

A020725


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


28



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, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 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
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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. AdamsWatters, 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 = m1 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 m1 = 48 safeties + 1 field goal = 99 points.  Cino Hilliard, Feb 03 2006
Possible nonnegative values of (a*bc*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=mn+1, c=m+n+1, d=mn, where m>n; for odd values 2n+1 take a=m+n, b=mn, c=m+n+1, d=mn1, 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 n1 (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 2ary 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


LINKS

Table of n, a(n) for n=1..74.
M. El Bachraoui, Primes in the interval (2n, 3n), International Journal of Contemporary Mathematical Sciences 1:13 (2006), pp. 617621.
Tanya Khovanova, Recursive Sequences
Eric Weisstein's World of Mathematics, Pierce Expansion
Index entries for linear recurrences with constant coefficients, signature (2,1).


FORMULA

From Franklin T. AdamsWatters, 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(s1).
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

Column 1 of A210976.
Sequence in context: A119972 A131738 A199969 A303502 A000027 A001477 A087156
Adjacent sequences: A020722 A020723 A020724 * A020726 A020727 A020728


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane


EXTENSIONS

Edited by Jon E. Schoenfield, Sep 20 2013


STATUS

approved



