

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
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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 nd1 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, A050515A050520.
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



