|
| |
|
|
A277173
|
|
Numbers m such that b^sigma(m) == b^phi(m) == b^numdiv(m) == b^m (mod m) for every integer b.
|
|
1
|
|
|
|
1, 2, 6, 12, 24, 60, 120, 126, 240, 420, 480, 504, 672, 780, 1248, 1260, 2340, 2520, 3360, 4680, 5040, 5460, 6240, 6552, 8160, 8736, 9360, 10080, 11424, 16380, 21216, 26208, 27360, 32760, 38304, 43680, 57120, 65520, 71136, 74592, 106080, 131040, 147168, 148512, 171360, 191520, 202464, 325920, 355680, 372960
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
Are terms products products of primes of the form 2^i*3^j + 1, A058383, for some nonnegative i and j? This is true for all terms up to 7.6*10^6. 7600320 is divisible by 29, which isn't of the form 2^j*3^i+1. Up to 10^8, all of the terms are divisible by only 16 distinct prime factors. That is: omega(lcm(all terms up to 10^8)) = 16.
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
6 is a term because for the primes up to 6, (2, 3 and 5), b^sigma(6) == b^phi(6) == b^numdiv(6) == b^6 (mod 6). This is sufficient to prove for all values b up to 6.
|
|
|
MATHEMATICA
|
fQ[n_] := Block[{b = 2, s = DivisorSigma[1, n], e = EulerPhi[n], d = DivisorSigma[0, n]}, While[b < n && PowerMod[b, s, n] == PowerMod[b, e, n] == PowerMod[b, d, n] == PowerMod[b, n, n], b = NextPrime@ b]; b >= n]; lst = {1}; k = 2; While[k < 400000, If[ fQ@ k, AppendTo[lst, k]]; k ++]; lst (* Robert G. Wilson v, Nov 04 2016 *)
|
|
|
PROG
|
(PARI) isk(n, k) = {Mod(k, n)^sigma(n)==Mod(k, n)^n && Mod(k, n)^eulerphi(n)==Mod(k, n)^n && Mod(k, n)^numdiv(n)==Mod(k, n)^n}
is(n) = my(i); forprime(i=2, n, if(isk(n, i)==0, return(0))) ; 1
upto(lim) = my(l=List()); for(n=1, lim, if(is(n), listput(l, n))); l
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
