A061422 -id:A061422 - cp's OEIS frontend

A051015 Zeisel numbers.

105, 1419, 1729, 1885, 4505, 5719, 15387, 24211, 25085, 27559, 31929, 54205, 59081, 114985, 207177, 208681, 233569, 287979, 294409, 336611, 353977, 448585, 507579, 982513, 1012121, 1073305, 1242709, 1485609, 2089257, 2263811, 2953711, 3077705, 3506371, 3655861, 3812599

Offset: 1

Eric W. Weisstein

nonn

Pick any integers A and B and consider the linear recurrence relation given by p(0) = 1, p(i + 1) = A*p(i) + B. If for some n > 2, p(1), p(2), ..., p(n) are distinct primes, then the product of these primes is called a Zeisel number.

M. F. Hasler and Lars Blomberg, Table of n, a(n) for n = 1..9607 (first 70 terms from _M. F. Hasler_)
Kevin S. Brown, Zeisel Numbers, MathPages website.
OEIS Wiki, Zeisel numbers.
Eric Weisstein's World of Mathematics, Zeisel Number.
Wikipedia, Zeisel number.
Helmut Zeisel, Primes of the form 2^(k-1)+k, sci.math newsgroup, February 24, 1994.

Cf. A027746, A061422, A252094 (A values), A252095 (B values).

a051015 n = a051015_list !! (n-1)
a051015_list = filter zeisel [3, 5 ..] where
   zeisel x = 0 `notElem` ds && length ds > 2 &&
         all (== 0) (zipWith mod (tail ds) ds) && all (== q) qs
         where q:qs = (zipWith div (tail ds) ds)
               ds = zipWith (-) (tail ps) ps
               ps = 1 : a027746_row x
-- Reinhard Zumkeller, Dec 15 2014

maxTerm = 3*10^7; ZeiselQ[n_] := Module[{a, b, pp, eq, r}, If[PrimeQ[n] || ! SquareFreeQ[n], False, pp = Join[{1}, FactorInteger[n][[All, 1]]]; If[Length[pp] <= 3, False, eq = Thread[Rest[pp] == b + a*Most[pp]]; r = Reduce[eq, {a, b}, Integers]; r =!= False]]]; p = 3; A051015 = Reap[While[p^3 < maxTerm, q = NextPrime[p]; While[p*q^2 < maxTerm, If[ ! IntegerQ[a = (q - p)/(p - 1)] || !IntegerQ[b = (p^2 - q)/(p - 1)], q = NextPrime[q]; Continue[]]; r = b + a*q; n = r*p*q; While[PrimeQ[r] && n < maxTerm, Sow[n]; r = b + a*r; n *= r]; q = NextPrime[q]]; p = NextPrime[p]]][[2, 1]]; A051015 = Select[Sort[A051015], ZeiselQ] (* Jean-François Alcover, Oct 31 2012, with much help from Giovanni Resta *)

is_A051015(n)={#(n=factor(n)~)>2 & vecmax(n[2,])==1 & denominator(n[2,1]=(n[1,3]-n[1,2])/(n[1,2]-n[1,1]))==1 & #Set(n[1,]-n[2,1]*concat(1,vecextract(n[1,],"^-1")))==1} \\ - M. F. Hasler, Oct 31 2012

More terms from David Wasserman, Feb 19 2002
Extended by Karsten Meyer, Jun 08 2006, but values were incorrect. M. F. Hasler, Oct 31 2012
Values up to a(70) computed by Jean-François Alcover and double-checked by M. F. Hasler, Oct 31 2012
Values < 10^15 by Lars Blomberg, Nov 02 2012

A061421 Primes of the form 2^n+n+1.

Original entry on oeis.org

2, 7, 71, 110427941548649020598956093796432407239217743554726184882600387580788973

Offset: 1

Jason Earls, May 02 2001

nonn

Next term is 2^1884+1884+1, with 568 digits and is too large to include. - Emeric Deutsch, May 13 2006

The Wikipedia article "Zeisel number" gives a historical connection to A051015. - Jonathan Sondow, Oct 17 2017

K. S. Brown, The Sum of the Prime Factors of N (Part 2: Primes of the Form 2k-1 + k), Math Pages
Wikipedia, Zeisel number

Cf. A001580, A069539, A052007, A048744, A100357, A100358, A100359.

Cf. A061422.

a:=proc(n) if isprime(2^n+n+1)=true then 2^n+n+1 else fi end: seq(a(n),n=0..1000); # Emeric Deutsch, May 13 2006

{ta={{0}}, tb={{0}}};Do[g=n;s=2^n+n+1; If[PrimeQ[s], Print[n];ta=Append[ta, n]; tb=Append[tb, s]], {n, 1, 10000}];{ta, tb, g} (* Labos Elemer, Nov 19 2004 *)

Edited by N. J. A. Sloane, May 04 2007

A100359 Numbers k such that 2^k + k + 1 is prime.

Original entry on oeis.org

0, 2, 6, 236, 1884, 51380, 75764

Offset: 1

Labos Elemer, Nov 19 2004

a(8) > 500000. - Robert Price, May 24 2014

Cf. A001580, A061422, A069539, A052007, A048744, A100357, A100358.

{ta={{0}}, tb={{0}}};Do[g=n;s=2^n+n+1; If[PrimeQ[s], Print[n];ta=Append[ta, n]; tb=Append[tb, s]], {n, 1, 10000}];{ta, tb, g}

is(n)=ispseudoprime(2^n+n+1) \\ Charles R Greathouse IV, Feb 20 2017

a(n) = A061422(n) - 1.

a(6) from A061422 Max Alekseyev, Feb 08 2009

a(7) from Giovanni Resta, Mar 19 2014

A173053 Numbers n such that 2^(2n)+2n+1 is a prime.

Original entry on oeis.org

0, 1, 3, 118, 942, 25690, 37882

Offset: 1

Vincenzo Librandi, Feb 08 2010

Studying primes of the form 2^(x-1)+x for x=2n+1 leads to A061422. The six odd x in A061422 give the known solutions shown here. [R. J. Mathar]
The associated primes are 1+1 = 2, 2^2+3 = 7, 2^6+7 = 71,
2^236+237 = 110427941548649020598956093796432407239217743554726184882600387580788973;
2^1884+1885 = 1382012053...8525348701 (Most inner digits omitted. The number of digits of the prime grows roughly as log_10(4^n) = 0.61*n.)

Cf. A061422.

Select[Range[0, 2000], PrimeQ[2^(2 #) + 2 # + 1] &] (* Vincenzo Librandi, Jun 07 2014 *)

a(n) = floor( A061422(n) / 2). - Michel Marcus, Jun 07 2014

Display of very long primes truncated by R. J. Mathar, Feb 15 2010

a(7) from Vincenzo Librandi, Jun 07 2014

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A051015 Zeisel numbers.

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A061421 Primes of the form 2^n+n+1.

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A100359 Numbers k such that 2^k + k + 1 is prime.

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A173053 Numbers n such that 2^(2*n)+2*n+1 is a prime.

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A173053 Numbers n such that 2^(2n)+2n+1 is a prime.