A000979 -id:A000979 - cp's OEIS frontend

A127955 Composite numbers of the form (2^p+1)/3 where p is a prime.

178956971, 45812984491, 733007751851, 46912496118443, 3002399751580331, 192153584101141163, 49191317529892137643, 787061080478274202283, 3148244321913096809131, 3223802185639011132549803

Offset: 1

Artur Jasinski, Feb 09 2007

nonn

If p-1 is squarefree, the terms are overpseudoprimes (see A141232). - Vladimir Shevelev, Jul 15 2008

G. C. Greubel, Table of n, a(n) for n = 1..440

Cf. A000979, A000978, A124400, A126614, A127956, A127957.

a = {}; Do[c = (2^Prime[x] + 1)/3; If[PrimeQ[c] == False, AppendTo[a, c]], {x, 2, 30}]; a
Select[(2^Prime[Range[2,30]]+1)/3,CompositeQ] (* Harvey P. Dale, Feb 04 2015 *)

A127957 Numbers k such that (2^prime(k) + 1)/3 is composite.

Original entry on oeis.org

10, 12, 13, 15, 16, 17, 19, 20, 21, 23, 24, 25, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89

Offset: 1

Artur Jasinski, Feb 09 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1175

Cf. A000979, A000978, A124400, A126614, A127955, A127956.

a = {}; Do[c = (2^Prime[x] + 1)/3; If[PrimeQ[c] == False, AppendTo[a, x]], {x, 2, 300}]; a
Select[Range[2,100],!PrimeQ[(2^Prime[#]+1)/3]&] (* Harvey P. Dale, Nov 24 2013 *)

isok(n) = (n!=1) && !isprime((2^prime(n)+1)/3); \\ Michel Marcus, Jul 07 2018

A127956 Prime numbers p such that (2^p+1)/3 is composite.

Original entry on oeis.org

29, 37, 41, 47, 53, 59, 67, 71, 73, 83, 89, 97, 103, 107, 109, 113, 131, 137, 139, 149, 151, 157, 163, 173, 179, 181, 193, 197, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 317, 331, 337, 349, 353, 359, 367, 373

Offset: 1

Artur Jasinski, Feb 09 2007

nonn

If p-1 is squarefree, 2a(n) is the multiplicative order of 2 modulo every divisor d>1 of (2^p+1)/3. - Vladimir Shevelev, Jul 15 2008

Cf. A000979, A000978, A124400, A126614, A127955, A127957.

a = {}; Do[c = (2^Prime[x] + 1)/3; If[PrimeQ[c] == False, AppendTo[a, Prime[x]]], {x, 2, 100}]; a
Select[Prime[Range[2,100]],CompositeQ[(2^#+1)/3]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 07 2021 *)

A127958 Numbers x such that 1 + Sum_{k=1..n} 2^(2k-1) is not prime for n=1,2,...,x.

Original entry on oeis.org

4, 7, 10, 12, 13, 14, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 84, 85

Offset: 1

Artur Jasinski, Feb 09 2007

nonn

Numbers x such that 1 + Sum_{k=1..n} 2^(2k-1) is prime for n=1,2,...,x gives A127936.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000979, A000978, A124400, A126614, A127955, A127957, A127936.

a = {}; Do[If[PrimeQ[1 + Sum[2^(2n - 1), {n, 1, x}]] == False, AppendTo[a, x]], {x, 1, 1000}]; a

isok(x) = !isprime(1+sum(k=1, x, 2^(2*k-1))); \\ Michel Marcus, May 09 2018

A049883 Primes in the Jacobsthal sequence (A001045).

Original entry on oeis.org

3, 5, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, 201487636602438195784363, 845100400152152934331135470251, 56713727820156410577229101238628035243

Offset: 1

Judson Neer

nonn

All terms, except a(2) = 5, are of the form (2^p + 1)/3 - the Wagstaff primes A000979 = {3, 11, 43, 683, 2731, 43691, 174763, ...}.
Indices of prime Jacobsthal numbers are listed in A107036 = {3, 4, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, ...}.
For n > 1, A107036(n) = A000978(n) (numbers m such that (2^m + 1)/3 is prime). - Alexander Adamchuk, Oct 10 2006

Vincenzo Librandi, Table of n, a(n) for n = 1..23

Cf. A001045, A000978, A000979, A107036.

Select[Table[(2^n + (-1)^(n - 1))/3, {n, 200}], PrimeQ] (* Vladimir Joseph Stephan Orlovsky, Mar 29 2011 *)

A123627 Smallest prime q such that (q^p+1)/(q+1) is prime, where p = prime(n); or 0 if no such prime q exists.

