A051015 -id:A051015 - cp's OEIS frontend

A252094 First member of a pair (A,B) to define the n-th Zeisel number, cf. A051015.

1, 4, 1, 2, 3, 2, 10, 2, 6, 4, 13, 8, 3, 2, 25, 5, 9, 28, 1, 6, 5, 17, 34, 15, 9, 23, 8, 49, 55, 12, 4, 33, 14, 2, 24, 36, 25, 2, 26, 4, 2, 42, 29, 11, 8, 2, 10, 4, 88, 6, 47, 32, 48, 20, 94, 37, 23, 57, 24, 3, 5, 115, 6, 118, 9, 44, 3, 46, 68, 5, 30, 139, 50, 10, 51, 14, 77, 20, 54, 2

Offset: 1

Reinhard Zumkeller, Dec 15 2014

nonn

Let p(0) = 1 and p(i+1) = A*p(i) + B, if p(i) is prime for i = 1..k, then z = p(1) * ... * p(k) is called a Zeisel number.

			.  n |   A=a(n)  B=A252095(n) | p(1)  p(2)  p(3) | A051015(n)
. ---+------------------------+------------------------------
.  1 |   1       2            |   3     5     7  |        105
.  2 |   4      -1            |   3    11    43  |       1419
.  3 |   1       6            |   7    13    19  |       1729
.  4 |   2       3            |   5    13    29  |       1885
.  5 |   3       2            |   5    17    53  |       4505
.  6 |   2       5            |   7    19    43  |       5719
.  7 |  10      -7            |   3    23   223  |      15387
.  8 |   2       9            |  11    31    71  |      24211
.  9 |   6      -1            |   5    29   177  |      25085
. 10 |   4       3            |   7    31   141  |      27449
. 11 |  13     -10            |   3    29   367  |      31929
. 12 |   8      -3            |   5    37   295  |      54205 .

OEIS Wiki, Zeisel numbers
Eric Weisstein's World of Mathematics, Zeisel Number.
Wikipedia, Zeisel number

Cf. A051015, A027746, A252095 (B values).

a252094 n = a252094_list !! (n-1)
(a252094_list, a252095_list) = unzip $ f 3 where
   f x = if z then (q, p - q) : f (x + 2) else f (x + 2)  where
         z = 0 `notElem` ds && length ds > 2 &&
             all (== 0) (zipWith mod (tail ds) ds) && all (== q) qs
         q:qs = (zipWith div (tail ds) ds)
         ds = zipWith (-) (tail ps) ps
         ps = 1 : ps'; ps'@(p:_) = a027746_row x

A252095 Second member of a pair (A,B) to define the n-th Zeisel number, cf. A051015.

Original entry on oeis.org

2, -1, 6, 3, 2, 5, -7, 9, -1, 3, -10, -3, 8, 3, -22, 6, -2, -25, 36, 5, 8, -12, -31, -8, 2, -18, 5, -46, -52, -1, 27, -28, -3, 9, -17, -31, -18, 59, -19, 33, 65, -37, -22, 6, 15, 69, 9, 39, -85, 25, -42, -25, -43, -9, -91, -30, -12, -52, -13, 68, 42, -112, 35, -115, 20, -37, 8, -39, -63, 48, -19, -136, -43, 21, -44, 9, -72, -3, -47, 125

Offset: 1

Reinhard Zumkeller, Dec 15 2014

sign

			 See A252094.

OEIS Wiki, Zeisel numbers
Eric Weisstein's World of Mathematics, Zeisel Number.
Wikipedia, Zeisel number

Cf. A051015, A027746, A252094 (A values).

a252095 n = a252095_list !! (n-1)
-- where a252095_list is defined in A252094.

A051200 Except for initial term, primes of form "n 3's followed by 1".

Original entry on oeis.org

3, 31, 331, 3331, 33331, 333331, 3333331, 33333331, 333333333333333331, 3333333333333333333333333333333333333331, 33333333333333333333333333333333333333333333333331

Offset: 1

N. J. A. Sloane

"A remarkable pattern that is entirely accidental and leads nowhere" - M. Gardner, referring to the first 8 terms.
a(2)*a(3)*a(4) = 34179391, a Zeisel number (A051015) with coefficients (10,21).
a(2)*a(3)*a(4)*a(5) = 1139233281421, a Zeisel number with coefficients (10,21).
a(2)*a(3)*..*a(6) = 379741768929343351, a Zeisel number with coefficients (10,21).
a(2)*a(3)*..*a(7) = 1265805010367017001532181, a Zeisel number with coefficients (10,21).
a(2)*a(3)*..*a(8) = 42193497392022209194699696424911, a Zeisel number with coefficients (10,21).
Besides first 3, primes of the form (10^n-7)/3, n>1. See A123568. - Vincenzo Librandi, Aug 06 2010
The integer lengths of the terms of the sequence are 1, 2, 3, 4, 5, 6, 7, 8, 18, 40, 50, 60, 78, 101, 151, 319, 382, etc. - Harvey P. Dale, Dec 01 2018

Martin Gardner, The Last Recreations, Chapter 12: Strong Laws of Small Primes, Springer-Verlag, 1997, pp. 191-205, especially p. 194.
W. Sierpiński, 200 Zadan z Elementarnej Teorii Liczb, Warsaw, 1964; Problem 88 [in English: 200 Problems from the Elementary Theory of Numbers]
W. Sierpiński, 250 Problems in Elementary Number Theory. New York: American Elsevier, Warsaw, 1970, pp. 8, 56-57.
F. Smarandache, Properties of numbers, University of Craiova, 1973

R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
Eric Weisstein's World of Mathematics, 3.

