A008864 -id:A008864 - cp's OEIS frontend

A299379 Numbers k such that k * 16^k - 1 is prime.

167, 189, 639

Offset: 1

Tim Johannes Ohrtmann, Feb 08 2018

a(4) > 500000.

Steven Harvey, Generalized Woodall Search
Wikipedia, Woodall number

Numbers n such that n * b^n - 1 is prime: A008864 (b=1), A002234 (b=2), A006553 (b=3), A086661 (b=4), A059676 (b=5), A059675 (b=6), A242200 (b=7), A242201 (b=8), A242202 (b=9), A059671 (b=10), A299374 (b=11), A299375 (b=12), A299376 (b=13), A299377 (b=14), A299378 (b=15), this sequence (b=16), A299380 (b=17), A299381 (b=18), A299382 (b=19), A299383 (b=20).

[n: n in [1..10000] |IsPrime(n*16^n-1)];

Select[Range[1, 10000], PrimeQ[n*16^n-1] &]

for(n=1, 10000, if(isprime(n*16^n-1), print1(n", ")))

A299380 Numbers k such that k * 17^k - 1 is prime.

Original entry on oeis.org

2, 18, 20, 38, 68, 3122, 3488, 39500

Offset: 1

Tim Johannes Ohrtmann, Feb 08 2018

a(9) > 400000.

Steven Harvey, Generalized Woodall Search
Wikipedia, Woodall number

Numbers n such that n * b^n - 1 is prime: A008864 (b=1), A002234 (b=2), A006553 (b=3), A086661 (b=4), A059676 (b=5), A059675 (b=6), A242200 (b=7), A242201 (b=8), A242202 (b=9), A059671 (b=10), A299374 (b=11), A299375 (b=12), A299376 (b=13), A299377 (b=14), A299378 (b=15), A299379 (b=16), this sequence (b=17), A299381 (b=18), A299382 (b=19), A299383 (b=20).

[n: n in [1..10000] |IsPrime(n*17^n-1)];

Select[Range[1, 10000], PrimeQ[n*17^n-1] &]

for(n=1, 10000, if(isprime(n*17^n-1), print1(n", ")))

A299381 Numbers k such that k * 18^k - 1 is prime.

Original entry on oeis.org

1, 2, 6, 8, 10, 28, 30, 39, 45, 112, 348, 380, 458, 585, 17559, 38751, 43346, 46984, 92711

Offset: 1

Tim Johannes Ohrtmann, Feb 08 2018

a(20) > 400000.

Steven Harvey, Generalized Woodall Search
Wikipedia, Woodall number

Numbers n such that n * b^n - 1 is prime: A008864 (b=1), A002234 (b=2), A006553 (b=3), A086661 (b=4), A059676 (b=5), A059675 (b=6), A242200 (b=7), A242201 (b=8), A242202 (b=9), A059671 (b=10), A299374 (b=11), A299375 (b=12), A299376 (b=13), A299377 (b=14), A299378 (b=15), A299379 (b=16), A299380 (b=17), this sequence (b=18), A299382 (b=19), A299383 (b=20).

[n: n in [1..10000] |IsPrime(n*18^n-1)];

Select[Range[1, 10000], PrimeQ[n*18^n-1] &]

for(n=1, 10000, if(isprime(n*18^n-1), print1(n", ")))

A299382 Numbers k such that k * 19^k - 1 is prime.

Original entry on oeis.org

12, 410, 33890, 91850, 146478, 189620, 280524

Offset: 1

Tim Johannes Ohrtmann, Feb 08 2018

a(8) > 400000.

Steven Harvey, Generalized Woodall Search.
Wikipedia, Woodall number.

Numbers n such that n * b^n - 1 is prime: A008864 (b=1), A002234 (b=2), A006553 (b=3), A086661 (b=4), A059676 (b=5), A059675 (b=6), A242200 (b=7), A242201 (b=8), A242202 (b=9), A059671 (b=10), A299374 (b=11), A299375 (b=12), A299376 (b=13), A299377 (b=14), A299378 (b=15), A299379 (b=16), A299380 (b=17), A299381 (b=18), this sequence (b=19), A299383 (b=20).

[n: n in [1..10000] |IsPrime(n*19^n-1)];

Select[Range[1, 10000], PrimeQ[n*19^n-1] &]

for(n=1, 10000, if(isprime(n*19^n-1), print1(n", ")))

A299383 Numbers k such that k * 20^k - 1 is prime.

Original entry on oeis.org

1, 18, 44, 60, 80, 123, 429, 1166, 2065, 8774, 35340, 42968, 50312, 210129

Offset: 1

Tim Johannes Ohrtmann, Feb 08 2018

a(15) > 400000.

