A094475 -id:A094475 - cp's OEIS frontend

A074600 a(n) = 2^n + 5^n.

2, 7, 29, 133, 641, 3157, 15689, 78253, 390881, 1953637, 9766649, 48830173, 244144721, 1220711317, 6103532009, 30517610893, 152587956161, 762939584197, 3814697527769, 19073486852413, 95367432689201, 476837160300277

Offset: 0

Robert G. Wilson v, Aug 25 2002

Digital root of a(n) is A010697(n). - Peter M. Chema, Oct 24 2016

Miller, Steven J., ed. Benford's Law: Theory and Applications. Princeton University Press, 2015. See page 14.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
D. Suprijanto, I. W. Suwarno, Observation on Sums of Powers of Integers Divisible by 3k-1, Applied Mathematical Sciences, Vol. 8, 2014, no. 45, pp. 2211-2217.
Index entries for linear recurrences with constant coefficients, signature (7,-10).

Cf. A000051, A034472, A052539, A034474, A062394, A034491, A062395, A062396, A007689, A063376, A063481, A074601-A074624, A010697, A094475, A337429.

[2^n + 5^n: n in [0..35]]; // Vincenzo Librandi, Apr 30 2011

Table[2^n + 5^n, {n, 0, 25}]
LinearRecurrence[{7,-10},{2,7},30] (* Harvey P. Dale, May 09 2019 *)

a(n)=2^n+5^n \\ Charles R Greathouse IV, Sep 24 2015

a(n) = 5*a(n-1)-3*2^(n-1) = 7*a(n-1)- 10*a(n-2). [Corrected by Zak Seidov, Oct 24 2009]

G.f.: 1/(1-2*x)+1/(1-5*x). E.g.f.: e^(2*x)+e^(5*x). - Mohammad K. Azarian, Jan 02 2009

A294133 Sorted list of prime factors of numbers of the form 5^(2^m) + 2^(2^m) with m >= 0.

Original entry on oeis.org

7, 17, 29, 97, 193, 257, 641, 12289, 22993, 65537, 102593, 115201, 152833, 211457, 993793, 5189633, 26411009, 79280897, 93847553, 167772161, 230686721, 1364951041, 1573071713, 3221225473, 5488091137, 186678460417, 206158430209, 274568286337

Offset: 1

Arkadiusz Wesolowski, Oct 23 2017

nonn

Primes p other than 3 such that the multiplicative order of 5/2 (mod p) is a power of 2.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..34
Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), no. 221, pp. 441-446.
Anders Björn and Hans Riesel, Table errata to “Factors of generalized Fermat numbers”, Math. Comp. 74 (2005), no. 252, p. 2099.
Anders Björn and Hans Riesel, Table errata 2 to "Factors of generalized Fermat numbers", Math. Comp. 80 (2011), pp. 1865-1866.

Cf. A094475, A294132, A294134, A294135, A294136.

print1(7, ", "); forprime(p=17, 274568286337, z=znorder(Mod(5/2, p)); if(2^ispower(z)==z, print1(p, ", ")));

A337429 a(n) is the largest prime factor of 2^n + 5^n.

Original entry on oeis.org

2, 7, 29, 19, 641, 41, 541, 1597, 22993, 397, 5521, 303293, 380881, 25117, 210466621, 508771, 1573071713, 108991369171, 1343341, 2724783836059, 39558401, 2525293, 4807441, 215038823, 1173553, 61001, 16463734208221, 3813697527769, 58116853330557841, 327866809, 99901

Offset: 0

Hugo Pfoertner, Sep 23 2020

nonn

Amiram Eldar, Table of n, a(n) for n = 0..387 (terms 0..200 from Hugo Pfoertner)

Cf. A006530, A074600, A094474, A094475, A122118.

a[n_] := FactorInteger[2^n + 5^n][[-1, 1]]; Array[a, 31, 0] (* Amiram Eldar, Mar 30 2023 *)

for(n=0,30,my(p=2^n+5^n);print1(vecmax(factor(p)[,1]),", "))

a(n) = A006530(A074600(n)).

A122118 Least prime factor of 2^n + 5^n.

Original entry on oeis.org

2, 7, 29, 7, 641, 7, 29, 7, 17, 7, 29, 7, 641, 7, 29, 7, 97, 7, 29, 7, 641, 7, 29, 7, 17, 7, 29, 7, 641, 7, 29, 7, 193, 7, 29, 7, 73, 7, 29, 7, 17, 7, 29, 7, 641, 7, 29, 7, 97, 7, 29, 7, 641, 7, 29, 7, 17, 7, 29, 7, 641, 7, 29, 7, 274568286337, 7, 29, 7, 137, 7, 29, 7, 17, 7, 29, 7, 457

Offset: 0

Zak Seidov, Oct 19 2006

nonn

a(n_odd)=7, a(n=2+4k,k=0,1,...)=29, a(64)=274568286337 is unusually large.

Hans Havermann, Table of n, a(n) for n = 0..1023 (terms 0..319 from Antti Karttunen)

Cf. A020639, A074600 (2^n + 5^n), A094475 (primes of form 2^n + 5^n), A122119, A337429.

Cf. also A094473.

