A020725 - OEIS

%I #115 Dec 19 2023 13:40:38

%S 2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,

%T 27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,

%U 50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75

%N Integers >= 2. a(n) = n+1.

%C This sequence is closed under multiplication by any integer k > 0. The primitive elements of the sequence (those not divisible by any smaller element) are the primes, A000040. - _Franklin T. Adams-Watters_, May 22 2006

%C Possible sums of the final scores of completed Chicago Bears football games. 1 point only is an impossible score in American football. But with the safety 2 and the field goal 3, we can construct the set of integers greater than 1. We can prove this by noting that if a score is even, we can build it with a series of safeties. Of course the other allowed scorings of 3, 6, and 1 after a touchdown, could also be used. Now if a score is odd it is of the form 2k+3. So for any odd number 2m+1, we subtract 3 (or 1 field goal) from it to make it even and divide by 2 to get the number of safeties we need to add back to the field goal. Symbolically, let the odd number be 2m+1; then (2m+1 - 3)/2 = m-1 safeties are needed. Add this to 3 and you will have the number. For example, say we want a score of 99. 99 = 2m+1 and m = 49. So m-1 = 48 safeties + 1 field goal = 99 points. - _Cino Hilliard_, Feb 03 2006

%C Possible nonnegative values of (a*b-c*d) where a,b,c and d are distinct positive integers and a+b=c+d. All positive values >=2 are possible: for even values 2n take a=m+n, b=m-n+1, c=m+n+1, d=m-n, where m>n; for odd values 2n+1 take a=m+n, b=m-n, c=m+n+1, d=m-n-1, where m>n+1. Elementary algebra shows that the values 0 and 1 are not possible without violating the assumption that a,b,c and d are distinct. - _John Grint_, Sep 26 2011

%C Also numbers n such that a semiprime is equal to the sum of n primes. Bachraoui proved that there is a prime between 2n and 3n for every n > 1, so every n > 1 is in this sequence since any number in that range is the sum of n integers each of which is either 2 or 3. - _Charles R Greathouse IV_, Oct 27 2011

%C From _Jason Kimberley_, Oct 30 2011: (Start)

%C Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), A198306 (k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10), A094626 (k=11); columns: this sequence (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7).

%C Digit string 12 read in base n-1 (for n>3 or by extending notation). (End)

%C Positive integers whose number of divisors is not 1. - _Omar E. Pol_, Aug 11 2012

%C Positive integers where the number of parts function on the set of 2-ary partitions is equidistributed mod 2. - _Tom Edgar_, Apr 26 2016

%C This sequence is also the Pierce Expansion of 1/exp(1). - _G. C. Greubel_, Nov 15 2016

%C Natural numbers with at least one prime factor. - _Michal Bozon_, Apr 24 2017

%C a(n) is the reciprocal of the Integral_{x=0..1} x^n dx. - _Felix Huber_, Aug 19 2023

%H M. El Bachraoui, <a href="http://www.m-hikari.com/ijcms-password/ijcms-password13-16-2006/elbachraouiIJCMS13-16-2006.pdf">Primes in the interval (2n, 3n)</a>, International Journal of Contemporary Mathematical Sciences 1:13 (2006), pp. 617-621.

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H Antonio Gracia Llorente, <a href="https://doi.org/10.31219/osf.io/gdqt2">Arithmetic Progression-Representing Constants</a>, OSF Preprint, 2023.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PierceExpansion.html">Pierce Expansion</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).

%F From _Franklin T. Adams-Watters_, May 22 2006: (Start)

%F O.g.f.: (2*x - x^2)/(1 - x)^2.

%F E.g.f.: (1 + x)*exp(x)-1.

%F Dirichlet g.f.: zeta(s) + zeta(s-1).

%F a(n) = n + 1 for n>0. (End)

%t Range[2,100] (* _Harvey P. Dale_, Aug 31 2015 *)

%t PierceExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@ NestList[{Floor[1/Expand[1 - #[[1]] #[[2]]]], Expand[1 - #[[1]] #[[2]]]} &, {Floor[1/(A - Floor[A])], A - Floor[A]}, n - 1]]; PierceExp[N[1/E , 7!], 50] (* _G. C. Greubel_, Nov 14 2016 *)

%o (PARI) a(n)=n+1 \\ _Charles R Greathouse IV_, Aug 23 2011

%Y Column 1 of A210976.

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_

%E Edited by _Jon E. Schoenfield_, Sep 20 2013