Cf. A055520, A089017, A089018, A093671, A056698, A105427, A105428, A033175, A123568.

Join[{3},Select[Rest[FromDigits/@Table[PadLeft[{1},n,3], {n,50}]], PrimeQ]]  (* Harvey P. Dale, Apr 20 2011 *)

Union of 3 and A123568.

More terms from James Sellers

Cross reference added by Harvey P. Dale, May 21 2014

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

A110704 Primes of form "n 42's followed by 43".

Original entry on oeis.org

43, 4243, 424243, 42424243, 4242424243, 4242424242424243, 424242424242424242424242424242424243, 42424242424242424242424242424242424242424242424243

Offset: 1

Ray Chandler, Aug 04 2005

nonn

Products of first three (77402711107), four (3283751424862167001) and five (13931066652821050218557005243) terms are Zeisel numbers (A051015) with coefficients (100,-57).

Cf. A110705.

t = 1; Do[ t = t*100 - 57; If[PrimeQ[t], Print[t]], {n, 0, 24}]

A259172 Numbers in A259145 that are neither prime nor semiprime.

Original entry on oeis.org

561, 595, 1105, 1235, 1245, 1495, 1547, 1885, 2405, 2555, 2717, 2849, 3115, 3495, 3655, 3657, 3689, 3815, 4521, 4795, 4945, 5035, 5385, 5395, 5453, 5457, 5709, 5865, 6083, 6141, 6251, 6285, 6365, 6391, 6501, 6695, 6755, 6969, 7021, 7887, 8113, 8255, 8355

Offset: 1

Carlos Eduardo Olivieri, Jun 19 2015

Regarding the distribution: Let K be the union of primes and semiprimes in A259145. Let S be the set of other terms. The growth rate of the cardinality of S with respect to the cardinality of K is significantly slower. For instance, if we take the first 50000 terms of A259145, about 32.5 percent are contained in S. If we take the first 350000 terms, about 38.2 percent are contained in S.
a(n) that are in A002997 (Carmichael numbers) for a(n) <= 10^6 are 561, 1105, 8911, 10585, 29341, 825265.
a(n) that are in A051015 (Zeisel numbers) for a(n) <= 3*10^6 are 1885, 353977, 2953711.

Carlos Eduardo Olivieri, Table of n, a(n) for n = 1..8906
Eric Naslund, Integers with a predetermined prime factorization, arXiv:1203.2363 [math.NT], 2012.

Cf. A001221, A001358, A002997, A007053, A051015, A125527, A259145.

Subsequence of A000469, A033942, A050384 (conjuctered).

Select[Range[25000], PrimeQ[#^2 - EulerPhi[#]] && PrimeNu[#] > 2 &]

A001221(a(n)) > 2.

A000005(a(n)) = 2^k, k >= 3.

A200525 Zeisel numbers with p(0)=4.

Original entry on oeis.org

385, 2585, 7315, 8911, 27001, 39905, 48391, 87283, 192211, 196285, 319705, 410089, 425585, 441091, 624605, 679855, 1310185, 1899163, 2460439, 2586971, 2777041, 6654005, 7042411, 7501261, 8291459, 9516637, 10484585, 11141671, 12527281, 13015891, 13788319

Offset: 1

Karsten Meyer, Nov 18 2011

nonn

Pick any integers A and B and consider the linear recurrence relation given by p(0) = 4, 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.

			a=2, b=-3 => p(1) = (4*2)+(-3) = 5; p(2) = (5*2)+(-3) = (7); p(3) = (7*2)+(-3) = 11 => 5*7*11 = 385.
a=2, b=5 => p(1) = (4*2)+5 = 13; p(2) = (13*2)+5 = 31; p(3) = (31*2)+5 = 67 => 13*31*67 = 27001.

Eric Weisstein's World of Mathematics, Zeisel Number.
Wikipedia, Zeisel number

Cf. A051015.

n0=4
do m=1 to 53
  a=2*m
  do b=(1-(4*a)) to 999
    n1=(n0*a)+b
    n2=(n1*a)+b
    n3=(n2*a)+b
    z=n1*n2*n3
    say n0 a b
    lineout("zeisel_4.txt",z||" = "||n1||"*"||n2||"*"||n3||"      "||a||" "||b||" n0="||n0)
    end
  end

A225217 Smallest Zeisel number with n distinct prime factors.

Original entry on oeis.org

105, 114985, 1136972771, 717429818501, 42193497392022209194699696424911, 2259851975647498450164729386247520625304964981, 6496837611815817806848181714391334227919720933013147552348535557303, 13839515413469463429656389971119647159455019743381019128792232352396451034929601

Offset: 3

Arkadiusz Wesolowski, May 02 2013

a(11) > 10^105. - Bert Dobbelaere, Apr 19 2019

Carlos Rivera, Puzzle 325
Wikipedia, Zeisel number

Cf. A051015.

a(7)-a(10) from Bert Dobbelaere, Apr 19 2019

cp's OEIS Frontend

A252094 First member of a pair (A,B) to define the n-th Zeisel number, cf. A051015.

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Haskell

A252095 Second member of a pair (A,B) to define the n-th Zeisel number, cf. A051015.

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Haskell

A051200 Except for initial term, primes of form "n 3's followed by 1".

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Mathematica

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

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Maple

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A110704 Primes of form "n 42's followed by 43".

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Mathematica

A259172 Numbers in A259145 that are neither prime nor semiprime.

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Mathematica

Formula

A200525 Zeisel numbers with p(0)=4.

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Rexx

A225217 Smallest Zeisel number with n distinct prime factors.

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