A074606 -id:A074606 - cp's OEIS frontend

A120978 2n+5^n-3^n.

0, 4, 20, 104, 552, 2892, 14908, 75952, 384080, 1933460, 9706596, 48651000, 243609208, 1219108828, 6098732684, 30503229248, 152544843936, 762810312996, 3814309845172, 19072324066696, 95363944856264, 476826697849964

Offset: 0

Mohammad K. Azarian, Aug 19 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10, -32, 38, -15).

Cf. A074606, A005058.

Table[2 n + 5^n - 3^n, {n, 0, 30}] (* or *) LinearRecurrence[{10, -32, 38, -15},{0, 4, 20, 104}, 30] (* Harvey P. Dale, Jun 15 2011 *)
CoefficientList[Series[2 x (2 - 10 x + 16 x^2)/((1 - x)^2(1 - 5 x)(1 - 3 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Feb 23 2013 *)

a(0)=0, a(1)=4, a(2)=20, a(3)=104, a(n) = 10*a(n-1)-32*a(n-2)+38*a(n-3)-15*a(n-4) [Harvey P. Dale, Jun 15 2011]

G.f.: 2*x*(2-10*x+16*x^2)/((1-x)^2*(1-5*x)*(1-3*x)). - Vincenzo Librandi, Feb 23 2013

Edited by Ray Chandler, Sep 06 2006

A120990 5^n-3^n-2n.

Original entry on oeis.org

0, 0, 12, 92, 536, 2872, 14884, 75924, 384048, 1933424, 9706556, 48650956, 243609160, 1219108776, 6098732628, 30503229188, 152544843872, 762810312928, 3814309845100, 19072324066620, 95363944856184, 476826697849880

Offset: 0

Mohammad K. Azarian, Aug 19 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A074606, A005058.

CoefficientList[Series[4 x^2 (3 - 7 x)/((1 - x)^2 (1 - 5 x)(1 - 3 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Feb 23 2013 *)

G.f.: 4*x^2*(3-7*x)/((1-x)^2*(1-5*x)*(1-3*x)) . - Vincenzo Librandi, Feb 23 2013

A346992 Numbers occurring as divisors of 3^k + 5^k.

Original entry on oeis.org

1, 2, 4, 8, 13, 17, 19, 23, 26, 29, 31, 34, 37, 38, 41, 46, 47, 53, 58, 62, 73, 74, 76, 79, 82, 83, 89, 92, 94, 97, 101, 106, 107, 113, 124, 137, 139, 146, 149, 151, 152, 157, 158, 166, 167, 169, 178, 184, 188, 193, 194, 199, 202, 211, 212, 214, 221, 226, 227

Offset: 1

Hugo Pfoertner, Aug 11 2021

nonn

If n is a term, then so are all divisors of n. - Robert Israel, Dec 08 2022

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A074606, A081186.

filter:= proc(n) local v;
   if igcd(n,15) <> 1 then return false fi;
   q:= 5/3 mod n;
   traperror(NumberTheory:-ModularLog(-1,q,n)) <> lasterror
end proc:
filter(1):= true:
select(filter, [$1..300]); # Robert Israel, Dec 08 2022

A045585 Numbers k that divide 5^k + 3^k.

Original entry on oeis.org

1, 2, 34, 578, 4658, 9826, 79186, 167042, 638146, 1346162, 2839714, 5109826, 10848482, 22884754, 48275138, 86867042, 87426002, 184424194, 389040818, 459469778, 700046162, 820677346, 1476739714, 1486242034, 3135211298, 5605479122, 6613693906, 7810986226, 11900784754, 11977362274

Offset: 1

David W. Wilson

nonn

Cf. A074606.

Select[Range[10^6], Divisible[PowerMod[3, #, #] + PowerMod[5, #, #], #] &] (* Amiram Eldar, Oct 23 2021 *)

a(26)-a(30) from Amiram Eldar, Oct 23 2021

A268511 Odd integers n such that 3^n + 5^n = x^2 + y^2 (x and y integers) is solvable.

Original entry on oeis.org

1, 5, 13, 17, 29, 89, 109, 149, 157, 193, 373

Offset: 1

Altug Alkan, Feb 06 2016

Corresponding 3^n + 5^n values are 8, 3368, 1222297448, 763068593288, 186264583553473068008, ...

445 <= a(12) <= 509. 509, 661, 709 are terms. - Chai Wah Wu, Jul 22 2020

			1 is a term because 3^1 + 5^1 = 8 = 2^2 + 2^2.
5 is a term because 3^5 + 5^5 = 3368 = 2^2 + 58^2.
13 is a term because 3^13 + 5^13 = 1222297448 = 4118^2 + 34718^2.

Cf. A001481, A074606.

Select[Range[1, 110, 2], Resolve@ Exists[{x, y}, Reduce[3^# + 5^# == (x^2 + y^2), {x, y}, Integers]] &] (* Michael De Vlieger, Feb 07 2016 *)

is(n) = #bnfisintnorm(bnfinit(z^2+1), n);
for(n=1, 1e3, if(n%2==1 && is(3^n + 5^n), print1(n, ", ")));

from sympy import factorint
A268511_list = []
for n in range(1,50,2):
    m = factorint(3**n+5**n)
    for d in m:
        if d % 4 == 3 and m[d] % 2:
            break
    else:
        A268511_list.append(n) # Chai Wah Wu, Dec 26 2018

a(8)-a(9) from Giovanni Resta, Apr 10 2016

a(10)-a(11) from Chai Wah Wu, Jul 22 2020

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A120978 2n+5^n-3^n.

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A120990 5^n-3^n-2n.

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A346992 Numbers occurring as divisors of 3^k + 5^k.

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A045585 Numbers k that divide 5^k + 3^k.

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A268511 Odd integers n such that 3^n + 5^n = x^2 + y^2 (x and y integers) is solvable.

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