A130072 -id:A130072 - cp's OEIS frontend

A130073 Numbers k such that k divides 5^k - 3^k - 2^k = A130072(k).

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 23, 24, 25, 27, 29, 31, 32, 36, 37, 41, 43, 44, 45, 47, 48, 53, 54, 59, 61, 64, 67, 71, 72, 73, 75, 79, 81, 83, 89, 95, 96, 97, 101, 103, 107, 108, 109, 113, 125, 127, 128, 131, 133, 135, 137, 139, 144, 149, 151

Offset: 1

Alexander Adamchuk, May 06 2007

nonn

All primes are the terms of a(n). Quotients A130072(p)/p for p = Prime(n) are listed in A130075(n) = {6,30,570,10830,4422630,93776970,44871187170,1003806502230,...}. p^(k+1) divides A130072(p^k) for prime p = {2,3,5,19} = A130076(n) and all k>0. Nonprimes n such that n divides A130072(n) are listed in A130074(n) = {1,4,6,8,9,12,15,16,18,24,25,27,32,36,44,45,48,54,64,72,75,81,95,96,...} which apparently include all powers p^k of primes p = {2,3,5,19} for k>1 and all powers of numbers of the form 2^k*3^m, 3^k*5^m, 5^k*19^m.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A130072, A130074, A130075, A130076.

Select[Range[1000],IntegerQ[(PowerMod[5,#,# ]-PowerMod[3,#,# ]-PowerMod[2,#,# ])/# ]&]

is(n)=Mod(5,n)^n==Mod(3,n)^n+Mod(2,n)^n \\ Charles R Greathouse IV, Nov 04 2016

A130074 Nonprimes k such that k divides 5^k - 3^k - 2^k = A130072(k).

Original entry on oeis.org

1, 4, 6, 8, 9, 12, 15, 16, 18, 24, 25, 27, 32, 36, 44, 45, 48, 54, 64, 72, 75, 81, 95, 96, 108, 125, 128, 133, 135, 144, 162, 175, 192, 216, 225, 243, 256, 264, 288, 324, 325, 361, 375, 384, 405, 432, 475, 486, 512, 561, 576, 594, 618, 625, 648, 675, 704, 729, 768

Offset: 1

Alexander Adamchuk, May 06 2007

nonn

Numbers k such that k divides A130072(k) are listed in A130073(n) = {1,2,3,4,5,6,7,8,9,11,12,13,15,16,17,18,19,23,24,25,27,29,31,32,36,37,41,43,...}, which includes all primes. a(n) includes nonprimes in A130073(n). p^(k+1) divides A130072(p^k) for prime p = {2,3,5,19} = A130076(n) and all k>0. It appears that a(n) includes all powers p^k of primes p = {2,3,5,19} for k>1 and all powers of numbers of the form 2^k*3^m, 3^k*5^m, 5^k*19^m.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A130072, A130073, A130075, A130076.

Select[Range[10000],!PrimeQ[ # ]&&IntegerQ[(PowerMod[5,#,# ]-PowerMod[3,#,# ]-PowerMod[2,#,# ])/# ]&]

A135158 a(n) = 5^n - 3^n - 2^n.

Original entry on oeis.org

-1, 0, 12, 90, 528, 2850, 14832, 75810, 383808, 1932930, 9705552, 48648930, 243605088, 1219100610, 6098716272, 30503196450, 152544778368, 762810181890, 3814309582992, 19072323542370, 95363943807648, 476826695752770, 2384154405761712, 11920834803510690, 59604362329076928, 298022376554789250

Offset: 0

Omar E. Pol, Nov 21 2007

Essentially the same as A130072. - Zak Seidov, Oct 03 2011

			a(4) = 528 because 5^4 = 625, 3^4 = 81, 2^4 = 16 and 625 - 81 - 16 = 528.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-31,30).

Cf. A000079, A000244, A000351, A001047, A130072.

[5^n-3^n-2^n: n in [0..50]]; // Vincenzo Librandi, Dec 15 2010

A135158:=n->5^n-3^n-2^n; seq(A135158(n), n=0..30); # Wesley Ivan Hurt, Feb 26 2014

lst={};Do[p=5^n-3^n-2^n;AppendTo[lst, p], {n, 0, 7^2}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 19 2008 *)
LinearRecurrence[{10,-31,30}, {-1, 0, 12}, 25] (* or *) Table[5^n - 3^n - 2^n, {n,0,25}] (* G. C. Greubel, Sep 30 2016 *)

a(n)=5^n-3^n-2^n \\ Charles R Greathouse IV, Oct 07 2015

G.f.: ( 1+19*x^2-10*x ) / ( (3*x-1)*(2*x-1)*(5*x-1) ).
a(n) = 10*a(n-1) - 31*a(n-2) + 30*a(n-3). - Zak Seidov, Oct 03 2011
E.g.f.: exp(5*x) - exp(3*x) - exp(2*x). - G. C. Greubel, Sep 30 2016

More terms from Vincenzo Librandi, Dec 15 2010

A130076 Primes p such that p^2 divides 5^p - 3^p - 2^p.

