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%I #42 Nov 10 2021 01:16:27
%S 2,3,2,5,3,7,2,5,3,11,7,13,2,5,3,17,11,19,7,13,2,23,5,3,17,11,19,29,7,
%T 31,13,2,23,5,3,37,17,11,19,41,29,43,7,31,13,47,2,23,5,3,37,53,17,11,
%U 19,41,29,59,43,61,7,31,13,47,2,67,23,5,3,71,37,73,53,17,11
%N If n is prime, a(n) = n, else a(n) = a(n-pi(n)), n >= 2; where pi is the prime counting function A000720.
%C A fractal sequence in which every term is prime. The proper subsequence a(k), for composite numbers k = 4,6,8,9... is identical to the original, and the records subsequence is A000040.
%C Regarding this sequence as an irregular triangle T(m,j) where the rows m terminate with 2 exhibits row length A338237(m). In such rows m, we have a permutation of the least A338237(m) primes. - _Michael De Vlieger_, Nov 04 2021
%H Michael De Vlieger, <a href="/A348907/b348907.txt">Table of n, a(n) for n = 2..10238</a> (as an irregular triangle, rows 1 <= n <= 35 flattened).
%H Michael De Vlieger, <a href="/A348907/a348907.png">Log-log scatterplot of a(n)</a>, for n=1..2^16.
%e 2 is prime so a(2) = 2.
%e 3 is prime so a(3) = 3.
%e 4 is not prime so a(4) = a(4-pi(4)) = 2.
%e 5 is prime so a(5) = 5.
%e 6 is composite so a(6) = a(6-pi(6)) = 3.
%e From _Michael De Vlieger_, Nov 04 2021: (Start)
%e Table showing pi(a(n)) for the first rows m of this sequence seen as an irregular triangle T(m,j). "New" primes introduced for prime n are shown in parentheses:
%e m\j 1 2 3 4 5 6 7 8 9 10 11 A338237(m)
%e ------------------------------------------------------------
%e 1: (1) 1
%e 2: (2) 1 2
%e 3: (3) 2 (4) 1 4
%e 4: 3 2 (5) 4 (6) 1 6
%e 5: 3 2 (7) 5 (8) 4 6 1 8
%e 6: (9) 3 2 7 5 8 (10) 4 (11) 6 1 11
%e ... (End)
%t a[n_]:=If[PrimeQ@n,n,a[n-PrimePi@n]];Array[a,75,2] (* _Giorgos Kalogeropoulos_, Nov 03 2021 *)
%o (PARI) a(n) = if (isprime(n), n, a(n-primepi(n))); \\ _Michel Marcus_, Nov 03 2021
%o (Python)
%o from sympy import isprime
%o def aupton(nn):
%o alst, primepi = [], 0
%o for n in range(2, nn+1):
%o if isprime(n): an, primepi = n, primepi + 1
%o else: an = alst[n - primepi - 2]
%o alst.append(an)
%o return alst
%o print(aupton(76)) # _Michael S. Branicky_, Nov 04 2021
%Y Cf. A000040, A002808, A000720, A010051, A338237.
%K nonn,look
%O 2,1
%A _David James Sycamore_, Nov 03 2021
%E More terms from _Michel Marcus_, Nov 03 2021