Steven Harvey, Generalized Woodall Search
Wikipedia, Woodall number

Numbers n such that n * b^n - 1 is prime: A008864 (b=1), A002234 (b=2), A006553 (b=3), A086661 (b=4), A059676 (b=5), A059675 (b=6), A242200 (b=7), A242201 (b=8), A242202 (b=9), A059671 (b=10), A299374 (b=11), A299375 (b=12), A299376 (b=13), A299377 (b=14), A299378 (b=15), A299379 (b=16), A299380 (b=17), A299381 (b=18), A299382 (b=19), this sequence (b=20).

[n: n in [1..10000] |IsPrime(n*20^n-1)];

Select[Range[1, 10000], PrimeQ[n*20^n-1] &]

for(n=1, 10000, if(isprime(n*20^n-1), print1(n", ")))

A078892 Numbers n such that phi(n) - 1 is prime, where phi is Euler's totient function (A000010).

Original entry on oeis.org

5, 7, 8, 9, 10, 12, 13, 14, 15, 16, 18, 19, 20, 21, 24, 25, 26, 27, 28, 30, 31, 33, 35, 36, 38, 39, 42, 43, 44, 45, 49, 50, 51, 52, 54, 56, 61, 62, 64, 65, 66, 68, 69, 70, 72, 73, 77, 78, 80, 81, 84, 86, 90, 91, 92, 93, 95, 96, 98, 99, 102, 103, 104, 105, 109, 111, 112, 117

Offset: 1

Reinhard Zumkeller, Dec 12 2002

For all primes p: p is in the sequence iff p is the greater member of a twin prime pair (A006512), see A078893.

Union of A006512 and A078893. - Ray Chandler, May 26 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000010, A000040, A008864, A109606, A006512, A039698, A072281, A078893.

[n: n in [1..200] | IsPrime(EulerPhi(n)-1)]; // Vincenzo Librandi, Aug 13 2013

Select[Range[200], PrimeQ[EulerPhi[#] - 1]&] (* Vincenzo Librandi, Aug 13 2013 *)

is(n)=isprime(eulerphi(n)-1) \\ Charles R Greathouse IV, Feb 21 2013

A246397 Numbers n such that Phi(12, n) is prime, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

2, 3, 4, 5, 9, 10, 12, 13, 17, 25, 27, 30, 31, 36, 38, 39, 43, 48, 52, 55, 56, 61, 62, 65, 83, 92, 94, 99, 100, 104, 105, 109, 114, 118, 126, 131, 166, 168, 169, 172, 183, 185, 190, 194, 196, 198, 209, 224, 225, 229, 231, 239, 244, 257, 260, 261, 263, 269, 270, 272, 278, 291, 296, 299, 300, 302, 308, 311

Offset: 1

Eric Chen, Nov 13 2014

nonn

Numbers n such that n^4-n^2+1 is prime, or numbers n such that A060886(n) is prime.

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

Cf. A008864 (1), A006093 (2), A002384 (3), A005574 (4), A049409 (5), A055494 (6), A100330 (7), A000068 (8), A153439 (9), A246392 (10), A162862 (11), this sequence (12), A217070 (13), A006314 (16), A217071 (17), A164989 (18), A217072 (19), A217073 (23), A153440 (27), A217074 (29), A217075 (31), A006313 (32), A097475 (36), A217076 (37), A217077 (41), A217078 (43), A217079 (47), A217080 (53), A217081 (59), A217082 (61), A006315 (64), A217083 (67), A217084 (71), A217085 (73), A217086 (79), A153441 (81), A217087 (83), A217088 (89), A217089 (97), A006316 (128), A153442 (243), A056994 (256), A056995 (512), A057465 (1024), A057002 (2048), A088361 (4096), A088362 (8192), A226528 (16384), A226529 (32768), A226530 (65536).

Cf. A060886, A125258, A085398, A124990.

A246397:=n->`if`(isprime(n^4-n^2+1),n,NULL): seq(A246397(n),n=1..300); # Wesley Ivan Hurt, Nov 14 2014

Select[Range[350], PrimeQ[Cyclotomic[12, #]] &] (* Vincenzo Librandi, Jan 17 2015 *)

for(n=1,10^3,if(isprime(polcyclo(12,n)),print1(n,", "))); \\ Joerg Arndt, Nov 13 2014

A346868 Sum of divisors of the numbers with no middle divisors.