Original entry on oeis.org

2, 3, 5, 19

Offset: 1

Alexander Adamchuk, May 06 2007

For a prime p, p divides A130072(p) = 5^p - 3^p - 2^p. Quotients A130072(p)/p are listed in A130075.
If p^2 divides A130072(p), then p^(k+1) divides A130072(p^k) for every k>0. For p = 19, even 19^(k+2) divides A130072(p^k).
Numbers n such that n divides A130072(n) are listed in A130073. Nonprimes n such that n divides A130072(n) are listed in A130074, which apparently include all powers p^k of primes p = {2,3,5,19} for k>1 and all powers of numbers of the form 2^k*3^m, 3^k*5^m, 5^k*19^m.
No other terms below 10^11. - Max Alekseyev, Dec 06 2010

			p^2 divides A130072(p) = 5^p - 3^p - 2^p for prime p = {2,3,5,19}, quotients A130072(p)/p^2 are {3,10,114,52831921170}.

Cf. A130072, A130073, A130074, A130075.

fQ[p_]:=Mod[PowerMod[5,p,p^2]-PowerMod[3,p,p^2]-PowerMod[2,p,p^2],p^2]==0 (* Robert G. Wilson v, Mar 14 2011 *)

forprime(p=2,1e9,if(Mod(5,p^2)^p==Mod(3,p^2)^p+Mod(2,p^2)^p,print1(p", "))) \\ Charles R Greathouse IV, Mar 14 2011

Edited by Max Alekseyev, Dec 05 2010

A130075 a(n) = (5^p - 3^p - 2^p)/p, where p = prime(n).

Original entry on oeis.org

6, 30, 570, 10830, 4422630, 93776970, 44871187170, 1003806502230, 518297165370030, 6422911941109705770, 150213298561349961630, 1966475018690546370358170, 1109139879321302763891656370

Offset: 1

Alexander Adamchuk, May 06 2007

nonn

p divides 5^p - 3^p - 2^p = A130072(p) for prime p.
p^(k+1) divides A130072(p^k) for prime p = {2,3,5,19} = A130076(n) and all k>0.
2 divides a(n). 3 divides a(n). 5 divides a(n) for n>1. 19 divides a(n) for n>2. 19^2 divides a(n) for n in A091178(n) or prime(n) in A002476.

Cf. A130072, A130073, A130074, A130076, A091178, A002476.

Table[(5^Prime[n]-3^Prime[n]-2^Prime[n])/Prime[n],{n,1,20}]
(5^#-3^#-2^#)/#&/@Prime[Range[20]] (* Harvey P. Dale, May 02 2012 *)

a(n) = (5^prime(n) - 3^prime(n) - 2^prime(n))/prime(n).

a(n) = A130072(prime(n))/prime(n).

A343115 Numbers k such that k^2 divides 5^k - 3^k - 2^k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 12, 15, 19, 24, 95, 361, 876

Offset: 1

Thomas Ordowski, Apr 05 2021

Are there only finitely many such numbers?
Primes in the sequence are A130076.
Next term, if it exists, exceeds 5*10^8. - Jon E. Schoenfield, May 07 2021

Cf. A130072, A130073, A130076.

q:= k-> is(0=5&^k-3&^k-2&^k mod k^2):
select(q, [$1..1000])[];  # Alois P. Heinz, May 07 2021

Select[Range[1000], Mod[PowerMod[2, #, #^2] + PowerMod[3, #, #^2],#^2] == PowerMod[5, #, #^2] &] (* Amiram Eldar, Apr 05 2021 *)

cp's OEIS Frontend

A130073 Numbers k such that k divides 5^k - 3^k - 2^k = A130072(k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A130074 Nonprimes k such that k divides 5^k - 3^k - 2^k = A130072(k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A135158 a(n) = 5^n - 3^n - 2^n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A130076 Primes p such that p^2 divides 5^p - 3^p - 2^p.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A130075 a(n) = (5^p - 3^p - 2^p)/p, where p = prime(n).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A343115 Numbers k such that k^2 divides 5^k - 3^k - 2^k.

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica