OFFSET
1,1
COMMENTS
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
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
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
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
From Jason Kimberley, Oct 30 2011: (Start)
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).
Digit string 12 read in base n-1 (for n>3 or by extending notation). (End)
Positive integers whose number of divisors is not 1. - Omar E. Pol, Aug 11 2012
Positive integers where the number of parts function on the set of 2-ary partitions is equidistributed mod 2. - Tom Edgar, Apr 26 2016
This sequence is also the Pierce Expansion of 1/exp(1). - G. C. Greubel, Nov 15 2016
Natural numbers with at least one prime factor. - Michal Bozon, Apr 24 2017
a(n) is the reciprocal of the Integral_{x=0..1} x^n dx. - Felix Huber, Aug 19 2023
LINKS
M. El Bachraoui, Primes in the interval (2n, 3n), International Journal of Contemporary Mathematical Sciences 1:13 (2006), pp. 617-621.
Tanya Khovanova, Recursive Sequences
Antonio Gracia Llorente, Arithmetic Progression-Representing Constants, OSF Preprint, 2023.
Eric Weisstein's World of Mathematics, Pierce Expansion
Index entries for linear recurrences with constant coefficients, signature (2,-1).
FORMULA
From Franklin T. Adams-Watters, May 22 2006: (Start)
O.g.f.: (2*x - x^2)/(1 - x)^2.
E.g.f.: (1 + x)*exp(x)-1.
Dirichlet g.f.: zeta(s) + zeta(s-1).
a(n) = n + 1 for n>0. (End)
MATHEMATICA
Range[2, 100] (* Harvey P. Dale, Aug 31 2015 *)
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 *)
PROG
(PARI) a(n)=n+1 \\ Charles R Greathouse IV, Aug 23 2011
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Edited by Jon E. Schoenfield, Sep 20 2013
STATUS
approved