

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, 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
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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(n1) = 2^(p1).


LINKS



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[n1]), va)); if (nb == 1, va[n] = numdiv(va[n1]), va[n] = nb*va[n1]); ); 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


CROSSREFS



KEYWORD

nonn,changed


AUTHOR



EXTENSIONS



STATUS

approved



