A355476 a(1)=1. For a(n) a novel term, a(n+1) = A000005(a(n)). For a(n) seen already k > 1 times, a(n+1) = k*a(n).

1, 1, 2, 2, 4, 3, 2, 6, 4, 8, 4, 12, 6, 12, 24, 8, 16, 5, 2, 8, 24, 48, 10, 4, 16, 32, 6, 18, 6, 24, 72, 12, 36, 9, 3, 6, 30, 8, 32, 64, 7, 2, 10, 20, 6, 36, 72, 144, 15, 4, 20, 40, 8, 40, 80, 10, 30, 60, 12, 48, 96, 12, 60, 120, 16, 48, 144, 288, 18, 36, 108, 12, 72, 216, 16

Offset

1,3

Comments

1 is the only number to appear twice, since it has just one divisor.

Consequently 2 is the only prime whose first occurrence is a multiple of prior terms (2*1), all other occurrences of 2 being subsequent to a novel prime (including itself).

All composite numbers appear as multiples of prior terms, and also as the number of divisors of novel terms, whereas all odd prime terms are the result of novel (square) terms whose number of divisors is prime (A009087).

Conjectures: (i) Every integer > 1 appears infinitely many times. (ii) The first occurrences of primes appear in natural order, starting with 2 as described above, and continuing with odd primes a(n) = p, following a(n-1) = 2^(p-1).

Links

Michel Marcus, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Annotated log-log scatterplot of a(n), n = 1..2^16, showing records in red and highlighting terms with predecessors that have appeared for the first time in green.

Example

a(1)=1 a novel term, so a(2)=d(1)=1, then a(3)=2+1=2.
a(15)=24, a novel term, therefore a(16)=d(24)=8, and since this is the second occurrence of 8 (a(10=8), a(17)=2*8=16. Since 16 is a novel term with 5 divisors, a(18)=5, and so on.

Mathematica

nn = 120; c[_] = 0; a[1] = c[1] = 1; Do[If[c[#] == 1, Set[k, DivisorSigma[0, #]]; c[k]++, Set[k, c[#]*#]; c[k]++] &@ a[n - 1]; a[n] = k, {n, 2, nn}]; Array[a, nn] (* Michael De Vlieger, Jul 05 2022 *)

Prog

(PARI) lista(nn) = my(va = vector(nn)); va[1] = 1; for (n=2, nn, my(nb = #select(x->(x==va[n-1]), va)); if (nb == 1, va[n] = numdiv(va[n-1]), va[n] = nb*va[n-1]); ); va; \\ Michel Marcus, Jul 05 2022
(Python)
from sympy import divisor_count
from collections import Counter
from itertools import count, islice
def agen():
    an, c = 1, Counter()
    for n in count(2):
        yield an; c[an] += 1
        an = divisor_count(an) if c[an] == 1 else c[an]*an
print(list(islice(agen(), 75))) # Michael S. Branicky, Jul 06 2022

Crossrefs

Cf. A000005, A000040, A009087.

Sequence in context: A112153 A112154 A112155 * A328932 A341148 A209749

Adjacent sequences: A355473 A355474 A355475 * A355477 A355478 A355479

Keyword

nonn,changed

Author

David James Sycamore, Jul 03 2022

Extensions

More terms from Michel Marcus, Jul 05 2022

Status

approved