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A334872
Number of steps needed to reach either 1 or one of the "Fermi-Dirac primes" (A050376) when starting from n and iterating with A334870.
6
0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 2, 0, 0, 1, 0, 4, 2, 1, 0, 3, 0, 1, 2, 8, 0, 3, 0, 1, 2, 1, 4, 2, 0, 1, 2, 5, 0, 3, 0, 16, 4, 1, 0, 2, 0, 1, 2, 32, 0, 3, 4, 9, 2, 1, 0, 6, 0, 1, 8, 2, 4, 3, 0, 64, 2, 5, 0, 3, 0, 1, 2, 128, 8, 3, 0, 4, 0, 1, 0, 10, 4, 1, 2, 17, 0, 5, 8, 256, 2, 1, 4, 3, 0, 1, 16, 2, 0, 3, 0, 33, 6
OFFSET
1,12
LINKS
FORMULA
If n = 1 or A302777(n) = 1, a(n) = 0, otherwise a(n) = 1 + a(A334870(n)).
For all n >= 1, a(n) <= A334871(n).
PROG
(PARI)
A209229(n) = (n && !bitand(n, n-1));
A302777(n) = A209229(isprimepower(n));
A334870(n) = if(issquare(n), sqrtint(n), my(c=core(n), m=n); forprime(p=2, , if(!(c % p), m/=p; break, m*=p)); (m));
A334872(n) = { my(s=0); while(n>1 && !A302777(n), s++; n = A334870(n)); (s); };
(PARI)
\\ Much faster, A302777 like in above:
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A052126(n) = if(1==n, n, (n/vecmax(factor(n)[, 1])));
A334872(n) = { my(s=0); while(n>1 && !A302777(n), if(issquarefree(n), return(s+A048675(A052126(n)))); if(issquare(n), s++; n = sqrtint(n), s += A048675(core(n)); n /= core(n))); (s); };
CROSSREFS
Cf. A050376 (positions of zeros after 1), A302777, A334859, A334865, A334870, A334871.
Sequence in context: A101661 A322989 A322823 * A263844 A079644 A072705
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jun 08 2020
STATUS
approved