A006127 -id:A006127 - cp's OEIS frontend

A226200 a(n) = 6^n + n.

1, 7, 38, 219, 1300, 7781, 46662, 279943, 1679624, 10077705, 60466186, 362797067, 2176782348, 13060694029, 78364164110, 470184984591, 2821109907472, 16926659444753, 101559956668434, 609359740010515, 3656158440062996, 21936950640377877, 131621703842267158, 789730223053602839

Offset: 0

Vincenzo Librandi, Jun 16 2013

After 7, the next prime of this form has 238 digits (see A058828). - Bruno Berselli, Jun 18 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (8,-13,6).

Cf. numbers of the form k^n + n: A006127 (k=2), A104743 (k=3), A158879 (k=4), A104745 (k=5), this sequence (k=6), A226199 (k=7), A226201 (k=8), A226202 (k=9), A081552 (k=10), A226737 (k=11).

Cf. A058828, A199320 (first differences).

Magma
```
[6^n+n: n in [0..30]];
```

I:=[1, 7, 38]; [n le 3 select I[n] else 8*Self(n-1)-13*Self(n-2)+6*Self(n-3): n in [1..30]];

Table[6^n + n, {n, 0, 30}] (* or *) CoefficientList[Series[(-1 + x + 5 x^2) / ((6 x - 1) (x - 1)^2), {x, 0, 30}], x]

a(n)=6^n+n \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (-1+x+5*x^2)/((6*x-1)*(x-1)^2).
a(n) = 8*a(n-1) - 13*a(n-2) + 6*a(n-3).
E.g.f.: exp(x)*(exp(5*x) + x). - Elmo R. Oliveira, Mar 05 2025

A226737 a(n) = 11^n + n.

Original entry on oeis.org

1, 12, 123, 1334, 14645, 161056, 1771567, 19487178, 214358889, 2357947700, 25937424611, 285311670622, 3138428376733, 34522712143944, 379749833583255, 4177248169415666, 45949729863572177, 505447028499293788, 5559917313492231499, 61159090448414546310, 672749994932560009221

Offset: 0

Vincenzo Librandi, Jun 16 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (13,-23,11).

Cf. numbers of the form k^n + n: A006127 (k=2), A104743 (k=3), A158879 (k=4), A104745 (k=5), A226200 (k=6), A226199 (k=7), A226201 (k=8), A226202 (k=9), A081552 (k=10), this sequence (k=11).

Cf. A199764 (first differences).

Magma
```
[11^n+n: n in [0..30]];
```

I:=[1, 12, 123]; [n le 3 select I[n] else 13*Self(n-1)-23*Self(n-2)+11*Self(n-3): n in [1..30]];

Table[11^n + n, {n, 0, 30}] (* or *) CoefficientList[Series[(- 1 + x + 10 x^2) / ((11 x - 1) (x - 1)^2), {x, 0, 30}], x]
LinearRecurrence[{13,-23,11},{1,12,123},20] (* Harvey P. Dale, Nov 14 2018 *)

a(n)=11^n+n \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (-1+x+10*x^2)/((11*x-1)*(x-1)^2).
a(n) = 13*a(n-1) - 23*a(n-2) + 11*a(n-3).
E.g.f.: exp(x)*(exp(10*x) + x). - Elmo R. Oliveira, Mar 06 2025

A247248 a(n) is the least k such that n divides 2^k + k.

Original entry on oeis.org

1, 2, 1, 4, 4, 2, 6, 8, 7, 4, 3, 8, 12, 6, 7, 16, 16, 14, 18, 4, 19, 8, 22, 8, 33, 12, 7, 40, 11, 26, 23, 32, 8, 16, 6, 32, 5, 18, 37, 24, 40, 38, 42, 8, 7, 22, 10, 32, 61, 84, 38, 12, 35, 32, 46, 40, 32, 28, 24, 44, 17, 30, 61, 64, 66, 8, 66, 16, 67, 6, 11, 32

Offset: 1

Michel Lagneau, Dec 01 2014

nonn

For every positive integer n, there exists an integer k such that 2^k + k is divisible by n. The proof is given in the link, p. 63.

			a(7) = 6 because 2^6 + 6 = 70 is divisible by 7.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
International Mathematical Olympiad, Problem N7, IMO-2006, p. 63.

