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

1,2

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.

Links

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

Formula

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

Example

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.

Prog

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

Crossrefs

Cf. A303845.

Sequence in context: A139094 A159081 A141285 * A286531 A351453 A331280

Adjacent sequences: A342162 A342163 A342164 * A342166 A342167 A342168

Keyword

base,nonn

Author

Jean-Marc Falcoz and Eric Angelini, Mar 03 2021

Extensions

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

Status

approved