A050515 -id:A050515 - cp's OEIS frontend

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

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.

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

cp's OEIS Frontend

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

Views

Author

Keywords

Crossrefs

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

Views

Author

Keywords

Crossrefs

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Extensions