Table[FactorInteger[2^n+5^n][[1,1]],{n,0,80}] (* or *) Riffle[Table[ FactorInteger[2^n+5^n][[1,1]],{n,0,80,2}],7] (* The second program is faster *) (* Harvey P. Dale, Mar 02 2015 *)

A122118(n) = { my(k=(2^n+5^n)); forprime(p=if(64==n,274568286337,2),k,if(!(k%p),return(p))); }; \\ Antti Karttunen, Nov 02 2018

a(n) = A020639(A074600(n)). - Antti Karttunen, Nov 02 2018

A122119 Least prime factor of 2^(2^n) + 5^(2^n).

Original entry on oeis.org

7, 29, 641, 17, 97, 193, 274568286337, 257, 211457, 12289

Offset: 0

Zak Seidov, Oct 19 2006

a(11) = 5189633, a(12) > 10^9, a(13) > 10^9, a(14) = 26411009, a(15) = 65537, a(16) > 10^9, a(17) > 10^9, a(18) = 93847553, a(19) > 10^9, a(20) = 230686721. - Robert Price, May 01 2013
10^15 < a(10) <= 1431459829606245207173905329679749121. - Max Alekseyev, Jun 28 2013
a(n) == 1 (mod 2^(n+1)). - Max Alekseyev, Jun 23 2013
The sequence of highest powers of 2 dividing a(n)-1 is {0, 2, 7, 4, 5, 6, 7, 8, 9, 12, ?, 12, ?, ?, 16, 16, ?, ?, 19, ?, 22}.  a(2)-1 = 640 = 5*2^7, a(3)-1 = 16 = 2^4, a(4)-1 = 96 = 3*2^5, a(5)-1 = 192 = 3*2^6, a(6)-1 = 274568286336 = 3*109*6559831*2^7, a(7)-1 = 256 = 2^8,.... - Robert Price, May 02 2013
a(12) = 5488091137, a(13) = 1364951041. - Chai Wah Wu, Jul 16 2019
All the known Fermat primes > 5 are in the sequence. a(22) = 167772161, a(26) = 1352914698241, a(28) = 11726871330817, a(29) = 3221225473, a(33) = 206158430209, a(38) = 19340409532579841, a(40) = 46179488366593, a(41) = 87930143896502273, a(46) = 19703248369745921, a(48) = 26747441136906797057, a(50) = 38280596832649217, a(57) = 639871435056800071681, a(66) = 1328165573307087716353, a(71) = 188894659314785808547841, a(75) = 441106808431887820120915969, a(77) = 1272917285561380533496310661121, a(83) = 380028249247497910327235837953, a(91) = 34662321099990647697175478273, a(107) = 48028745941447155563907091045285889, a(110) = 77884452878022414427957444938301441. - Chai Wah Wu, Oct 14 2019

FactorDB, Status of 2^(2^n)+5^(2^n).
Mersenneforum, C680 of A122119, post by user swellman, 2013-06-17.

Cf. A074600 (2^n + 5^n), A094475 (primes of form 2^n+5^n).

Table[FactorInteger[2^(2^n)+5^(2^n)][[1,1]],{n,0,7}] (* James C. McMahon, Oct 26 2024 *)

A386618 Primes of the form 2^k + 13^k.

Original entry on oeis.org

2, 173, 815730977

Offset: 1

Vincenzo Librandi, Aug 17 2025

If 13^k + 2^k is prime then k is either 0 or a power of 2. The corresponding values of k for a(1)-a(4) are 0, 2, 8 and 512. The fourth value is too long to enter.

Vincenzo Librandi, Table of n, a(n) for n = 1..4

Cf. A082101, A094475, A094476, A094477, A176946, A176947.

[a: n in [0..200] | IsPrime(a) where a is 13^n+2^n ];

Select[Table[2^n+13^n,{n,0,600}],PrimeQ]

A122116 Numbers k such that 2^k+5^k is semiprime.

Original entry on oeis.org

3, 6, 8, 12, 14, 16, 17, 19, 28, 38, 47, 52, 64, 101, 274, 466, 1709, 2539

Offset: 1

Zak Seidov, Oct 19 2006

Cf. A074600 (2^n + 5^n), A094475 (primes of form 2^k+5^k).

IsSemiprime:=func< n|&+[ k[2]: k in Factorization(n)] eq 2 >;[ n: n in [2..100]|IsSemiprime(2^n+5^n)]; // Vincenzo Librandi, Dec 16 2010

Select[Range[100],PrimeOmega[2^#+5^#]==2&] (* James C. McMahon, Oct 26 2024 *)

a(14)-a(15) from D. S. McNeil, Dec 20 2010

a(16)-a(18) from Sean A. Irvine, Nov 13 2024

cp's OEIS Frontend

A074600 a(n) = 2^n + 5^n.

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A294133 Sorted list of prime factors of numbers of the form 5^(2^m) + 2^(2^m) with m >= 0.

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A337429 a(n) is the largest prime factor of 2^n + 5^n.

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A122118 Least prime factor of 2^n + 5^n.

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A122119 Least prime factor of 2^(2^n) + 5^(2^n).

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A386618 Primes of the form 2^k + 13^k.

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A122116 Numbers k such that 2^k+5^k is semiprime.

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