A342165 - OEIS

%I #19 Mar 04 2021 14:59:30

%S 1,2,3,2,4,3,5,2,4,3,6,5,7,2,4,3,8,6,9,5,7,2,10,4,3,8,6,9,11,5,12,7,2,

%T 10,4,3,13,8,6,9,14,11,15,5,12,7,16,2,10,4,3,13,17,8,6,9,14,11,18,15,

%U 19,5,12,7,16,2,20,10,4,3,21,13,22,17,8,6,9,14

%N A fractal-like sequence: erase the terms that have a prime index, the non-erased terms rebuild the original sequence.

%C To build the sequence, we start with a(1) = 1 and always extend it with the smallest integer not yet used, except in the case where the number is imposed by the constraint (i.e. if the index is nonprime). This fractal-like sequence takes arbitrarily large values.

%H Michael S. Branicky, <a href="/A342165/b342165.txt">Table of n, a(n) for n = 1..10000</a>

%F a(prime(i)) = i + 1. - _Michael S. Branicky_, Mar 04 2021

%e Original sequence: 1,2,3,2,4,3,5,2,4,3,6,5,7,2,4,3,8,6,9,5,7,2,10,4,3

%e Erasing: 1,(2,3,)2,(4,)3,(5,)2,4,3,(6,)5,(7,)2,4,3,(8,)6,(9,)5,7,2,(10,)4,3

%e Non-erased: 1,( )2,(  )3,(  )2,4,3,(  )5,(  )2,4,3,(  )6,(  )5,7,2,(   )4,3

%e The non-erased terms rebuild the original sequence.

%o (Python)

%o from sympy import isprime

%o def aupton(terms):

%o   alst, idx = [1], 1

%o   for n in range(2, terms+1):

%o     if isprime(n): an = max(alst) + 1

%o     else: an, idx = alst[idx], idx + 1

%o     alst.append(an)

%o   return alst

%o print(aupton(78)) # _Michael S. Branicky_, Mar 03 2021

%o (PARI) a(n) = while(n>1 && !isprime(n), n-=primepi(n)); primepi(n)+1; \\ _Kevin Ryde_, Mar 03 2021

%Y Cf. A303845.

%K base,nonn

%O 1,2

%A _Jean-Marc Falcoz_ and _Eric Angelini_, Mar 03 2021

%E a(26) and beyond from _Michael S. Branicky_, Mar 03 2021