A047845 -id:A047845 - cp's OEIS frontend

A104278 Numbers m such that 2m+1 and 2m-1 are not primes.

13, 17, 25, 28, 32, 38, 43, 46, 47, 58, 59, 60, 61, 62, 67, 71, 72, 73, 77, 80, 85, 88, 92, 93, 94, 101, 102, 103, 104, 107, 108, 109, 110, 118, 122, 123, 124, 127, 130, 133, 137, 143, 144, 145, 148, 149, 150, 151, 152, 160, 161, 162, 163, 164, 167, 170, 171, 172

Offset: 1

Alexandre Wajnberg, Apr 17 2005

Complement of A147820. - Omar E. Pol, Nov 17 2008

m is in the sequence iff A177961(m)Vladimir Shevelev, May 16 2010

			a(1)=13 is the first number satisfying simultaneously the two rules.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Intersection of A047845 and A104275.
Cf. A040040, A147820, A025583.
Cf. A010051, A002808, A099047.

a104278 n = a104278_list !! (n-1)
a104278_list = [m | m <- [1..],
                    a010051' (2 * m - 1) == 0 && a010051' (2 * m + 1) == 0]
-- Reinhard Zumkeller, Aug 04 2015

Select[ Range[300], !PrimeQ[2# + 1] && !PrimeQ[2# - 1] &] (* Robert G. Wilson v, Apr 18 2005 *)
Select[Range[300],NoneTrue[2#+{1,-1},PrimeQ]&] (* The program uses the NoneTrue function from Mathematica version 10 *)  (* Harvey P. Dale, Jul 07 2015 *)

select( {is_A104278(n)=!isprime(2*n-1)&&!isprime(2*n+1)}, [1..222]) \\ M. F. Hasler, Apr 29 2024

a(n) = (A025583-1)/2. - Bill McEachen, Feb 05 2025

More terms from Robert G. Wilson v, Apr 18 2005

A193773 Number of ways to write n as 2xy - x - y with 1 <= x <= y.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 3, 1, 2, 2, 1, 2, 2, 1, 1, 3, 2, 1, 2, 1, 1, 3, 2, 1, 3, 1, 2, 2, 1, 2, 2, 2, 1, 3, 1, 1, 4, 1, 1, 2, 1, 2, 3, 2, 2, 2, 2, 1, 2, 1, 2, 4, 1, 1, 2, 2, 2, 3, 1, 1, 3, 2, 1, 2, 2, 1, 4, 1, 2, 3

Offset: 0

Reinhard Zumkeller, Jan 02 2013

nonn

a(A005097(n)) = 1; for n > 1: a(A047845(n)) > 1. - Reinhard Zumkeller, Jan 02 2013

Number of ways to write 2*n+1 as a difference of two squares. Note that 2*(2*x*y - x - y) + 1 = (2*x - 1) * (2*y - 1) = (y + x - 1)^2 - (y - x)^2. - Michael Somos, Dec 23 2018

			G.f. = 1 + x + x^2 + x^3 + 2*x^4 + x^5 + x^6 + 2*x^7 + x^8 + x^9 + 2*x^10 + ... - _Michael Somos_, Dec 23 2018

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

Cf. A000005, A005097, A047845, A125203.

a193773 n = length [() | x <- [1 .. n + 1],
                         let (y,m) = divMod (x + n) (2 * x - 1),
                         x <= y, m == 0]

a[ n_] := If[ n < 0, 0, Ceiling[ DivisorSigma[0, 2 n + 1] / 2]]; (* Michael Somos, Dec 23 2018 *)

{a(n) = if(n < 0, 0, (numdiv(2*n+1) + 1)\2)}; /* Michael Somos, Dec 23 2018 */

a(n) = ceiling(A000005(2*n+1) / 2). - Michael Somos, Dec 23 2018

A153144 Numbers n such that 2*n+19 is not a prime.

Original entry on oeis.org

1, 3, 4, 7, 8, 10, 13, 15, 16, 18, 19, 22, 23, 25, 28, 29, 31, 33, 34, 36, 37, 38, 40, 43, 46, 48, 49, 50, 51, 52, 53, 55, 57, 58, 61, 62, 63, 64, 67, 68, 70, 71, 73, 75, 76, 78, 79, 82, 83, 84, 85, 88, 91, 92, 93, 94, 95, 97, 98, 99, 100, 101

Offset: 1

Vincenzo Librandi, Dec 19 2008

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

Cf. A153143, A153081, A155551.

