A104066 -id:A104066 - cp's OEIS frontend

A257272 a(n) = 2^(n-1)*(2^n+7).

4, 9, 22, 60, 184, 624, 2272, 8640, 33664, 132864, 527872, 2104320, 8402944, 33583104, 134275072, 536985600, 2147713024, 8590393344, 34360655872, 137440788480, 549759483904, 2199030595584, 8796107702272, 35184401448960, 140737547075584, 562950070861824, 2251800048566272, 9007199724503040

Offset: 0

M. F. Hasler, Apr 27 2015

For n in A057195, a(n) is of deficiency 8, i.e., in A125247.

Also, the third column (k=2) of the table given in A181444.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-8).

Cf. A000079, A007582, A028403, A256873, A257273, A256871.

Cf. A168415, A057195, A104066, A125247.

[2^(n-1)*(2^n+7): n in [0..25]]; // Vincenzo Librandi, Apr 27 2015

Table[2^(n - 1) (2^n + 7), {n, 0, 30}] (* Bruno Berselli, Apr 27 2015 *)
CoefficientList[Series[(4 - 15 x)/((1 - 4 x) (1 - 2 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Apr 27 2015 *)

PARI
```
a(n)=2^(n-1)*(2^n+7)
```

Vec((4-15*x)/((1-4*x)*(1-2*x)) + O(x^100)) \\ Colin Barker, Apr 27 2015

a(n) = 2^(n-1)*A168415(n).
n in A057195 <=> A168415(n) in A104066 <=> a(n) in A125247.
G.f.: (4-15*x)/((1-4*x)*(1-2*x)). - Vincenzo Librandi, Apr 27 2015

A217349 Numbers k such that 4^k + 7 is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 10, 14, 15, 19, 22, 39, 44, 49, 63, 80, 87, 102, 107, 294, 305, 399, 463, 595, 599, 903, 944, 1324, 1727, 1755, 1932, 1935, 4485, 6165, 6665, 9438, 11169, 19859, 27503, 55392, 86235, 98217, 117855, 123640, 134204, 139660, 150437, 157634, 186475, 236129, 283248, 390142, 410178

Offset: 1

Vincenzo Librandi, Oct 01 2012

nonn

The next terms are > 4.1*10^5. - Elmo R. Oliveira, Nov 29 2023

			For k = 14, 4^14 + 7 = 268435463 is prime.

Cf. A057195, A059266, A089437, A104066 (associated primes).

Select[Range[0, 5000], PrimeQ[4^# + 7] &]

is(n)=ispseudoprime(4^n+7) \\ Charles R Greathouse IV, Jun 06 2017

a(n) = A057195(n)/2.

Extended using A057195 terms by Michel Marcus, Aug 28 2015

a(51)-a(54) derived from A057195 by Elmo R. Oliveira, Nov 29 2023

A228027 Primes of the form 4^k + 9.

Original entry on oeis.org

13, 73, 1033, 262153, 1073741833, 73786976294838206473, 4835703278458516698824713

Offset: 1

Vincenzo Librandi, Aug 11 2013

Subsequence of A104070. - Elmo R. Oliveira, Nov 28 2023

			262153 is a term because 4^9 + 9 = 262153 is prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..12

Cf. A000040, A217350 (corresponding k's).

Cf. Primes of the form r^k + h: A092506 (r=2, h=1), A057733 (r=2, h=3), A123250 (r=2, h=5), A104066 (r=2, h=7), A104070 (r=2, h=9), A057735 (r=3, h=2), A102903 (r=3, h=4), A102870 (r=3, h=8), A102907 (r=3, h=10), A290200 (r=4, h=1), A228026 (r=4, h=3), this sequence (r=4, h=9), A182330 (r=5, h=2), A228029 (r=5, h=6), A102910 (r=5, h=8), A182331 (r=6, h=1), A104118 (r=6, h=5), A104115 (r=6, h=7), A104065 (r=7, h=4), A228030 (r=7, h=6), A228031 (r=7, h=10), A228032 (r=8, h=3), A228033 (r=8, h=5), A144360 (r=8, h=7), A145440 (r=8, h=9), A228034 (r=9, h=2), A159352 (r=10, h=3), A159031 (r=10, h=7).

[a: n in [0..200] | IsPrime(a) where a is 4^n+9];

Select[Table[4^n + 9, {n, 0, 200}],PrimeQ]

a(n) = 4^A217350(n) + 9. - Elmo R. Oliveira, Nov 28 2023

Corrected cross-references - Robert Price, Aug 01 2017

A228033 Primes of the form 8^k + 5.