Original entry on oeis.org

4, 6, 8, 18, 12, 14, 24, 18, 20, 32, 36, 24, 42, 40, 30, 32, 48, 54, 38, 60, 56, 42, 44, 84, 72, 48, 72, 98, 54, 72, 80, 90, 60, 62, 96, 84, 68, 126, 96, 72, 74, 114, 124, 140, 168, 80, 126, 84, 108, 132, 120, 90, 168, 128, 144, 120, 98, 102, 216, 104, 192, 162, 108, 110

Offset: 1

Omar E. Pol, Aug 18 2021

nonn

The characteristic shape of the symmetric representation of a(n) consists in that in the main diagonal of the diagram the width is equal to zero.
So knowing this characteristic shape we can know if a number has middle divisors  (or not) just by looking at the diagram, even ignoring the concept of middle divisors.
Therefore we can see a geometric pattern of the distribution of the numbers with no middle divisors in the stepped pyramid described in A245092.
For the definition of "width" see A249351.
All terms are even numbers.

			a(4) = 18 because the sum of divisors of the fourth number with no middle divisors (i.e., 10) is 1 + 2 + 5 + 10 = 18.
On the other hand we can see that in the main diagonal of every diagram the width is equal to zero as shown below.
Illustration of initial terms:
m(n) = A071561(n).
.
   n   m(n) a(n)   Diagram
.                      _   _   _     _ _   _ _     _   _   _ _ _     _
                      | | | | | |   | | | | | |   | | | | | | | |   | |
                   _ _|_| | | | |   | | | | | |   | | | | | | | |   | |
   1    3    4    |_ _|  _|_| | |   | | | | | |   | | | | | | | |   | |
                   _ _ _|    _|_|   | | | | | |   | | | | | | | |   | |
   2    5    6    |_ _ _|  _|    _ _| | | | | |   | | | | | | | |   | |
                   _ _ _ _|     |  _ _|_| | | |   | | | | | | | |   | |
   3    7    8    |_ _ _ _|  _ _|_|    _ _|_| |   | | | | | | | |   | |
                            |  _|     |  _ _ _|   | | | | | | | |   | |
                   _ _ _ _ _| |      _|_|    _ _ _|_| | | | | | |   | |
   4   10   18    |_ _ _ _ _ _|  _ _|       |    _ _ _|_| | | | |   | |
   5   11   12    |_ _ _ _ _ _| |  _|      _|   |  _ _ _ _|_| | |   | |
                   _ _ _ _ _ _ _| |      _|  _ _| | |  _ _ _ _|_|   | |
   6   13   14    |_ _ _ _ _ _ _| |  _ _|  _|    _| | |    _ _ _ _ _| |
   7   14   24    |_ _ _ _ _ _ _ _| |     |     |  _|_|   |  _ _ _ _ _|
                                    |  _ _|  _ _|_|       | |
                   _ _ _ _ _ _ _ _ _| |  _ _|  _|        _|_|
   8   17   18    |_ _ _ _ _ _ _ _ _| | |_ _ _|         |
                   _ _ _ _ _ _ _ _ _ _| |  _ _|        _|
   9   19   20    |_ _ _ _ _ _ _ _ _ _| | |        _ _|
                   _ _ _ _ _ _ _ _ _ _ _| |  _ _ _|
  10   21   32    |_ _ _ _ _ _ _ _ _ _ _| | |  _ _|
  11   22   36    |_ _ _ _ _ _ _ _ _ _ _ _| | |
  12   23   24    |_ _ _ _ _ _ _ _ _ _ _ _| | |
                                            | |
                   _ _ _ _ _ _ _ _ _ _ _ _ _| |
  13   26   42    |_ _ _ _ _ _ _ _ _ _ _ _ _ _|
.

Cf. A000203, A067742, A071090, A071561, A071562, A237591, A237593, A245092, A249351, A262626, A281007, A299777, A346864.

Some sequences that gives sum of divisors: A000225 (of powers of 2), A008864 (of prime numbers), A065764 (of squares), A073255 (of composites), A074285 (of triangular numbers, also of generalized hexagonal numbers), A139256 (of perfect numbers), A175926 (of cubes), A224613 (of multiples of 6), A346865 (of hexagonal numbers), A346866 (of second hexagonal numbers), A346867 (of numbers with middle divisors).

s[n_] := Module[{d = Divisors[n]}, If[AnyTrue[d, Sqrt[n/2] <= # < Sqrt[n*2] &], 0, Plus @@ d]]; Select[Array[s, 110], # > 0 &] (* Amiram Eldar, Aug 19 2021 *)

is(n) = fordiv(n, d, if(sqrt(n/2) <= d && d < sqrt(2*n), return(0))); 1; \\ A071561 apply(sigma, select(is, [1..150])) \\ Michel Marcus, Aug 19 2021

a(n) = A000203(A071561(n)).