Original entry on oeis.org

0, 2, 2, 2, 2, 2, 2, 2, 2, 7, 2, 19, 61, 2, 7, 839, 89, 2, 5, 409, 571, 2, 809, 227, 317, 2, 5, 79, 23, 4073, 2, 281, 89, 739, 1427, 727, 19, 19, 2, 281, 11, 2143, 2, 1013, 4259, 2, 661, 1879, 401, 5, 4099, 1579, 137, 43, 487, 307, 547, 1709, 43, 3, 463, 2161, 353, 443, 2

Offset: 1

Alexander Adamchuk, Oct 04 2006, Aug 05 2008

nonn

a(1) = 0 because such a prime does not exist, Mod[n^2+1,n+1] = 2 for n>1.
Corresponding primes (q^p+1)/(q+1), where prime q = a(n) and p = Prime[n], are listed in A123628[n] = {1,3,11,43,683,2731,43691,174763,2796203,402488219476647465854701,715827883,...}.
a(n) coincides with A103795[n] when A103795[n] is prime.
a(n) = 2 for n = PrimePi[A000978[k]] = {2,3,4,5,6,7,8,9,11,14,18,22,26,31,39,43,46,65,69,126,267,380,495,762,1285,1304,1364,1479,1697,4469,8135,9193,11065,11902,12923,13103,23396,23642,31850,...}.
Corresponding primes of the form (2^p + 1)/3 are the Wagstaff primes that are listed in A000979[n] = {3,11,43,683,2731,43691,174763,2796203,715827883,...}.

Robert Israel, Table of n, a(n) for n = 1..240

Cf. A123628, A103795, A123487, A123488, A000978, A000979.

f:= proc(n) local p,q;
  p:= ithprime(n);
  q:= 1;
  do
   q:= nextprime(q);
   if isprime((q^p+1)/(q+1)) then return q fi
  od
end proc:
f(1):= 0:
map(f, [$1..70]); # Robert Israel, Jul 31 2019

a(1) = 0, for n>1 Do[p=Prime[k]; n=1; q=Prime[n]; cp=(q^p+1)/(q+1); While[ !PrimeQ[cp], n=n+1; q=Prime[n]; cp=(q^p+1)/(q+1)]; Print[q], {k, 2, 61}]
Do[p=Prime[k]; n=1; q=Prime[n]; cp=(q^p+1)/(q+1); While[ !PrimeQ[cp], n=n+1; q=Prime[n]; cp=(q^p+1)/(q+1)]; Print[{k,q}], {k, 1, 134}]
spq[n_]:=Module[{p=Prime[n],q=2},While[!PrimeQ[(q^p+1)/(q+1)],q=NextPrime[ q]]; q]; Join[{0},Array[spq,70,2]] (* Harvey P. Dale, Mar 23 2019 *)

A123628(n) = (a(n)^prime(n) + 1) / (a(n) + 1).

A123628 Smallest prime of the form (q^p+1)/(q+1), where p = prime(n) and q is also prime (q = A123627(n)); or 1 if such a prime does not exist.

Original entry on oeis.org

1, 3, 11, 43, 683, 2731, 43691, 174763, 2796203, 402488219476647465854701, 715827883, 10300379826060720504760427912621791994517454717, 254760179343040585394724919772965278539769280548173566545431025735121201

Offset: 1

Alexander Adamchuk, Oct 03 2006

nonn

a(1) = 1 because such a prime does not exist; (n^2+1) mod (n+1) = 2 for n > 1. a(n) = (A103795(n)^prime(n)+1)/(A103795(n)+1) when A103795(n) is prime. Corresponding smallest primes q such that (q^p+1)/(q+1) is prime, where p = prime(n), are listed in A123627(n) = {0, 2, 2, 2, 2, 2, 2, 2, 2, 7, 2, 19, 61, 2, 7, 839, 1459, 2, 5, 409, 571, 2, ...}. All Wagstaff primes or primes of form (2^p + 1)/3 belong to a(n). Wagstaff primes are listed in A000979(n) = {3, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, ...}. Corresponding indices n such that a(n) = (2^prime(n) + 1)/3 are PrimePi(A000978(n)) = {2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 18, 22, 26, 31, 39, 43, 46, 65, 69, 126, 267, 380, 495, 762, 1285, 1304, 1364, 1479, 1697, 4469, 8135, 9193, 11065, 11902, 12923, 13103, 23396, 23642, 31850, ...}. All primes with prime indices in the Jacobsthal sequence A001045(n) belong to a(n).

Cf. A123627, A103795, A123487, A123488, A000978, A000979, A001045, A049883, A107036.

a(n) = (A123627(n)^prime(n) + 1) / (A123627(n) + 1).

A127963 Number of 1's in A127962(n).

Original entry on oeis.org

2, 3, 4, 6, 7, 9, 10, 12, 16, 22, 31, 40, 51, 64, 84, 96, 100, 157, 174, 351, 855, 1309, 1770, 2904, 5251, 5346, 5640, 6196, 7240, 21369, 41670, 47685, 58620, 63516, 69469, 70540, 133509, 134994, 187161, 493096, 2015700

Offset: 1

Artur Jasinski, Feb 09 2007

Cf. A000120, A000978, A000979, A007953, A124400, A126614, A127936, A127955, A127956, A127957, A127958, A127961, A127962.

b = {}; Do[c = 1 + Sum[2^(2n - 1), {n, 1, x}]; If[PrimeQ[c], AppendTo[b, c]], {x, 0, 1000}]; a = {}; Do[AppendTo[a, FromDigits[IntegerDigits[b[[x]], 2]]], {x, 1, Length[b]}]; d = {}; Do[AppendTo[d, DigitCount[a[[x]], 10, 1]], {x, 1, Length[a]}]; d (* Artur Jasinski, Feb 09 2007 *)
DigitCount[#, 2, 1]& /@ Select[Table[(2^p + 1)/3, {p, Prime[Range[300]]}], PrimeQ] (* Amiram Eldar, Jul 23 2023 *)

a(n) = A000120(A000979(n)). - Michel Marcus, Nov 07 2013

a(n) = A007953(A127962(n)). - Amiram Eldar, Jul 23 2023

a(22)-a(29) from Vincenzo Librandi, Mar 31 2012

a(30)-a(41) from Amiram Eldar, Jul 23 2023

A127964 Number of 0's in the binary expansion of A127962(n).

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 8, 10, 14, 20, 29, 38, 49, 62, 82, 94, 98, 155, 172, 349, 853, 1307, 1768, 2902, 5249, 5344, 5638, 6194, 7238, 21367, 41668, 47683, 58618, 63514, 69467, 70538, 133507, 134992, 187159, 493094, 2015698

Offset: 1

Artur Jasinski, Feb 09 2007

Apparently numbers k such that (2^(2*k+3)+1)/3 is prime. - James R. Buddenhagen, Apr 14 2011 [This is true. See the second formula. - Amiram Eldar, Oct 13 2024]

Cf. A000978, A000979, A023416, A127962, A127961, A000979, A000978, A124400, A126614, A127955, A127956, A127957, A127958, A127936.

b = {}; Do[c = 1 + Sum[2^(2n - 1), {n, 1, x}]; If[PrimeQ[c], AppendTo[b, c]], {x, 0, 1000}]; a = {}; Do[AppendTo[a, FromDigits[IntegerDigits[b[[x]], 2]]], {x, 1, Length[b]}]; d = {}; Do[AppendTo[d, DigitCount[a[[x]], 10, 0]], {x, 1, Length[a]}]; d
(Select[Prime[Range[200]], PrimeQ[(2^# + 1)/3] &] - 3)/2 (* Amiram Eldar, Oct 13 2024 *)

a(n) = A023416(A000979(n)). - Michel Marcus, Nov 07 2013

a(n) = (A000978(n)-3)/2. - Amiram Eldar, Oct 13 2024

a(22)-a(29) from Vincenzo Librandi, Mar 31 2012

a(30)-a(41) from Amiram Eldar, Oct 13 2024

A127965 Number of bits in A127962(n).

Original entry on oeis.org

2, 4, 6, 10, 12, 16, 18, 22, 30, 42, 60, 78, 100, 126, 166, 190, 198, 312, 346, 700, 1708, 2616, 3538, 5806, 10500, 10690, 11278, 12390, 14478, 42736, 83338, 95368, 117238, 127030, 138936, 141078, 267016, 269986, 374320, 986190, 4031398

Offset: 1

Artur Jasinski, Feb 09 2007

Cf. A127962, A127963, A127964, A127961, A000979, A000978, A124400, A126614, A127955, A127956, A127957, A127958, A127936.

b = {}; Do[c = 1 + Sum[2^(2n - 1), {n, 1, x}]; If[PrimeQ[c], AppendTo[b, c]], {x, 0, 1000}]; a = {}; Do[AppendTo[a, FromDigits[IntegerDigits[b[[x]], 2]]], {x, 1, Length[b]}]; d = {}; Do[AppendTo[d, DigitCount[a[[x]], 10, 0]+DigitCount[a[[x]], 10, 1]], {x, 1, Length[a]}]; d

a(n) = A127964(n) + A127963(n).

a(n) = 1 + floor(log_2(A000979(n))) = 1 + floor(log_2(2^A000978(n)+1) - A020857) = A000978(n) - 1. - R. J. Mathar, Feb 01 2008

a(22)-a(29) from Vincenzo Librandi, Mar 30 2012

a(30)-a(41) from Amiram Eldar, Oct 19 2024

cp's OEIS Frontend

A127955 Composite numbers of the form (2^p+1)/3 where p is a prime.

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A127957 Numbers k such that (2^prime(k) + 1)/3 is composite.

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A127956 Prime numbers p such that (2^p+1)/3 is composite.

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A127958 Numbers x such that 1 + Sum_{k=1..n} 2^(2k-1) is not prime for n=1,2,...,x.

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A049883 Primes in the Jacobsthal sequence (A001045).

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A123627 Smallest prime q such that (q^p+1)/(q+1) is prime, where p = prime(n); or 0 if no such prime q exists.

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A123628 Smallest prime of the form (q^p+1)/(q+1), where p = prime(n) and q is also prime (q = A123627(n)); or 1 if such a prime does not exist.

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A127963 Number of 1's in A127962(n).

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A127964 Number of 0's in the binary expansion of A127962(n).

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