e, OddQ] && ! AllTrue[e, # == 1 &]]; Select[Range[10000], q[#] && q[# + 1] &]

%o (PARI) is1(k) = {my(e = factor(k)[, 2]); for(i = 1, #e, if(!(e[i] %2), return(0))); for(i = 1, #e, if(e[i] >1, return(1))); 0;}

%o lista(kmax) = {my(q1 = is1(1), q2); for(k = 2, kmax, q2 = is1(k); if(q1 && q2, print1(k-1, ", ")); q1 = q2);}

%Y Intersection of A068781 and A367695.

%Y Subsequence of A268335 and A374459.

%K nonn

%O 1,1

%A _Amiram Eldar_, Jul 09 2024