Original entry on oeis.org

13, 2787593149816327892691964784081045188247557, 15177100720513508366558296147058741458143803430094840009779784451085189728165691397

Offset: 1

Vincenzo Librandi, Aug 11 2013

a(4) = 8^64655 + 5 = 1.919...*10^58389 is too large to include. - Amiram Eldar, Jul 23 2025

Cf. A217355 (associated n).

Cf. Primes of the form k^n + h: A092506 (k=2, h=1), A057733 (k=2, h=3), A123250 (k=2, h=5), A104066 (k=2, h=7), A104070 (k=2, h=9), A057735 (k=3, h=2), A102903 (k=3, h=4), A102870 (k=3, h=8), A102907 (k=3, h=10), A290200 (k=4, h=1), A182330 (k=5, h=2), A102910 (k=5, h=8), A182331 (k=6, h=1), A104118 (k=6, h=5), A104115 (k=6, h=7), A104065 (k=7, h=4), this sequence (k=8, h=5), A144360 (k=8, h=7), A145440 (k=8, h=9), A228034 (k=9, h=2), A159352 (k=10, h=3), A159031 (k=10, h=7).

[a: n in [1..300] | IsPrime(a) where a is 8^n+5];

Select[Table[8^n + 5, {n, 4000}], PrimeQ]

A156973 Primes of the form 2^k + 17.

Original entry on oeis.org

19, 8209, 2097169, 8589934609, 2417851639229258349412369, 680564733841876926926749214863536422929, 62165404551223330269422781018352605012557018849668464680057997111644937126566671941649

Offset: 1

Edwin Dyke (ed.dyke(AT)btinternet.com), Feb 19 2009

nonn

			19 = 2^1 + 17 is in the sequence;
8209 = 2^13 + 17 is in the sequence.

Cf. A000040, A057200, A057733 (2^k + 3), A123250 (2^k + 5), A104066 (2^k + 7), A156940 (2^k + 11), A104067 (2^k + 13).

[ a: n in [1..400] | IsPrime(a) where a is 2^n+17 ]; // Vincenzo Librandi, Nov 27 2010

Delete[Union[Table[If[PrimeQ[2^n + 17], 2^n + 17, 0], {n, 1, 300}]],1]

a(n) = 2^A057200(n) + 17. - Elmo R. Oliveira, Nov 08 2023

a(7) from Vincenzo Librandi, Apr 29 2010

A228028 Primes of the form 5^n + 4.

Original entry on oeis.org

5, 29, 15629, 9765629

Offset: 1

Vincenzo Librandi, Aug 11 2013

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..8

Cf. A124621 (associated n).

Cf. Primes of the form k^n + h: A092506 (k=2, h=1), A057733 (k=2, h=3), A123250 (k=2, h=5), A104066 (k=2, h=7), A104070 (k=2, h=9), A057735 (k=3, h=2), A102903 (k=3, h=4), A102870 (k=3, h=8), A102907 (k=3, h=10), A290200 (k=4, h=1), A228027 (k=4, h=9), A182330 (k=5, h=2), this sequence (k=5, h=4), A228029 (k=5, h=6), A102910 (k=5, h=8), A182331 (k=6, h=1), A104118 (k=6, h=5), A104115 (k=6, h=7), A104065 (k=7, h=4), A228030 (k=7, h=6), A228031 (k=7, h=10), A228032 (k=8, h=3), A228033 (k=8, h=5), A144360 (k=8, h=7), A145440 (k=8, h=9), A228034 (k=9, h=2), A159352 (k=10, h=3), A159031 (k=10, h=7).

[a: n in [0..200] | IsPrime(a) where a is  5^n+4];

Select[Table[5^n + 4, {n, 0, 200}], PrimeQ]

Corrected cross-references - Robert Price, Aug 01 2017

A156974 Primes of the form 2^k + 29.

Original entry on oeis.org

31, 37, 61, 157, 541, 8221, 32797, 131101, 8388637, 134217757, 8589934621, 137438953501, 8796093022237, 9223372036854775837, 590295810358705651741, 9444732965739290427421, 604462909807314587353117, 618970019642690137449562141, 166153499473114484112975882535043101, 170141183460469231731687303715884105757, 883423532389192164791648750371459257913741948437809479060803100646309917

Offset: 1

Edwin Dyke (ed.dyke(AT)btinternet.com), Feb 19 2009

nonn

Cf. A000040, A156982.

Cf. A057733 (2^k+3), A123250 (2^k+5), A104066 (2^k+7), A156940 (2^k+11).