Numbers n such that 2n+k is not prime: A047845 (k=1), A153238 (k=3), A153052 (k=5), A153053 (k=7), A153723 (k=9), A153083 (k=11), A153082 (k=13), A241571 (k=15), A241572 (k=17), this sequence (k=19).

[n: n in [1..120] | not IsPrime(2*n + 19)]; // Vincenzo Librandi, Dec 13 2012

Select[Range[0, 500], !PrimeQ[2# + 19] &] (* Vincenzo Librandi, Dec 13 2012 *)

A125713 Smallest odd prime p such that (n+1)^p - n^p is prime.

Original entry on oeis.org

3, 3, 3, 3, 5, 3, 7, 7, 3, 3, 3, 17, 3, 3, 43, 5, 3, 1607, 5, 19, 127, 229, 3, 3, 3, 13, 3, 3, 149, 3, 5, 3, 23, 3, 5, 83, 3, 3, 37, 7, 3, 3, 37, 5, 3, 5, 58543, 3, 3, 7, 29, 3, 479, 5, 3, 19, 5, 3, 4663, 54517, 17, 3, 3, 5, 7, 3, 3, 17, 11, 47, 61, 19, 23, 3, 5, 19, 7, 5, 7, 3, 3

Offset: 1

Alexander Adamchuk, Dec 01 2006, Feb 15 2007

Corresponding smallest primes of the form (n+1)^p - n^p, where p = a(n) is an odd prime, are listed in A121091(n+1) = {7, 19, 37, 61, 4651, 127, 1273609, 2685817, 271, 331, 397, 6431804812640900941, 547, 631, ...}. a(n) = A058013(n) for n = {4, 7, 10, 12, 13, 16, 17, 19, 22, 24, 25, 27, 28, 31, ...} = A047845(n) = (n-1)/2, where n runs through odd nonprimes (A014076), for n>1. a(97) = 7. a(99)..a(112) = {5, 43, 5, 13, 7, 5, 3, 6529, 59, 3, 5, 5, 113, 5}. a(114) = 139. a(117)..a(129) = {7, 13, 3, 5, 5, 7, 3, 5167, 3, 41, 59, 3, 3}. a(131) = 101. a(n) is currently unknown for n = {113, 115, 116, 130, 132, ...}.
a(96) = 1307, a(98) = 709.
a(137) is probably 196873 from a prime of this form discovered by Jean-Louis Charton in December 2009 and reported to Henri Lifchitz's PRP Top. - Robert Price, Feb 17 2012
a(138) through a(150) are 113, >32401, 3, 7, 3, 8839, 5, 7, 13, 3, 5, 271, 13. - Robert Price, Feb 17 2012
a(137) = 196873 confirmed by Fischer link; a(139) > 260000. - Ray Chandler, Feb 26 2017

Robert Price and Ray Chandler, Table of n, a(n) for n = 1..138 (first 136 terms from Robert Price)
Richard Fischer, Generalized primes of the form (B+1)^N - B^N.

Cf. A058013 (smallest prime p such that (n+1)^p - n^p is prime).
Cf. A065913 (smallest prime of form (n+1)^k - n^k).
Cf. A121091 (smallest nexus prime of the form n^p - (n-1)^p, where p is odd prime).
Cf. A047845, A014076.
Cf. A062585 (numbers n such that k^n - (k-1)^n is prime, where k is 19).
Cf. A000043, A057468, A059801, A059802, A062572-A062666.

A266410 a(n) = (A266419(n) - 1) / 2; numbers n such that 2n+1 is a nonludic number (in A192607).

Original entry on oeis.org

4, 7, 9, 10, 13, 15, 16, 17, 19, 22, 24, 25, 27, 28, 29, 31, 32, 34, 36, 37, 39, 40, 42, 43, 46, 47, 49, 50, 51, 52, 54, 55, 56, 58, 61, 62, 64, 66, 67, 68, 69, 70, 72, 73, 75, 76, 77, 79, 81, 82, 83, 84, 85, 88, 91, 92, 93, 94, 95, 97, 98, 99, 100, 101, 102, 103, 106, 107, 108, 109, 112, 114, 115, 118, 120, 121

Offset: 1

Antti Karttunen, Jan 28 2016

nonn

Odd nonludic numbers (A266419) decremented by one, then halved.