Cf. A135366, A006127 (2^n+n).

f:= proc(n) local k;
for k from 1 do
  if 2^k + k mod n = 0 then return k fi
od
end proc:
seq(f(n), n=1..100); # Robert Israel, Dec 01 2014

Table[s=0; k=0; While[k++; s=Mod[2^k+k, n]; s>0]; k, {n, 50}]
lk[n_]:=Module[{k=1},While[Mod[2^k+k,n]!=0,k++];k]; Array[lk,120] (* Harvey P. Dale, Jun 18 2022 *)

a(n) = for(m=1, oo, if(Mod(2, n)^m==-m, return(m))); \\ Jinyuan Wang, Mar 15 2020

def A247248(n):
    if n == 1:
        return 1
    else:
        x, k, kr = 1,0,0
        while (x+kr) % n:
            x, kr = (2*x) % n, (kr+1) % n
            k += 1
        return k
# Chai Wah Wu, Dec 03 2014

A057663 Primes p such that p + 2^p is also a prime.

Original entry on oeis.org

3, 5, 89, 317, 701

Offset: 1

Labos Elemer, Oct 16 2000

Different from A056206, where, e.g., at n=89, 89 is not minimal, A056206(89)=29 and not 89.
a(6) > 27479. - Ralf Stephan, Oct 23 2002
Intersection of A000040 and A052007. - Iain Fox, Nov 08 2017
a(6) > 678561. - Iain Fox, Nov 08 2017
Every term other than 3 is congruent to 5 (mod 6). - Arkadiusz Wesolowski, Nov 14 2017
These terms satisfy phi(k + 2^k) = phi(k) + 2^k, where phi is A000010, the Euler totient function. Conjecture: this sequence gives all numbers k that satisfy the condition phi(k + 2^k) = phi(k) + 2^k. - Juri-Stepan Gerasimov, May 23 2019

			q=3, 2^3 + 3 = 11 a prime.

Cf. A000010, A006127, A056206, A056208, A057664, A057665.

[p: p in PrimesUpTo(1000) | IsPrime(2^p+p) ] // Vincenzo Librandi, Aug 07 2010

Select[Prime@ Range[10^3], PrimeQ[# + 2^#] &] (* Michael De Vlieger, Nov 08 2017 *)

lista(nn) = forprime(p=3, nn, if(ispseudoprime(p + 2^p), print1(p, ", "))) \\ Iain Fox, Nov 13 2017

[n for n in (1..1000) if is_prime(n) and is_prime(2^n+n)] # G. C. Greubel, May 24 2019

A132925 a(n) = 2^n - 1 + n*(n-1)/2.

Original entry on oeis.org

1, 4, 10, 21, 41, 78, 148, 283, 547, 1068, 2102, 4161, 8269, 16474, 32872, 65655, 131207, 262296, 524458, 1048765, 2097361, 4194534, 8388860, 16777491, 33554731, 67109188, 134218078, 268435833, 536871317, 1073742258, 2147484112

Offset: 1

Gary W. Adamson, Sep 05 2007

Row sums of triangle A132924. n-th Mersenne number + (n-1)-th triangular number.

Partial sums of A006127. - Jaroslav Krizek, Oct 16 2009

			a(4) = 21 = sum of row 4 terms of triangle A132924: (4 + 4 + 5 + 8).
a(4) = 21 = (1, 3, 3, 1) dot (1, 3, 3, 2) = (1 + 9 + 9 + 2).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2).

Cf. A006127, A132924.

Cf. A000225, A000217.

[2^n - 1 + n*(n-1)/2: n in [1..40]]; // Vincenzo Librandi, Jun 21 2011

A132925 := proc(n) 2^n-1+n*(n-1)/2 ; end proc; # R. J. Mathar, Oct 23 2009

Table[2^n-1+n (n-1)/2,{n,40}] (* or *) LinearRecurrence[{5,-9,7,-2},{1,4,10,21},40] (* Harvey P. Dale, Jun 19 2011 *)

a(n)=2^n+binomial(n,2)-1 \\ Charles R Greathouse IV, Jun 20 2011

Binomial transform of [1, 3, 3, 2, 2, 2, 2, ...].
a(n) = A000225(n) + A000217(n-1). - Jaroslav Krizek, Oct 16 2009
From Harvey P. Dale, Jun 19 2011: (Start)
a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4); a(1)=1, a(2)=4, a(3)=10, a(4)=21.
G.f.: -x*(x^2+x-1)/((x-1)^3*(2*x-1)). (End)
E.g.f.: exp(x)*(2*exp(x) + x^2 - 2)/2. - Stefano Spezia, Feb 01 2025

A188915 Union of squares and powers of 2.

