A050518 An arithmetic progression of at least 6 terms having the same value of phi starts at these numbers.

583200, 1166400, 1749600, 2332800, 2916000, 3499200, 4082400, 4665600, 5248800, 5832000, 6415200, 6998400, 7581600, 8164800, 8748000, 9331200, 9914400, 10497600, 11080800, 11664000, 12247200, 12830400, 13413600, 13996800, 14580000, 15163200, 15746400

Offset

1,1

Comments

From Mauro Fiorentini, Apr 12 2015 (Start):

The following are all the terms between 13413600 and 10^9 with increment <= 1000:

13996800, 14580000, 15163200, 15746400, 16329600, 16912800, 17496000, 18079200, 18662400, 19245600, 65621220, 85731240, 131242440, 165488430, 171462480, 196863660, 257193720, 262484880, 330976860, 342924960, 496465290, 504932430, 544924830, 661953720, 827442150, 892306830, 992930580.

(End)

If phi is constant on the arithmetic progression A = [x, x+d, ..., x+m*d], and k is an integer such that each prime factor of k divides either all members of A or no members of A, then phi is also constant on the arithmetic progression k*A = [x*k, x*k+d*k, ..., x*k+m*(d*k)]. - Robert Israel, Apr 12 2015

The a.p. of 7 terms starting at 1158419010 with increment 210 have the same value of phi. - Robert Israel, Apr 15 2015

a(n) = 583200*n for n <= 112, but a(113) = 65621220. - Robert Israel, May 10 2015

Links

Robert Israel, Table of n, a(n) for n = 1..114 (all the terms <= 6.6*10^7).

Tanya Khovanova, Non Recursions

Eric Weisstein's World of Mathematics, Totient function.

Maple

N:= 10^7: # to get all terms <= N
with(numtheory):
Res:= NULL:
phis:= {seq(phi(i), i=2..N)}:
for m in phis do
   S:= convert(invphi(m), set);
   if nops(S) < 6 then next fi;
   for d from 0 to 4 do
     Sd[d]:= select(t-> (t mod 5 = d), S, d);
     nd:= nops(Sd[d]);
     for i0 from 1 to nd-1 do
       s0:= Sd[d][i0];
       if s0 > N then break fi;
       for i5 from i0+1 to nd do
         s5:= Sd[d][i5];
         incr:= (s5 - s0)/5;
         if {s0+incr, s0+2*incr, s0+3*incr, s0+4*incr} subset S then
           Res:= Res, [s0, incr];
         fi
       od
     od;
   od;
od:
sort([Res], (s, t)->s[1]<t[1]); # gives both A050518 and A050519 entries
map2(op, 1, %); # Robert Israel, Apr 16 2015

Crossrefs

Cf. A000010, A050495, A050496, A050497, A050515-A050520.

The increments are in A050519. The values of phi are in A050520.

Sequence in context: A190682 A231253 A212468 * A249610 A254551 A254558

Adjacent sequences: A050515 A050516 A050517 * A050519 A050520 A050521

Keyword

nonn

Author

Jud McCranie, Dec 28 1999

Status

approved