A050496 -id:A050496 - cp's OEIS frontend

A050495 Numbers that are the first term of at least one arithmetic progression with 4 or more terms all having the same value of Euler's totient function phi(x).

72, 144, 216, 216, 288, 432, 432, 576, 648, 648, 792, 864, 864, 1080, 1152, 1224, 1296, 1296, 1368, 1446, 1512, 1584, 1656, 1728, 1728, 1944, 1944, 2088, 2160, 2232, 2304, 2376, 2376, 2448, 2592, 2592, 2664, 2736, 2892, 2952, 3024, 3096, 3168

Offset: 1

Jud McCranie, Dec 27 1999

nonn

A number can occur multiple times, corresponding to the number of these arithmetic progressions that it starts. - Robert Israel, Nov 29 2016

			phi(72) = phi(78) = phi(84) = phi(90) = 24, so 72 is a member of the sequence.

Robert Israel, Table of n, a(n) for n = 1..2704

Cf. A000010, A050496, A050497.

N:= 5000: # to get all terms <= N
AP4:= proc(S) local res, n, i1,i4;
   n:= nops(S); res:= NULL;
   for i1 from 1 to n-3 do
     for i4 from i1+3 to n do
       if (S[i1] - S[i4]) mod 3 = 0 and has(S, (2*S[i1]+S[i4])/3) and has(S, (S[i1]+2*S[i4])/3) then res:= res, S[i1]
       fi
   od od;
   [res]
end proc:
Res:= NULL:
for m from 1 to N-1 do
  Res:= Res, op(select(`<=`,AP4(numtheory:-invphi(m)),N));
od:
sort([Res]); # Robert Israel, Nov 29 2016

A050497 Values of phi(x) in arithmetic progressions of at least 4 terms having the same value of phi(x) in A050495.

Original entry on oeis.org

24, 48, 72, 72, 96, 144, 144, 192, 216, 216, 240, 288, 288, 288, 384, 384, 432, 432, 432, 480, 432, 480, 528, 576, 576, 648, 648, 672, 576, 720, 768, 720, 720, 768, 864, 864, 864, 864, 960, 960, 864, 1008, 960, 864, 1056, 1104, 1152, 1152, 1152, 1152, 1248

Offset: 1

Jud McCranie, Dec 27 1999

nonn

			phi(72)=phi(78)=phi(84)=phi(90)=24, so 24 is a member of the sequence.

Sean A. Irvine, Table of n, a(n) for n = 1..1000

Cf. A000010, A050495, A050496.

Missing repeated terms inserted by Sean A. Irvine, Aug 15 2021

A050515 Arithmetic progression of at least 5 terms having the same value of phi start at these numbers.

Original entry on oeis.org

353640, 550860, 583200, 707280, 1060920, 1101720, 1166400, 1414560, 1609260, 1652580, 1749600, 1768200, 2121840, 2203440, 2332800, 2475480, 2754300, 2829120, 2916000, 3182760, 3196860, 3218520, 3305160, 3499200, 3536400

Offset: 1

Jud McCranie, Dec 28 1999

nonn

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

A050520 Values of phi in arithmetic progression of at least 6 terms having the same value of phi in A050518.

Original entry on oeis.org

155520, 311040, 466560, 622080, 777600, 933120, 933120, 1244160, 1399680, 1555200, 1555200, 1866240, 1866240, 1866240, 2332800, 2488320, 2488320, 2799360, 2799360, 3110400, 2799360, 3110400, 3421440, 3732480, 3888000

Offset: 1

Jud McCranie, Dec 28 1999

nonn

The values of phi for terms between 13413600 and 10^9 (see comment on A050518) are 3732480, 3888000, 3732480, 4199040, 3732480, 4354560, 4665600, 4665600, 4976640, 4665600, 14999040, 19595520, 29998080, 44130240, 39191040, 44997120, 58786560, 59996160, 88260480, 78382080, 132390720, 134648640, 145313280, 176520960, 220651200, 237948480, 264781440. - Mauro Fiorentini, Apr 17 2015

Robert Israel, Table of n, a(n) for n = 1..114
Tanya Khovanova, Non Recursions

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

N:= 10^7: # to get all terms <= N in A050518
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, m];
         fi
       od
     od;
   od;
od:
map2(op,2,sort([Res], (s, t)->s[1]Robert Israel, May 10 2015

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

Original entry on oeis.org

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

Jud McCranie, Dec 28 1999

nonn

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

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.