Original entry on oeis.org

0, 1, 2, 4, 8, 9, 16, 25, 32, 36, 49, 64, 81, 100, 121, 128, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 512, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2048, 2116, 2209, 2304, 2401, 2500, 2601, 2704, 2809, 2916

Offset: 0

Reinhard Zumkeller, Apr 14 2011

nonn

A188916 and A188917 give positions where squares and powers of 2 occur:
n^2: a(A188916(n)) = A000290(n);
2^n: a(A188917(n)) = A000079(n);
4^n: a(A006127(n)) = A000302(n), A006127 is the intersection of A188916 and A188917.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Union of A000290 and A000079.
Disjoint union of A000290 and A004171.
Cf. A000302, A010052, A209229, A006127, A188916, A188917.

import Data.List.Ordered (union)
a188915 n = a188915_list !! n
a188915_list = union a000290_list a000079_list
-- Reinhard Zumkeller, May 19 2015, Apr 14 2011

seq[lim_] := Union[2^Range[1, Floor[Log2[lim]], 2], Range[0, Floor[Sqrt[lim]]]^2]; seq[3000] (* Amiram Eldar, Apr 13 2025 *)

from math import isqrt
def A188915(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-isqrt(x)-((m:=x.bit_length()-1)>>1)-(m&1)
    return bisection(f,n-1,n**2) # Chai Wah Wu, Sep 19 2024

A010052(a(n)) + A209229(a(n)) > 0. - Reinhard Zumkeller, May 19 2015

Sum_{n>=1} 1/a(n) = Pi^2/6 + 2/3. - Amiram Eldar, Apr 13 2025

A371928 T(n,k) is the total number of levels in all Dyck paths of semilength n containing exactly k path nodes; triangle T(n,k), n>=0, 1<=k<=n+1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 3, 5, 6, 1, 8, 15, 13, 11, 1, 23, 44, 43, 29, 20, 1, 71, 134, 138, 106, 62, 37, 1, 229, 427, 446, 371, 248, 132, 70, 1, 759, 1408, 1478, 1275, 941, 571, 283, 135, 1, 2566, 4753, 5017, 4410, 3437, 2331, 1310, 611, 264, 1, 8817, 16321, 17339, 15458, 12426, 9027, 5709, 3002, 1324, 521, 1

Offset: 0

Alois P. Heinz, Apr 14 2024

A Dyck path of semilength n has 2n+1 = A005408(n) nodes.

			In the A000108(3) = 5 Dyck paths of semilength 3 there are 3 levels with 1 node, 5 levels with 2 nodes, 6 levels with 3 nodes, and 1 level with 4 nodes.
  1
  2   /\      2           1           1
  2  /  \     3  /\/\     3  /\       3    /\     3
  2 /    \    2 /    \    3 /  \/\    3 /\/  \    4 /\/\/\    .
  So row 3 is [3, 5, 6, 1].
Triangle T(n,k) begins:
     1;
     1,     1;
     1,     3,     1;
     3,     5,     6,     1;
     8,    15,    13,    11,     1;
    23,    44,    43,    29,    20,    1;
    71,   134,   138,   106,    62,   37,    1;
   229,   427,   446,   371,   248,  132,   70,    1;
   759,  1408,  1478,  1275,   941,  571,  283,  135,    1;
  2566,  4753,  5017,  4410,  3437, 2331, 1310,  611,  264,   1;
  8817, 16321, 17339, 15458, 12426, 9027, 5709, 3002, 1324, 521, 1;
  ...

