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

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

Adjacent sequences: A376468 A376469 A376470 * A376472 A376473 A376474

Keyword

nonn

Author

Amiram Eldar, Sep 24 2024

Status

approved