A213529 -id:A213529 - cp's OEIS frontend

A222534 Smallest Sierpinski number that is divisible by the n-th prime.

7592506760633776533, 36293948155, 157957457, 603713, 422590909, 78557, 6134663, 1259779, 575041, 7892569, 2931991, 4095859, 2541601, 7892569, 29169451, 271577, 35193889, 12824269, 603713, 9454157, 575041, 7696009, 5455789, 41561687, 7400371, 2191531, 29046541, 2931991

Offset: 2

Arkadiusz Wesolowski, Feb 24 2013

nonn

For an odd prime p and odd k, if p divides k, then p does not divide k*2^n + 1 for any n.

			603713 is first Sierpinski number that is divisible by 11, the 5th prime - so a(5) = 603713.

Arkadiusz Wesolowski, Table of n, a(n) for n = 2..1000
Chris Caldwell, The Prime Glossary: Sierpinski number
Carlos Rivera, Problem 86. A smaller Sierpinski integer divided by 3, The Prime Puzzles & Problems Connection.

Cf. A076336, A187716, A213529, A364413.

Offset changed and initial term added by Arkadiusz Wesolowski, May 11 2017

a(2) corrected by Arkadiusz Wesolowski, Jul 27 2023

A305473 Let k be a Sierpiński or Riesel number divisible by 2n - 1, and let p be the largest number in a set of primes which cover every number of the form k2^m + 1 (or of the form k*2^m - 1) with m >= 1. a(n) = p if and only if there exists no number k that has a covering set with largest prime < p.

Original entry on oeis.org

73, 257, 151, 151, 257, 73, 151, 1321, 73, 109, 1321, 73, 151, 257, 73, 73, 331, 257, 109, 331, 73, 73, 1321, 73, 151, 331, 73, 241

Offset: 1

Arkadiusz Wesolowski, Jun 02 2018

nonn

R. G. Stanton found that a(2) = 257.
a(n) >= 73 for any n, see [Stanton].
There exist infinitely many Riesel numbers that are divisible by 15. The number 334437671621489828385689959795356586832846847109919809460835 is one such number.

			Examples of the covering sets:
- for n = 2, the set is {5, 7, 11, 13, 17, 19, 31, 37, 41, 61, 73, 97, 109, 151, 241, 257},
- for n = 3, the set is {3, 7, 11, 13, 19, 31, 37, 41, 61, 73, 109, 151},
- for n = 4, the set is {3, 5, 11, 13, 19, 31, 37, 41, 61, 73, 151},
- for n = 6, the set is {3, 5, 7, 13, 19, 37, 73},
- for n = 7, the set is {3, 5, 7, 11, 19, 31, 37, 41, 61, 73, 151},
- for n = 8, the set is {7, 11, 13, 17, 19, 29, 31, 37, 41, 43, 61, 71, 73, 97, 109, 113, 127, 151, 193, 211, 241, 257, 281, 331, 337, 421, 433, 577, 673, 1153, 1321},
- for n = 11, the set is {5, 11, 13, 17, 19, 31, 37, 41, 61, 73, 97, 109, 151, 181, 193, 241, 257, 331, 433, 577, 631, 673, 1153, 1321},
- for n = 17, the set is {5, 7, 13, 17, 19, 31, 37, 41, 61, 73, 97, 109, 151, 241, 257, 331},
- for n = 18, the set is {3, 11, 13, 17, 19, 31, 37, 41, 61, 73, 97, 109, 151, 241, 257},
- for n = 20, the set is {5, 7, 11, 17, 19, 31, 37, 41, 61, 73, 97, 109, 151, 241, 257, 331},
- for n = 26, the set is {5, 7, 11, 13, 19, 31, 37, 41, 61, 73, 97, 109, 151, 241, 257, 331},
- for n = 28, the set is {3, 7, 13, 17, 19, 37, 73, 109, 241}.

R. G. Stanton, Further results on covering integers of the form 1 + k * 2^n by primes, pp. 107-114 in: Kevin L. McAvaney (ed.), Combinatorial Mathematics VIII, Lecture Notes in Mathematics 884, Berlin: Springer, 1981.

Carlos Rivera, Problem 49. Sierpinski-like numbers, The Prime Puzzles and Problems Connection.

Cf. A076336, A101036, A187714, A187716, A213529, A222534, A244071, A244562, A306151.

a(((2*n-1)^b+1)/2) = a(n) for every b >= 2.
a((2*b-1)*n-b+1) >= a(n) for every b >= 2; n > 1.
a(n) = 73 if and only if gcd(2*n-1, 70050435) = 1.

A276458 Smallest odd number not of the form p + 2^k with p prime and k >= 0 that is divisible by the n-th prime.

Original entry on oeis.org

1719, 905, 959, 1199, 1807, 1207, 2983, 1541, 2465, 1271, 5143, 1271, 2279, 1927, 2279, 1829, 5917, 1541, 1207, 2263, 3239, 7387, 4717, 1649, 6161, 4841, 7169, 1199, 1243, 127, 10873, 959, 1529, 149, 11023, 2669, 12877, 2171, 1211, 1969, 905, 1719, 7913, 7289

Offset: 2

Arkadiusz Wesolowski, Sep 03 2016

nonn

a(n) <= A213529(n).

			a(3) = 905 because it is the smallest de Polignac number (A006285) divisible by the third prime.

Robert Israel, Table of n, a(n) for n = 2..3000

Cf. A006285, A098237, A213529.

lst:=[]; for r in [2..45] do p:=NthPrime(r); n:=-p; f:=0; while IsZero(f) do n:=n+2*p; k:=-1; repeat k+:=1; a:=n-2^k; until a lt 1 or IsPrime(a); if a lt 1 then Append(~lst, n); f:=1; end if; end while; end for; lst;

N:= 10^5: # to use de Polignac numbers <= N
P:= select(isprime,{2,seq(i,i=3..N,2)}):
dP:= {seq(i,i=1..N,2)}:
for k from 0 to ilog2(N) do
  dP:= dP minus map(`+`,P,2^k)
od:
for m from 2 do
   R:= ListTools:-SelectFirst(1, t -> t mod P[m] = 0, dP);
   if R = {} then break fi;
   A[m]:= R[1];
od:
seq(A[i],i=2..m-1); # Robert Israel, Sep 06 2016

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A222534 Smallest Sierpinski number that is divisible by the n-th prime.

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A276458 Smallest odd number not of the form p + 2^k with p prime and k >= 0 that is divisible by the n-th prime.

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