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

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

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]A050518 and A050519 entries
map2(op,1,%); # Robert Israel, Apr 16 2015

A050519 Increments of arithmetic progression of at least 6 terms having the same value of phi in A050518.

Original entry on oeis.org

30, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330, 360, 390, 420, 450, 480, 510, 540, 570, 600, 630, 660, 690, 720, 750, 780, 810, 840, 870, 900, 930, 960, 990, 1020, 1050, 1080, 1110, 1140, 1170, 1200, 1230, 1260, 1290, 1320, 1350, 1380, 1410, 1440, 1470

Offset: 1

Jud McCranie, Dec 28 1999

nonn

The first 112 terms are successive multiples of 30, but this sequence does not coincide with A249674: a(113) = 210. See the Khovanova link and comments in A050518.

The increments <= 1000 for terms of A050518 between 13413600 and 10^9 (see comment on A050518) are 720, 750, 780, 810, 840, 870, 900, 930, 960, 990, 210, 210, 420, 30, 420, 630, 630, 840, 60, 840, 90, 30, 30, 120, 150, 30, 18. - Mauro Fiorentini, Apr 12 2015

Robert Israel, Table of n, a(n) for n = 1..114
Tanya Khovanova, Non Recursions

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

N:= 10^7: # to get all terms <= N in A050518
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]A050518 and A050519 entries map2(op, 2, %); # Robert Israel, Apr 30 2015

More terms from Mauro Fiorentini, Apr 12 2015

A050516 Increments in arithmetic progression of at least 5 terms having the same value of phi in A050515.

Original entry on oeis.org

210, 30, 30, 420, 630, 60, 60, 840, 30, 90, 90, 1050, 1260, 120, 120, 1470, 150, 1680, 150, 1890, 30, 60, 180, 180, 2100, 210, 2310, 210, 2520, 240, 2730, 240, 90, 2940, 270, 270, 3150, 300, 3360, 300, 3570, 330, 3780, 60, 330, 120, 360, 3990, 360, 4200

Offset: 1

Jud McCranie, Dec 28 1999

nonn

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

A050517 Value of phi in arithmetic progression of at least 5 terms having the same value of phi in A050515.

Original entry on oeis.org

80640, 146880, 155520, 161280, 241920, 293760, 311040, 322560, 429120, 440640, 466560, 403200, 483840, 587520, 622080, 564480, 734400, 645120, 777600, 725760, 852480, 858240, 881280, 933120, 806400, 881280, 806400, 933120, 967680

Offset: 1

Jud McCranie, Dec 28 1999

nonn

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

cp's OEIS Frontend

A050495 Numbers that are the first term of at least one arithmetic progression with 4 or more terms all having the same value of Euler's totient function phi(x).

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A050497 Values of phi(x) in arithmetic progressions of at least 4 terms having the same value of phi(x) in A050495.

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A050515 Arithmetic progression of at least 5 terms having the same value of phi start at these numbers.

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A050520 Values of phi in arithmetic progression of at least 6 terms having the same value of phi in A050518.

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Maple

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

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Maple

A050519 Increments of arithmetic progression of at least 6 terms having the same value of phi in A050518.

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A050516 Increments in arithmetic progression of at least 5 terms having the same value of phi in A050515.

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A050517 Value of phi in arithmetic progression of at least 5 terms having the same value of phi in A050515.

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