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A050518
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An arithmetic progression of at least 6 terms having the same value of phi starts at these numbers.
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3
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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
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1,1
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COMMENTS
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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
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LINKS
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MAPLE
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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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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