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A175177
Conjectured number of numbers for which the iteration x -> phi(x) + 1 terminates at prime(n). Cardinality of rooted tree T_p (where p is n-th prime) in Karpenko's book.
5
2, 3, 4, 9, 2, 31, 6, 4, 2, 2, 2, 11, 24, 41, 2, 2, 2, 57, 2, 2, 58, 2, 2, 6, 17, 4, 2, 2, 39, 67, 2, 2, 2, 2, 2, 2, 25, 4, 2, 2, 2, 158, 2, 61, 2, 2, 2, 2, 2, 2, 54, 2, 186, 2, 10, 2, 2, 2, 18, 8, 2, 2, 2, 2, 96, 2, 2, 18, 2, 6, 15, 2, 2, 2, 2, 2, 2, 44, 34, 6, 2, 16, 2, 105, 2, 2, 60, 5, 4, 2, 2, 2, 4
OFFSET
1,1
REFERENCES
Richard K. Guy, Unsolved Problems in Number Theory, Third Edition, Springer, New York 2004. Chapter B41, Iterations of phi and sigma, page 148.
A. S. Karpenko, Lukasiewicz's Logics and Prime Numbers, (English translation), 2006. See Table 2 on p.125 ff.
A. S. Karpenko, Lukasiewicz's Logics and Prime Numbers, (Russian), 2000.
LINKS
EXAMPLE
a(3) = 4 because x = { 5, 8, 10, 12 } are the 4 numbers from which the iteration x -> phi(x) + 1 terminates at prime(3) = 5.
a(4) = 8 because x = { 7, 9, 14, 15, 16, 18, 20, 24, 30 } are the 9 numbers from which the iteration x -> phi(x) + 1 terminates at prime(4) = 7.
PROG
(PARI)
iterat(x) = {my(k, s); if ( isprime(x), return(x)); s=x;
for (k=1, 1000000000, s=eulerphi(s)+1; if(isprime(s), return(s)));
return(s); }
check(y, endrange) = {my(count, start); count=0;
for(start=1, endrange, if(iterat(start)==y, count++; ));
return(count); }
for (n=1, 93, x=prime(n); print1(check(x, 1000000), ", "))
\\ Hugo Pfoertner, Sep 23 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Artur Jasinski, Mar 01 2010
EXTENSIONS
Name clarified by Hugo Pfoertner, Sep 23 2017
STATUS
approved