A392880 - OEIS

%I #19 Jan 31 2026 21:09:12

%S 0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,0,1,1,0,0,0,1,0,0,

%T 0,2,0,0,0,1,0,0,0,1,0,0,0,2,0,0,0,1,0,1,0,1,0,0,0,1,0,0,0,2,0,0,0,1,

%U 0,0,0,2,0,0,0,1,0,0,0,1,1,0,0,1,0,0,0,1,0,0,0,1,0,0,0,2,0,0,0,1,0,0,0,1,0,0,0,2

%N a(n) is the number of iterations of map k -> A392879(k) needed to reach the fixed point A392878(n), when starting iterating from k=n.

%C The number of steps to reach a term of A048103, when starting from n, and applying succesively the function A392879, which guarantees that on every step the value of A276085 stays invariant.

%C Positions of the first occurrence of n, for n>=0 are: 1, 4, 36, 22500, 2647102500, ..., which seems to suggest a factorization pattern 2^2 * Product_{i>=2} prime(i)^(prime(i)-1).

%H Antti Karttunen, <a href="/A392880/b392880.txt">Table of n, a(n) for n = 1..67500</a>

%H <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>.

%e For n = 5, A392879(5) = A392878(5) = 5, therefore a(5) = 0.

%e For n = 27 = 3^3, A392879(27) = 5, thus a(27) = 1+a(5) = 1.

%e For n = 36 = 2^2 * 3^2, A392879(36) = 27, thus a(36) = 1+a(27) = 2.

%e For n = 823543 = 7^7, A392879(n) = 11 (which is one of the fixed points), thus a(823543) = 1.

%e For n = 367653125 = 5^5 * 7^6, A392879(n) = 7^7 = 823543, thus a(367653125) = 1+a(7^7) = 2.

%e For n = 6361161361558200394723968750000 [= A380459(5005)] = 2^4 * 3^4 * 5^9 * 7^13 * 11^10,

%e              A392879(n) = 3578153265876487722032232421875 = 3^6 * 5^9 * 7^13 * 11^10,

%e            A392879^2(n) =  122707587993020840947607421875 = 5^11 * 7^13 * 11^10,

%e            A392879^3(n) =        615697593513781371538715 = 5^1 * 7^15 * 11^10,

%e            A392879^4(n) =                 109844993185235 = 5^1 * 7^1 * 11^12,

%e            A392879^5(n) =                            5005 = 5 * 7 * 11 * 13,

%e     and as A392879(5005) = 5005, we have reached a fixed point in 5 steps, thus a(n) = 5.

%o (PARI)

%o A129252(n) = { my(pp); forprime(p=2, , pp = p^p; if(!(n%pp), return(p)); if(pp > n, return(1))); };

%o A392879(n) = { my(p=A129252(n), e); if(1==p, return(n), e = valuation(n, p); (n * p^((e%p)-e) * nextprime(1+p)^(e\p))); };

%o A392880(n) = { my(u, c=0); while((u=A392879(n))!=n, n = u; c++); (c); };

%Y Cf. A048103 (positions of 0's), A276085, A276086, A392878, A392879.

%K nonn

%O 1,36

%A _Antti Karttunen_, Jan 31 2026