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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]

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Last modified August 15 19:07 EDT 2024. Contains 375173 sequences. (Running on oeis4.)