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A376471 Lexicographically earliest strictly increasing sequence of numbers whose partial products are all exponentially 2^n-numbers (A138302).

%I A376471 #9 Sep 24 2024 09:29:00
%S A376471 1,2,3,5,6,7,9,11,13,17,19,20,23,25,29,31,37,41,43,47,53,59,61,67,71,
%T A376471 73,77,79,81,83,89,97,101,103,107,109,113,121,127,131,137,139,149,151,
%U A376471 157,163,167,173,179,181,191,193,197,199,208,211,223,227,229,233,239,241
%N A376471 Lexicographically earliest strictly increasing sequence of numbers whose partial products are all exponentially 2^n-numbers (A138302).
%C A376471 All the primes are terms.
%H A376471 Amiram Eldar, <a href="/A376471/b376471.txt">Table of n, a(n) for n = 1..10000</a>
%e A376471 1 * 2 = 2^1 and 1 = 2^0.
%e A376471 1 * 2 * 3 = 6 = 2^1 * 3^1 and 1 = 2^0.
%e A376471 1 * 2 * 3 * 5 * 6 = 180 = 2^2 * 3^2 * 5^1, 1 = 2^0 and 2 = 2^1.
%t A376471 expPow2Q[n_] := AllTrue[FactorInteger[n][[;; , 2]], # == 2^IntegerExponent[#, 2] &]; a[1] = 1; a[n_] := a[n] = Module[{prod = Times @@ Array[a, n - 1], k = a[n - 1] + 1}, While[! expPow2Q[prod*k], k++]; k]; Array[a, 100]
%o A376471 (PARI) ispow2(n) = if(n == 0, 1, n >> valuation(n, 2) == 1);
%o A376471 lista(pindmax) = {my(pmax = prime(pindmax), v = vector(pindmax), f, pind, prd); print1(1, ", "); for(k = 2, pmax, f = factor(k); pind = apply(x -> primepi(x), f[,1]); for(i = 1, #pind, v[pind[i]] += f[i, 2]); if(vecprod(apply(x -> ispow2(x), v)) > 0, print1(k, ", "), for(i = 1, #pind, v[pind[i]] -= f[i, 2])));}
%Y A376471 Disjoint union of A000040 and A376472.
%Y A376471 Similar sequences:
%Y A376471   Sequence      | Partial products are in | Exponents are in
%Y A376471   --------------+-------------------------+------------------------
%Y A376471   A050376       | A037992                 | A000225 \ {0} (2^n-1)
%Y A376471   A089237       | A268335                 | A005408 (odd numbers)
%Y A376471   {1} U A246551 | A246551                 | A000290 \ {0} (squares)
%Y A376471   this sequence | A138302                 | A000079 (powers of 2)
%K A376471 nonn
%O A376471 1,2
%A A376471 _Amiram Eldar_, Sep 24 2024