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A376471
Lexicographically earliest strictly increasing sequence of numbers whose partial products are all exponentially 2^n-numbers (A138302).
2
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, 73, 77, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 208, 211, 223, 227, 229, 233, 239, 241
OFFSET
1,2
COMMENTS
All the primes are terms.
LINKS
EXAMPLE
1 * 2 = 2^1 and 1 = 2^0.
1 * 2 * 3 = 6 = 2^1 * 3^1 and 1 = 2^0.
1 * 2 * 3 * 5 * 6 = 180 = 2^2 * 3^2 * 5^1, 1 = 2^0 and 2 = 2^1.
MATHEMATICA
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]
PROG
(PARI) ispow2(n) = if(n == 0, 1, n >> valuation(n, 2) == 1);
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]))); }
CROSSREFS
Disjoint union of A000040 and A376472.
Similar sequences:
Sequence | Partial products are in | Exponents are in
--------------+-------------------------+------------------------
A050376 | A037992 | A000225 \ {0} (2^n-1)
A089237 | A268335 | A005408 (odd numbers)
{1} U A246551 | A246551 | A000290 \ {0} (squares)
this sequence | A138302 | A000079 (powers of 2)
Sequence in context: A347498 A023884 A135607 * A358221 A026453 A026455
KEYWORD
nonn
AUTHOR
Amiram Eldar, Sep 24 2024
STATUS
approved