A248145 - OEIS

%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,changed

%O 1,1

%A _Vladimir Shevelev_, Oct 02 2014

%E a(43) and beyond from _Michael S. Branicky_, Oct 05 2024