Antti Karttunen, Table of n, a(n) for n = 1..10220

Complement: A266409.
Cf. A192490, A192607, A266419.
Cf. permutations A266418, A266638.
Cf. also A047845.

a(n) = (A266419(n) - 1) / 2.

A075555 Smallest prime p such that p+n is a square, or 0 if no such p exists.

Original entry on oeis.org

3, 2, 13, 5, 11, 3, 2, 17, 7, 71, 5, 13, 3, 2, 181, 0, 19, 7, 17, 5, 43, 3, 2, 97, 11, 23, 37, 53, 7, 19, 5, 17, 3, 2, 29, 13, 107, 11, 61, 41, 23, 7, 101, 5, 19, 3, 2, 73, 0, 31, 13, 29, 11, 67, 89, 113, 7, 23, 5, 61, 3, 2, 37, 17, 79, 103, 257, 13, 31, 11, 29, 97, 71, 7, 181, 5

Offset: 1

Amarnath Murthy, Sep 23 2002

nonn

If n=A047845(i)^2 for some i, i.e. if n has the form ((k-1)/2)^2 with k odd but not prime, then a(n)=0. It is conjectured that these are the only values of n for which a(n)=0; this would follow from Schinzel's hypothesis.

			a(8) = 17 because 8 + 17 is the first square that can be made by adding a prime to 8.
a(16) = 0 because 16 + p cannot be x^2, since then p = x^2 - 16 = (x-4)(x+4).

T. D. Noe, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Schinzel's Hypothesis.

Cf. A075556.

a(n) = A105016(n)^2 - n, if a(n) exists.

a[n_] := If[IntegerQ[s=Sqrt[n]]&&!PrimeQ[2s+1], 0, For[x=Ceiling[s], True, x++, If[PrimeQ[x^2-n], Return[x^2-n]]]]

for(n=1,100,f=0:forprime(p=2,10^7,if(issquare(p+n),f=p:break)): if(f,print1(f","),print1("0,")))

More terms from Ralf Stephan, Mar 28 2003

Edited by Dean Hickerson, Mar 31 2003

A090767 Numbers of the form 3xyz + 2(xy + yz + zx) + (x + y + z) for x, y, z positive integers.

Original entry on oeis.org

12, 20, 28, 33, 36, 44, 46, 52, 54, 59, 60, 64, 68, 72, 75, 76, 82, 84, 85, 92, 96, 98, 100, 104, 105, 108, 111, 116, 117, 118, 124, 128, 132, 133, 136, 137, 138, 140, 144, 148, 150, 151, 154, 156, 159, 162, 163, 164, 170, 172, 174, 176, 180, 184, 188, 189, 190

Offset: 1

John H. Mason, Feb 02 2004

nonn

This is the set of numbers which count the unit sticks or unit segments needed to construct a three-dimensional cubic lattice made up from unit cubes. This generalizes the two-dimensional version which is A047845 (numbers of the form 2*x*y + x + y for x and y positive integers) and is also the numbers of sticks needed to construct a rectangular lattice of unit squares.

			a(1) = 12 because there are 12 edges to a cube.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A047845.

SeqGen1 := proc(n,N) local a,b,c,F,V,v; # n specifies the search space; N specifies the maximal number to appear in the initial segment of the sequence F := 3*x*y*z + 2*(x*y+y*z+z*x)+x+y+z; V := {}; for a from 1 to n do for b from 1 to n do for c from b to n do v := subs(x=a,y=b,F); if v < N then V := V union {v};fi; od;od; sort(V) end:
# alternative:
N:= 1000: # to get all terms <= N
S:= {seq(seq(seq(3*x*y*z + 2*(x*y+y*z+z*x)+(x+y+z),
z = 1 .. min(y, (-2*x*y+N-x-y)/(3*x*y+2*x+2*y+1))),
y = 1 .. min(x, (N-3*x-1)/(5*x+3))),
x = 1 .. (N-4)/8)}:
sort(convert(S,list)); # Robert Israel, Feb 18 2016

M = 1000;
S = Table[3 x y z + 2(x y + y z + z x) + (x + y + z), {x, 1, (M - 4)/8}, {y, 1, Min[x, (M - 3 x - 1)/(5 x + 3)]}, {z, 1, Min[y, (-2 x y + M - x - y)/(3 x y + 2 x + 2 y + 1)]}] // Flatten // Union (* Jean-François Alcover, Apr 11 2019, after Robert Israel *)

More terms from Ray Chandler, Feb 04 2004

A065824 Smallest solution m to (n+1)phi(m) = nsigma(m), or -1 if no solution exists.