Alois P. Heinz, Rows n = 0..20, flattened
Wikipedia, Counting lattice paths

Columns k=1-2 give: A152880, A371903.
Row sums give A261003.
T(n+1,n+1) gives A006127.
Cf. A000108, A001700, A005408, A372014.

g:= proc(x, y, p) (h-> `if`(x=0, add(z^coeff(h, z, i)
          , i=0..degree(h)), b(x, y, h)))(p+z^y) end:
b:= proc(x, y, p) option remember; `if`(y+2<=x,
      g(x-1, y+1, p), 0)+`if`(y>0, g(x-1, y-1, p), 0)
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=1..n+1))(g(2*n, 0$2)):
seq(T(n), n=0..10);

Sum_{k=1..n+1} k * T(n,k) = A001700(n) = A005408(n) * A000108(n).

A085745 Numbers m such that 2^m + m is a semiprime.

Original entry on oeis.org

2, 11, 23, 34, 59, 69, 87, 95, 119, 123, 129, 171, 197, 239, 341, 425, 455, 471, 515, 519, 635, 765, 959, 1115, 1181, 1210

Offset: 1

Jason Earls, Jul 21 2003

a(27) >= 1239. If 1239 is not a term then a(27) = 1275. - Amiram Eldar, Oct 26 2024

factordb, 2^1239+1239.

Cf. A006127, A052007, A089535, A089536, A089537.

Sequences A165767, A165768, A165769 are the analog for 2^n-n. - M. F. Hasler, Oct 08 2009

Select[Range[1300],PrimeOmega[2^#+#]==2&] (* Harvey P. Dale, Dec 18 2014 *)

a(n) = 2*k <=> A089535(n) is even <=> A089536(n) = 2 <=> A089537(n) = 4^k/2 + k, and for any prime of this form, there is a term a(n) = 2*k in this sequence. - M. F. Hasler, Oct 08 2009

More terms from Ray Chandler, Nov 08 2003
a(15) from Donovan Johnson, Mar 06 2008
a(16)-a(26) from Sean A. Irvine, Oct 27 2009
Offset changed to 1 by Jinyuan Wang, Jul 30 2021

A089535 Semiprimes of the form 2^k + k.

Original entry on oeis.org

6, 2059, 8388631, 17179869218, 576460752303423547, 590295810358705651781, 154742504910672534362390615, 39614081257132168796771975263, 664613997892457936451903530140172407, 10633823966279326983230456482242756731, 680564733841876926926749214863536423041

Offset: 1

Ray Chandler, Nov 08 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..26

Intersection of A001358 and A006127.

Cf. A085745, A089536, A089537, 129962.

Select[Table[2^k+k,{k,200}],PrimeOmega[#]==2&] (* Harvey P. Dale, Aug 11 2024 *)

a(n) = 2^A085745(n) + A085745(n).

a(10)-a(11) from Jinyuan Wang, Jul 30 2021

A188916 Where squares occur in the union of squares and powers of 2.

Original entry on oeis.org

0, 1, 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100

Offset: 0

Reinhard Zumkeller, Apr 14 2011

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A000079, A000290.

Cf. A188915, A188917, A010052, A006127 (subsequence).

a188916 n = a188916_list !! n
a188916_list = filter ((== 1) . a010052. a188915) [0..]
-- Reinhard Zumkeller, May 19 2015

seq[lim_] := -1 + Position[Union[2^Range[1, Floor[Log2[lim]], 2], Range[0, Floor[Sqrt[lim]]]^2], ?(IntegerQ[Sqrt[#]] &)] // Flatten; seq[10000] (* _Amiram Eldar, Apr 13 2025 *)

A188915(a(n)) = A000290(n); A188915(A188917(n)) = A000079(n).

cp's OEIS Frontend

A226200 a(n) = 6^n + n.

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A226737 a(n) = 11^n + n.

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A247248 a(n) is the least k such that n divides 2^k + k.

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Maple

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A057663 Primes p such that p + 2^p is also a prime.

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A132925 a(n) = 2^n - 1 + n*(n-1)/2.

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Magma

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A188915 Union of squares and powers of 2.

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A371928 T(n,k) is the total number of levels in all Dyck paths of semilength n containing exactly k path nodes; triangle T(n,k), n>=0, 1<=k<=n+1, read by rows.

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