A363636 Indices of numbers of the form k^2+1, k >= 0, that can be written as a product of smaller numbers of that same form.

Original entry on oeis.org

0, 3, 7, 13, 17, 18, 21, 31, 38, 43, 47, 57, 68, 73, 91, 99, 111, 117, 123, 132, 133, 157, 183, 211, 241, 242, 253, 255, 268, 273, 293, 302, 307, 313, 322, 327, 343, 381, 413, 421, 438, 443, 463, 487, 507, 515, 553, 557, 577, 593, 601, 651, 693, 697, 703, 707

Offset: 1

Pontus von Brömssen, Jun 19 2023

nonn

For the corresponding sequence for numbers of the form k^3+1 instead of k^2+1, the only terms known to me are 0 and 26, with 26^3+1 = (2^3+1)^2*(6^3+1).

			0 is a term because 0^2+1 = 1 equals the empty product.
3 is a term because 3^2+1 = 10 = 2*5 = (1^2+1)*(2^2+1).
38 is a term because 38^2+1 = 1445 = 5*17*17 = (2^2+1)*(4^2+1)^2. (This is the first term that requires more than two factors.)

David Trimas, Table of n, a(n) for n = 1..2260

Sequences that list those terms (or their indices or some other key) of a given sequence that are products of smaller terms of the same sequence (in other words, the nonprimitive terms of the multiplicative closure of the sequence):
  A018252 (A000027),
  A034878 (A000142),
  A068143 (A000217),
  A363492 (A000041),
  A363634 (A000959),
  A363635 (A003309),
  this sequence (A002522),
  A363637 (A005563),
  A363638 (A008864),
  A363750 (A006093),
  A364151 (A000292),
  A374372 (A000326),
  A374373 (A000384),
  A374374 (A002378),
  A374375 (A007531),
  A374500 (A000330),
  A374501 (A002411),
  A374502 (A002412).

g[lst_, p_] :=
  Module[{t, i, j},
   Union[Flatten[Table[t = lst[[i]]; t[[j]] = p*t[[j]];
      Sort[t], {i, Length[lst]}, {j, Length[lst[[i]]]}], 1],
    Table[Sort[Append[lst[[i]], p]], {i, Length[lst]}]]];
multPartition[n_] :=
  Module[{i, j, p, e, lst = {{}}}, {p, e} =
    Transpose[FactorInteger[n]];
   Do[lst = g[lst, p[[i]]], {i, Length[p]}, {j, e[[i]]}]; lst];
output = Join[{0}, Flatten[Position[Table[
     test = Sqrt[multPartition[n^2 + 1][[2 ;; All]] - 1];
     Count[AllTrue[#, IntegerQ] & /@ test, True] > 0
     , {n, 707}], True]]]
(* David Trimas, Jul 23 2023 *)

A008331 a(n) = phi(prime(n)+1).

Original entry on oeis.org

2, 2, 2, 4, 4, 6, 6, 8, 8, 8, 16, 18, 12, 20, 16, 18, 16, 30, 32, 24, 36, 32, 24, 24, 42, 32, 48, 36, 40, 36, 64, 40, 44, 48, 40, 72, 78, 80, 48, 56, 48, 72, 64, 96, 60, 80, 104, 96, 72, 88, 72, 64, 110, 72, 84, 80, 72, 128, 138, 92, 140, 84, 120, 96, 156, 104, 164, 156, 112, 120, 116

Offset: 1

N. J. A. Sloane

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000010, A008864, A363700.

for i from 1 to 500 do if isprime(i) then print(phi(i+1)); fi; od;

EulerPhi[1 + Prime@ Range@ 71] (* Michael De Vlieger, Feb 17 2021 *)

a(n) = eulerphi(prime(n)+1); \\ Michel Marcus, Feb 17 2021

a(n) = A000010(A008864(n)). - Michel Marcus, Feb 17 2021

Offset 1 from Michel Marcus, Feb 17 2021

cp's OEIS Frontend

A299379 Numbers k such that k * 16^k - 1 is prime.

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Magma

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A299380 Numbers k such that k * 17^k - 1 is prime.

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A299381 Numbers k such that k * 18^k - 1 is prime.

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Magma

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A299382 Numbers k such that k * 19^k - 1 is prime.

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A299383 Numbers k such that k * 20^k - 1 is prime.

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A078892 Numbers n such that phi(n) - 1 is prime, where phi is Euler's totient function (A000010).

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A246397 Numbers n such that Phi(12, n) is prime, where Phi is the cyclotomic polynomial.

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A346868 Sum of divisors of the numbers with no middle divisors.

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