A050519 -id:A050519 - cp's OEIS frontend

A249674 a(n) = 30*n.

0, 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

Offset: 0

Kaylan Purisima, Nov 03 2014

Numbers divisible by 2, 3 and 5. - Robert Israel, Nov 19 2014

a(n) is the maximum score of a 10-pin n-frame bowling game and the maximum score of an n-pin 10-frame bowling game, given the rules: a strike is worth the number of pins in each frame plus the number of pins knocked down by the next two balls (except in the last frame), a spare is worth the number of pins in each frame plus the number of pins knocked down by the next ball (except in the last frame), and if a strike or spare is earned in the last frame then the player must continue to throw balls until they have thrown 3 balls in the last frame. - Iain Fox, Mar 02 2018

			a(7) = 7 * 30 = 210.

Iain Fox, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Bowling.
Wikipedia, Ten-pin bowling.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A005843, A008585, A008587, A008588, A008592, A008597, A050519, A169823.

[30*n : n in [0..50]]; // Wesley Ivan Hurt, Nov 18 2014

A249674:=n->30*n: seq(A249674(n), n=0..50); # Wesley Ivan Hurt, Nov 18 2014

30*Range[0, 59] (* Alonso del Arte, Nov 18 2014 *)

vector(100,n,30*(n-1)) \\ Derek Orr, Nov 18 2014

first(n) = Vec(30*x/(x-1)^2 + O(x^n), -n) \\ Iain Fox, Mar 02 2018

G.f.: 30*x/(x-1)^2; a(n) = 2*a(n-1) - a(n-2). - Wesley Ivan Hurt, Nov 18 2014
a(n) = 2*A008597(n) = 3*A008592(n) = 5*A008588(n) = 6*A008587(n) = 10*A008585(n) = 15*A005843(n). - Omar E. Pol, Nov 24 2014
From Elmo R. Oliveira, Apr 08 2025: (Start)
E.g.f.: 30*x*exp(x).
a(n) = A169823(n)/2. (End)

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

cp's OEIS Frontend

A249674 a(n) = 30*n.

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