%I #45 Oct 05 2024 08:56:28
%S 2,1,1,1,1,1,1,1,1,1,1,3,1,1,2,1,1,1,1,1,2,1,2,1,1,1,3,1,1,2,1,1,1,7,
%T 1,1,1,2,1,1,1,2,6,1,5,11,7,1,1,1,2,1,1,1,2,1,1,348,2,20,30,453,2,1,2,
%U 3,17,1,219,1,2,4,10,1,2,1,1,46,1303,4,2,1,2,2,1
%N Consider the partition of the positive odd integers into minimal blocks such that concatenation of numbers in each block is a number of the form 3^k*prime, k>=0. Sequence lists numbers of odd integers in the blocks.
%C 3^m, m>=1, is of the considered form 3^k*prime, k=m-1>=0, prime=3.
%C The first blocks of the partition are |1,3|,|5|,|7|,|9|,|11|,|13|,|15|,|17|,|19|,|21|,|23|,|25,27,29|,|31|,|33|,|35,37|,...
%e The 12th block of partition is |25,27,29|, since we have 25=5^2, 2527=7*19^2, 252729=3^2*28081, and only the last number is of the required form. So a(12)=3.
%o (Python)
%o from gmpy2 import is_prime
%o from itertools import count, islice
%o def c(n):
%o if n < 3: return False
%o while n%3 == 0: n //= 3
%o return n == 1 or is_prime(n)
%o def agen(): # generator of terms
%o i = 1
%o while True:
%o s, an = str(i), 1
%o while not c(t:=int(s)): i += 2; s += str(i); an += 1
%o yield an
%o i += 2
%o print(list(islice(agen(), 78))) # _Michael S. Branicky_, Oct 05 2024
%Y Cf. A103899, A248146.
%K nonn,base
%O 1,1
%A _Vladimir Shevelev_, Oct 02 2014
%E a(43) and beyond from _Michael S. Branicky_, Oct 05 2024