A127579 -id:A127579 - cp's OEIS frontend

A127578 Primes congruent to 31 mod 32.

31, 127, 191, 223, 383, 479, 607, 863, 991, 1087, 1151, 1279, 1439, 1471, 1567, 1663, 1759, 1823, 1951, 2111, 2143, 2207, 2239, 2399, 2591, 2687, 2719, 2879, 3167, 3359, 3391, 3583, 3967, 4127, 4159, 4447, 4639, 4703, 4799, 4831, 5023, 5087, 5119, 5279

Offset: 1

Artur Jasinski, Jan 19 2007

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

Cf. A000040, A127575, A127576, A127577, A127579, A127580, A127581.

[p: p in PrimesUpTo(6000) | p mod 32 eq 31 ]; // Vincenzo Librandi, Aug 14 2012

lst={};Do[p=Prime[n];If[Mod[p,32]==31,AppendTo[lst,p]],{n,7!}];lst (* Vladimir Joseph Stephan Orlovsky, Feb 26 2009 *)
Select[Prime[Range[6000]],Mod[#,32]==31&] (* Vincenzo Librandi, Aug 14 2012 *)
Select[Range[31,5300,32],PrimeQ] (* Harvey P. Dale, Oct 03 2012 *)

is(n)=isprime(n) && n%32==31 \\ Charles R Greathouse IV, Jul 01 2016

Corrected by N. J. A. Sloane, Jul 11 2008

Edited by N. J. A. Sloane at the suggestion of R. J. Mathar, Jul 20 2008

A127577 Numbers n for which 32n+31 is prime.

Original entry on oeis.org

3, 5, 6, 11, 14, 18, 26, 30, 33, 35, 39, 44, 45, 48, 51, 54, 56, 60, 65, 66, 68, 69, 74, 80, 83, 84, 89, 98, 104, 105, 111, 123, 128, 129, 138, 144, 146, 149, 150, 156, 158, 159, 164, 168, 170, 171, 180, 188, 189, 191, 195, 198

Offset: 1

Artur Jasinski, Jan 19 2007

nonn

Cf. A127575, A127576, A127578, A127579, A127580, A127581.

a = {}; Do[If[PrimeQ[32n + 31], AppendTo[a, n]], {n, 1, 200}]; a
Select[Range[200],PrimeQ[32#+31]&] (* Harvey P. Dale, Dec 14 2024 *)

A101999 Primes of the form 64k-1 such that 4k-1, 8k-1, 16k-1 and 32*k-1 are also primes.

Original entry on oeis.org

2879, 858239, 1014719, 2029439, 2034239, 4068479, 4737599, 5454719, 9717119, 12968639, 17107199, 17962559, 25579199, 25945919, 29135999, 29859839, 30602879, 30735359, 32725439, 34214399, 34526399, 35925119, 36449279

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 23 2004

			4*45-1 = 179, 8*45-1 = 359, 16*45-1 = 719, 32*45-1 = 1439 and 64*45-1 = 2879 are primes, so 2879 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A101994, A101995, A101996, A101997, A101998.

Subsequence of A127579.

64#-1&/@Select[Range[570000],AllTrue[#*2^Range[2,6]-1,PrimeQ]&] (* Harvey P. Dale, Aug 07 2021 *)

is(k) = if(k % 64 == 63, my(m = k\64 + 1); isprime(4*m-1) && isprime(8*m-1) && isprime(16*m-1) && isprime(32*m-1) && isprime(64*m-1), 0); \\ Amiram Eldar, May 13 2024

a(n) = 64*A101994(n) - 1 = 16*A101995(n) + 15 = 8*A101996(n) + 7 = 4*A101997(n) + 3 = 2*A101998(n) + 1. - Amiram Eldar, May 13 2024

A306431 Least number x > 1 such that n*x divides 1 + Sum_{k=1..x-1} k^(x-1).

Original entry on oeis.org

2, 3, 13, 7, 19, 31, 41, 31, 13, 19, 43, 31, 23, 83, 139, 31, 61, 67, 113, 79, 251, 43, 19, 31, 199, 23, 13, 167, 53, 139, 83, 127, 157, 67, 293, 431, 443, 151, 103, 79, 61, 251, 113, 47, 337, 19, 179, 31, 41, 199, 67, 23, 19, 499, 181, 367, 607, 139, 257, 359

Offset: 1

Paolo P. Lava, Apr 05 2019

If n = 1, all the solutions of x | 1 + Sum_{k=1..x-1} k^(x-1) should be prime numbers, according to Giuga's conjecture.
If n*x | 1 + Sum_{k=1..x-1} k^(x-1), then certainly x does, so Giuga's conjecture would say x must be prime. Similarly if x^n divides it, so does x, so again Giuga would say x is prime. - Robert Israel, Apr 26 2019
E.g., the first solution for x^2 | 1 + Sum_{k=1..x-1} k^(x-1) is x = 1277, that is prime.

			a(4) = 7 because (1 + 1^6 + 2^6 + 3^6 + 4^6 + 5^6 + 6^6) / (4*7) = 67172 / 28 = 2399 and it is the least prime to have this property.

Robert Israel, Table of n, a(n) for n = 1..2000
Eric Weisstein's World of Mathematics, Giuga's Conjecture

Cf. A191677. All the solutions for n = m: A000040 (m=1), A002145 (m=2), A007522 (m=4), A127576 (m=8), A141887 (m=10), A127578 (m=16), A142198 (m=20), A127579 (m=32), A095995 (m=50).

P:=proc(j) local k,n; for n from 2 to 10^6 do
if frac((add(k^(n-1),k=1..n-1)+1)/(j*n))=0
then RETURN(n); break; fi; od; end: seq(P(i),i=1..60);

a[n_] := For[x = 2, True, x++, If[Divisible[1+Sum[k^(x-1), {k, x-1}], n x], Return[x]]];
Array[a, 60] (* Jean-François Alcover, Oct 16 2020 *)

a(n) = my(x=2); while (((1 + sum(k=1, x-1, k^(x-1))) % (n*x)), x++); x; \\ Michel Marcus, Apr 27 2019

Least solution of n*x | 1 + Sum_{k=1..x-1} k^(x-1), for n = 1, 2, 3, ...

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A127578 Primes congruent to 31 mod 32.

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A127577 Numbers n for which 32n+31 is prime.

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A101999 Primes of the form 64*k-1 such that 4*k-1, 8*k-1, 16*k-1 and 32*k-1 are also primes.

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A306431 Least number x > 1 such that n*x divides 1 + Sum_{k=1..x-1} k^(x-1).

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A101999 Primes of the form 64k-1 such that 4k-1, 8k-1, 16k-1 and 32*k-1 are also primes.