[a: n in [0..300] | IsPrime(a) where a is 2^n+29 ]; // Vincenzo Librandi, Nov 27 2010

a := proc (n) if isprime(2^n+29) = true then 2^n+29 else end if end proc: seq(a(n), n = 1 .. 110); # Emeric Deutsch, Mar 14 2009

Delete[Union[Table[If[PrimeQ[2^n + 29], 2^n + 29, 0], {n, 1, 500}]],1]

a(n) = 2^A156982(n) + 29. - Elmo R. Oliveira, Nov 08 2023

More terms from Emeric Deutsch, Mar 14 2009

More terms from Vincenzo Librandi, Nov 27 2010

A195463 a(n) = 4^(n+1) + 7.

Original entry on oeis.org

11, 23, 71, 263, 1031, 4103, 16391, 65543, 262151, 1048583, 4194311, 16777223, 67108871, 268435463, 1073741831, 4294967303, 17179869191, 68719476743, 274877906951, 1099511627783, 4398046511111, 17592186044423, 70368744177671, 281474976710663, 1125899906842631

Offset: 0

Brad Clardy, Sep 19 2011

These are the even terms of A168415. Since the odd terms of A168415 are divisible by three the primes of this sequence are the same as A104066.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A104066, A168415, A188165.

[4^(n+1) + 7: n in [0..30]]; // Vincenzo Librandi, Sep 30 2011

4^Range[25] + 7 (* Paolo Xausa, Feb 21 2025 *)

a(n)=4^(n+1)+7 \\ Charles R Greathouse IV, Sep 19 2011

a(n) = 4^(n+1) + 7.
From Alexander R. Povolotsky, Sep 19 2011: (Start)
G.f.: (11 - 32*x)/(1 - 5*x + 4*x^2).
a(n+1) = 4*a(n) - 21. (End)
a(n) = A188165(2*n+2) - 2. - Bruno Berselli, Sep 26 2011
E.g.f.: exp(x)*(4*exp(3*x) + 7). - Elmo R. Oliveira, Feb 20 2025

A267413 Dropping any binary digit gives a prime number.

Original entry on oeis.org

6, 7, 11, 15, 35, 39, 63, 135, 255, 999, 2175, 8223, 16383, 57735, 131075, 131079, 262143, 524295, 1048575, 536870919, 1073735679, 2147483655, 4294967295, 17179770879, 4260641103903, 4611686018427387903, 4720069647059686260735, 1237940039285380274899124223

Offset: 1

Stanislav Sykora, Jan 14 2016

This is the binary analog of A034895. The sequence contains mostly numbers with very few binary digit runs (BDR, A005811). Those with one BDR are of the type 2^k-1, such that 2^(k-1)-1 is a Mersenne prime (A000668). Vice versa, if M is any Mersenne prime, then 2*M+1 is a term. The number 6 is the only term with an even number of BDRs. There are many terms with 3 BDRs. The first term with 5 BDRs is 57735. The next terms with at least 5 BDRs (if they exist at all) are larger than 10^10. So far, I could test that a(24) > 10^10.
From Robert Israel, Jan 14 2016: (Start)
For n >= 2, a(n) == 3 (mod 4).
2^k+3 is in the sequence if 2^(k-1)+1 and 2^(k-1)+3 are primes, i.e., 2^(k-1)+1 is in the intersection of A019434 and A001359.  The only known terms of the sequence in this class are 7, 11, 35, 131075.
2^k+7 is in the sequence if 2^(k-1)+3 and 2^(k-1)+7 are primes: thus 2^(k-1)+3 is in A057733 and 2^(k-1)+7 is in A104066.  Terms of the sequence in this class include 15, 39, 135, 131079, 524295, 536870919, 2147483655 (but no more for k <= 2000).
(End)
a(25) > 2^38. - Giovanni Resta, Apr 10 2016
For n > 1, a(n) = 2p+1 for some prime p. - Chai Wah Wu, Aug 27 2021
From Bert Dobbelaere, Aug 07 2023: (Start)
There are no more terms with an odd number of binary digits: from any number having an odd number of binary digits, one can always drop a digit and obtain a multiple of 3. Numbers of the form 2^k+3 (k even and k > 2) cannot be terms because 2^(k-1)+1 is a multiple of 3.
(End)

			Decimal and binary forms of the known terms:
   1           6                                110
   2           7                                111
   3          11                               1011
   4          15                               1111
   5          35                             100011
   6          39                             100111
   7          63                             111111
   8         135                           10000111
   9         255                           11111111
  10         999                         1111100111
  11        2175                       100001111111
  12        8223                     10000000011111
  13       16383                     11111111111111
  14       57735                   1110000110000111 <--- (a binary palindrome
  15      131075                 100000000000000011       with 5 digit runs)
  16      131079                 100000000000000111
  17      262143                 111111111111111111
  18      524295               10000000000000000111
  19     1048575               11111111111111111111
  20   536870919     100000000000000000000000000111
  21  1073735679     111111111111111110011111111111
  22  2147483655   10000000000000000000000000000111
  23  4294967295   11111111111111111111111111111111
  24 17179770879 1111111111111111100111111111111111