Original entry on oeis.org

3, 5, 7, 323, 11, 13, 899, 17, 19, 1763, 23, 5249, 3239, 29, 31, 979801, 5459, 37, 10763, 41, 43, 9179, 47, 9701, 10403, 53, 12319, 5646547, 59, 61, 24569, 19109, 67, 19043, 71, 73, 22499, 50819, 79, 41309, 83, 32639, 46979, 89, 34579, 39059, 125969

Offset: 1

Labos Elemer, Nov 23 2001

If p = a(n) is a prime solution, then (n+1)*(p-1) = n*(p+1) and p = 2n+1, so position for p if it is in fact a minimal solution is at n = (p-1)/2. E.g. 29 appears at 14th position shown by A005097. On the other hand large and (seemingly always composite) solutions arise at indices shown essentially by A047845. Also, differences between the sites of two consecutive small prime solutions appears to be d/2, half the difference between consecutive primes (A001223).

Donovan Johnson, Table of n, a(n) for n = 1..456

Cf. A000010, A000203, A062699, A065818, A065819, A065822, A065823.

See also A005097, A047845, A014076, A001223.

max = 10^7; a[n_] := For[m = 3, m <= max, m++, If[(n+1)*EulerPhi[m] == n*DivisorSigma[1, m], Print[m]; Return[m]]] /. Null -> -1; Array[a, 50] (* Jean-François Alcover, Oct 08 2016 *)

from itertools import count
from math import prod
from sympy import factorint
def A065824(n):
    for m in count(1):
        f = factorint(m)
        if (n+1)*m*prod((p-1)**2 for p in f)==n*prod(p**(e+2)-p for p,e in f.items()):
            return m # Chai Wah Wu, Aug 12 2024

(n+1)*A000010(a(n)) = n*A000203(a(n)), smallest x=a(n) solutions.

A121091 Smallest nexus prime of the form n^p - (n-1)^p, where p is an odd prime.

Original entry on oeis.org

7, 19, 37, 61, 4651, 127, 1273609, 2685817, 271, 331, 397, 6431804812640900941, 547, 631, 5613125740675652943160572913465695837595324940170321, 371281, 919

Offset: 2

Alexander Adamchuk, Aug 11 2006, revised Dec 01 2006, Feb 15 2007

nonn

a(19) = 19^1607 - 18^1607, which is too large to include. It has 2055 decimal digits. See A062585(1) = 1607.

a(20)-a(21) = {723901, 8005616640331026125580781}. a(n) is currently known for all n up to n = 96. Corresponding smallest odd primes p such that (n+1)^p - n^p is prime are listed in A125713(n) = {3,3,3,3,5,3,7,7,3,3,3,17,3,3,43,5,3,10957,5,19,127,229,3,3,3,13,3,3,149,3,5,3,23,3,5,83,3,3,37,7,3,3,37,5,3,5,58543,...}. a(n+1) = A065013(n) for n = {4, 7, 10, 12, 13, 16, 17, 19, 22, 24, 25, 27, 28, 31, ...} = A047845(n) = (n-1)/2, where n runs through odd nonprimes (A014076), for n>1.

Cf. A121620, A000043, A058765, A121616, A121618.
Cf. A125713 = Smallest odd prime p such that (n+1)^p - n^p is prime. Cf. A065913 = Smallest prime of form (n+1)^k - n^k. Cf. A058013 = Smallest prime p such that (n+1)^p - n^p is prime. Cf. A047845, A014076.
Cf. A062585 = numbers n such that k^n - (k-1)^n is prime, where k is 19. Cf. A000043, A057468, A059801, A059802, A062572-A062666.

a(n) = n^A125713(n) - (n-1)^A125713(n).

