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A050518 An arithmetic progression of at least 6 terms having the same value of phi starts at these numbers. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified May 23 02:46 EDT 2017. Contains 286909 sequences.