A342165 A fractal-like sequence: erase the terms that have a prime index, the non-erased terms rebuild the 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, 8, 6, 9, 11, 5, 12, 7, 2, 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, 19, 5, 12, 7, 16, 2, 20, 10, 4, 3, 21, 13, 22, 17, 8, 6, 9, 14

Offset: 1

Author

Jean-Marc Falcoz and Eric Angelini, Mar 03 2021

Keywords

• base
• nonn

Comments

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.

Examples

			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
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
Non-erased: 1,( )2,(  )3,(  )2,4,3,(  )5,(  )2,4,3,(  )6,(  )5,7,2,(   )4,3
The non-erased terms rebuild the original sequence.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A303845.

Programs

PARI

a(n) = while(n>1 && !isprime(n), n-=primepi(n)); primepi(n)+1; \\ Kevin Ryde, Mar 03 2021

Python

from sympy import isprime
def aupton(terms):
  alst, idx = [1], 1
  for n in range(2, terms+1):
    if isprime(n): an = max(alst) + 1
    else: an, idx = alst[idx], idx + 1
    alst.append(an)
  return alst
print(aupton(78)) # Michael S. Branicky, Mar 03 2021

Formula

a(prime(i)) = i + 1. - Michael S. Branicky, Mar 04 2021

Extensions

a(26) and beyond from Michael S. Branicky, Mar 03 2021