Cf. A000668, A001359, A005811, A019434, A034895 (base 10), A051362, A057733, A104066.

filter:= proc(n) local B,k,y;
   if not isprime(floor(n/2)) then return false fi;
   B:= convert(n,base,2);
   for k from 2 to nops(B) do
     if B[k] <> B[k-1] then
       y:= n mod 2^(k-1);
       if not isprime((y+n-B[k]*2^(k-1))/2) then return false fi
     fi
   od;
   true
end proc:
select(filter, [6, seq(i,i=7..10^6,4)]); # Robert Israel, Jan 14 2016

Select[Range[2^20], AllTrue[Function[w, Map[FromDigits[#, 2] &@ Drop[w, {#}] &, Range@ Length@ w]]@ IntegerDigits[#, 2], PrimeQ] &] (* Michael De Vlieger, Jan 16 2016, Version 10 *)

DroppingAnyDigitGivesAPrime(N,b) = {
\\ Property-testing function; returns 1 if true for N, 0 otherwise
\\ Works with any base b. Here used with b=2.
  my(k=b,m); if(N=(k\b), m=(N\k)*(k\b)+(N%(k\b));
    if ((m<2)||(!isprime(m)),return(0)); k*=b);
  return(1);
}

from sympy import isprime
def ok(n):
    if n < 7 or n%4 != 3: return n == 6
    b = bin(n)[2:]
    return all(isprime(int(b[:i]+b[i+1:], 2)) for i in range(len(b)))
print(list(filter(ok, range(2, 2**20)))) # Michael S. Branicky, Jun 07 2021

a(24) from Giovanni Resta, Apr 10 2016

a(25)-a(28) from Bert Dobbelaere, Aug 07 2023

A172183 a(n) is the smallest prime of the form p^q+n, where p and q are prime, or zero if no such prime exists.

Original entry on oeis.org

5, 11, 7, 13, 13, 31, 11, 17, 13, 19, 19, 37, 17, 23, 19, 41, 8209, 43, 23, 29, 29, 31, 31, 73, 29, 53, 31, 37, 37, 79, 0, 41, 37, 43, 43, 61, 41, 47, 43, 67, 73, 67, 47, 53, 53, 71, 79, 73, 53, 59, 59, 61, 61, 79, 59, 83, 61, 67, 67, 109, 0, 71, 67, 73, 73, 191, 71, 193, 73, 79

Offset: 1

Cheng Zhang (cz1(AT)rice.edu), Jan 28 2010, Mar 02 2010

nonn

If n mod 6 = 1, both p and q must be 2, and a(n)=0 if n + 4 is not a prime. The values of a(n) for n=257,297,353,383,557 are either greater than 176 000 or 0. Several large entries: a(87) = 2^25633 + 87, a(717) = 2^3217 + 717, a(773) = 2^2539 + 773, a(927) = 2^1117 + 927.

			a(1)=5 because 5=2^2+1 is the smallest prime of the form p^q+1. a(2)=11 because 11=3^2+2. a(3)=7, because 7=2^2+3. a(17)=8209, because 8209=2^13+17. a(31)=0, because p^q+31 cannot be a prime.

Cf. A123318, A056208, A056206, A057733, A123250, A104066, A156940, A104067, A156973

For[l = {}; n = 1, n <= 70, n++, found = False; If[Mod[n, 2] == 0, For[rm = Infinity; i = 1, i < 100, i++, For[j = 1, j < 100, j++, p = Prime[i]; q = Prime[j]; r = p^q + n; If[r >= rm, Break[], If[PrimeQ[r], rm = r; found = True]]; ]; ], (* if n is odd, r=2^q+n *) If[Mod[n, 6] == 1, r = 4 + n; If[PrimeQ[r], found = True], For[j = 1, j < 1000, j++, q = Prime[j]; r = 2^q + n; If[PrimeQ[r], found = True; rm = r; Break[]]; ]; ]; ]; If[ ! found, rm = 0]; l = Append[l, rm]; ]; l

cp's OEIS Frontend

A257272 a(n) = 2^(n-1)*(2^n+7).

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A217349 Numbers k such that 4^k + 7 is prime.

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A228027 Primes of the form 4^k + 9.

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A228033 Primes of the form 8^k + 5.

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A156973 Primes of the form 2^k + 17.

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A228028 Primes of the form 5^n + 4.

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A156974 Primes of the form 2^k + 29.

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