A379757 - OEIS

A379757

a(n) = a(n-1) + 1 with two exceptions: if a(n-1) is prime, a(n) = a(n-2) + a(n-1), or if a(n-1) is a power, a(n) = a(n-1) / (root factor), with initial three terms are 0, 1, 2.

0, 1, 2, 3, 5, 8, 4, 2, 6, 7, 13, 20, 21, 22, 23, 45, 46, 47, 93, 94, 95, 96, 97, 193, 290, 291, 292, 293, 585, 586, 587, 1173, 1174, 1175, 1176, 1177, 1178, 1179, 1180, 1181, 2361, 2362, 2363, 2364, 2365, 2366, 2367, 2368, 2369, 2370, 2371, 4741, 4742, 4743

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OFFSET

1,3

COMMENTS

The construction rules are very basic, but lead to somewhat surprising results. Terms that are perfect powers are extremely rare (only n=6,7 so far). Additionally, the sequence is nearly all composites. Comparing to A000045, eight early distinct terms are in common, but it is unclear when another intersection is seen.

LINKS

Table of n, a(n) for n=1..54.

FORMULA

Conjecture: log(a(n)) ~ k*sqrt(n).

EXAMPLE

We know a(1)=0, a(2)=1, a(3)=2. Since a(3) is prime, a(4)=a(2)+a(3)=3. Since a(4) is prime, a(5)=a(3)+a(4)=5. Similarly, a(6)=a(4)+a(5)=8. Since a(6) is a perfect power, a(7) = a(6)/2 since 8=2^3. Since a(7)=4 is another perfect power, a(8)=4/2=2. Since a(8) is prime, a(9)=a(7)+a(8)=6. For clarity, if a(n-1) = r^k, then a(n) = a(n-1)/r.

MATHEMATICA

a[n_] := a[n] = If[n < 4, n-1, If[PrimeQ[a[n-1]], a[n-1] + a[n-2], If[(g = GCD @@ FactorInteger[a[n-1]][[;; , 2]]) > 1, a[n-1]^(1 - 1/g), a[n-1] + 1]]]; Array[a, 54] (* Amiram Eldar, Apr 10 2025 *)

CROSSREFS

Cf. A000045, A001597, A089580.

Sequence in context: A246558 A307638 A098906 * A007887 A105472 A030132

Adjacent sequences: A379754 A379755 A379756 * A379758 A379759 A379760

KEYWORD

nonn,easy

AUTHOR

Bill McEachen, Jan 02 2025

STATUS

approved