A286680 Smallest nonnegative m such that (1 + n)^(2^m) + n is not prime.

Original entry on oeis.org

0, 5, 4, 2, 0, 3, 1, 0, 3, 3, 0, 1, 0, 0, 2, 4, 0, 0, 2, 0, 2, 1, 0, 2, 0, 0, 1, 0, 0, 2, 3, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 0, 1, 0, 1, 1, 0, 3, 2, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 1, 2, 0, 0, 0, 0, 1, 2, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 2

Offset: 0

Juri-Stepan Gerasimov, May 12 2017

nonn

Nonprimes: 1, 4294967297, 43046723, 259, 9, 1679621, 55, 15, 43046729, 100000009, 21, 155, 25, 27, 50639, 18446744073709551631, 33, 35, ...
Conjecture: a(n) <= 6 for all n.
This conjecture would contradict the generalized Bunyakovsky conjecture.  That is, the polynomials (1+n)^k+n for k=0..6 satisfy the conditions for that conjecture, and so there should be some n for which all seven are prime. - Robert Israel, May 17 2017
Smallest k such that (1 + k)^(2^n) + k is not prime: 0, 6, 3, 5, 2, 1, 54131988 (conjecturally finite). Last term found by Robert G. Wilson v, May 14 2017
From Robert G. Wilson v, May 18 2017: (Start)
m=
0: 0, 4, 7, 10, 12, 13, 16, 17, 19, 22, 24, 25, 27, 28, 31, 32, 34, 37, 38, etc.;
1: 6, 11, 21, 26, 33, 35, 36, 39, 41, 48, 50, 51, 56, 68, 74, 78, 81, 83, etc.;
2: 3, 14, 18, 20, 23, 29, 44, 54, 63, 65, 69, 75, 95, 99, 113, 114, 125, etc.;
3: 5, 8, 9, 30, 53, 119, 230, 308, 329, 350, 624, 638, 779, 785, 813, 1110, etc.;
4: 2, 15, 2100, 4223, 4773, 7868, 8744, 9339, 9540, 13178, 14589, 15884, etc.;
5: 1, 1432578, 1627035, 1737054, 1888094, 1959638, 2176139, 3172304, 3425069, etc.;
6: 54131988, 177386619, 229940778, 846372674, 2124404844, 2367307088, 2539775055, etc.;
(End)

			a(0) = 0 because (1 + 0)^(2^0) + 0 = 1 is not prime.

Robert G. Wilson v, Table of n, a(n) for n = 0..10000
Wikipedia, Bunyakovsky conjecture.

Cf. A019434, A047845, A057726, A160027.

f:= proc(n) local k;
  for k from 0 while isprime((1+n)^(2^k)+n) do od:
  k;
end proc:
map(f, [$0..100]); # Robert Israel, May 17 2017

f[n_] := Block[{k = 0}, While[ PrimeQ[(1 + n)^(2^k) + n], k++]; k]; Array[f, 105, 0] (* Robert G. Wilson v, May 14 2017 *)

a(n) = {my(m = 0); while (isprime((1 + n)^(2^m) + n), m++); m;} \\ Michel Marcus, May 19 2017

cp's OEIS Frontend

A104278 Numbers m such that 2m+1 and 2m-1 are not primes.

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A193773 Number of ways to write n as 2*x*y - x - y with 1 <= x <= y.

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A153144 Numbers n such that 2*n+19 is not a prime.

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A125713 Smallest odd prime p such that (n+1)^p - n^p is prime.

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A266410 a(n) = (A266419(n) - 1) / 2; numbers n such that 2n+1 is a nonludic number (in A192607).

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A075555 Smallest prime p such that p+n is a square, or 0 if no such p exists.

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A090767 Numbers of the form 3*x*y*z + 2(x*y + y*z + z*x) + (x + y + z) for x, y, z positive integers.

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A065824 Smallest solution m to (n+1)*phi(m) = n*sigma(m), or -1 if no solution exists.

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A193773 Number of ways to write n as 2xy - x - y with 1 <= x <= y.

A090767 Numbers of the form 3xyz + 2(xy + yz + zx) + (x + y + z) for x, y, z positive integers.

A065824 Smallest solution m to (n+1)phi(m) = nsigma(m), or -1 if no solution exists.