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A020725
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Integers >= 2.
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22
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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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. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), 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 scorings allowed 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 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 (hillcino368(AT)gmail.com), Feb 03 2006]
Possible non-negative 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]
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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
Eric Weisstein's World of Mathematics, Pierce Expansion
Index to sequences with linear recurrences with constant coefficients, signature (2,-1).
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FORMULA
| G.f. (2x-x^2)/(1-x)^2. E.g.f. (x+1)e^x-1. Dirichlet g.f. zeta(s) + zeta(s-1). a(n) = n + 1 (for n>0). - Frank Adams-Watters (FrankTAW(AT)Netscape.net), May 22 2006
Digit string 12 read in base n-1. (for n>3 or by extending notation) - Jason Kimberley, Oct 30 2011
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PROG
| (PARI) a(n)=n+1 \\ Charles R Greathouse IV, Aug 23 2011
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CROSSREFS
| 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). - Jason Kimberley, Oct 30 2011
Sequence in context: A114142 * A119972 A131738 A000027 A001477 A087156
Adjacent sequences: A020722 A020723 A020724 * A020726 A020727 A020728
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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