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%I M0972 N0363 #77 Mar 27 2024 13:06:42
%S 1,2,4,5,8,10,14,15,16,21,22,25,26,28,33,34,35,36,38,40,42,46,48,49,
%T 50,53,57,60,62,64,65,70,77,80,81,83,85,86,90,91,92,100,104,107,108,
%U 116,119,124,127,132,133,137,141,144,145,148,150,151,154,158,159,163,165
%N Segmented numbers, or prime numbers of measurement.
%C The segmented numbers are the positive integers excluding those equal to the sum of two or more consecutive smaller terms. The prime numbers of measurement are their partial sums, cf. A002049. - _M. F. Hasler_, Jun 26 2019
%C Without the requirement that the smaller terms be consecutive, the sequence becomes the sequence of powers of 2 (A000079). - _Alonso del Arte_, Jan 25 2020
%D R. K. Guy, Unsolved Problems in Number Theory, E30.
%D Š. Porubský, On MacMahon's segmented numbers and related sequences. Nieuw Arch. Wisk. (3) 25 (1977), no. 3, 403--408. MR0485763 (58 #5575)
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H R. J. Mathar, <a href="/A002048/b002048.txt">Table of n, a(n) for n = 1..7836</a>
%H G. E. Andrews, <a href="http://www.jstor.org/stable/2318498">MacMahon's prime numbers of measurement</a>, Amer. Math. Monthly, 82 (1975), 922-923.
%H Thomas Bloom, <a href="https://www.erdosproblems.com/474">Problem 474</a>, Erdős Problems.
%H R. L. Graham and C. B. A. Peck, <a href="http://www.jstor.org/stable/2315138">Problem E1910</a>, Amer. Math. Monthly, 75 (1968), 80-81.
%H R. K. Guy, <a href="/A002048/a002048.pdf">Letter to G. E. Andrews, Apr 14 1975</a>
%H P. A. MacMahon, <a href="https://www.biodiversitylibrary.org/item/88477#page/679/mode/1up">The prime numbers of measurement on a scale</a>, Proc. Camb. Phil. Soc. 21 (1923), 651-654; reprinted in Coll. Papers I, pp. 797-800.
%H Samuel B. Reid, <a href="/A002048/a002048_1.c.txt">C program for A002048</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeNumberofMeasurement.html">Prime Number of Measurement.</a>
%F Andrews conjectures that lim_{n -> oo} n log n / (a(n) loglog n) = 1. - _N. J. A. Sloane_, Dec 01 2013
%e Although 5 is the sum of the terms 1 and 4, those prior terms are not consecutive, and therefore 5 is in the sequence.
%e 6 is not in the sequence because it is the sum of consecutive prior terms 2 and 4.
%e 7 is not in the sequence either because it is also the sum of consecutive prior terms, in this case 1, 2, 4.
%e 8 is in the sequence because no sum whatsoever of distinct prior terms adds up to 8.
%p A002048 := proc(anmax::integer,printlist::boolean)
%p local a, asum,su,i,piv,j;
%p a := [];
%p for i from 1 to anmax do
%p a := [op(a),i];
%p od:
%p if printlist then
%p printf("%d %d\n",1,a[1]);
%p printf("%d %d\n",2,a[2]);
%p fi;
%p asum := [a[1]+a[2],a[2]];
%p for i from 3 to anmax do
%p asum := [op(asum),0];
%p od:
%p piv := 3;
%p while piv <= nops(a) do
%p for i from 1 to piv-2 do
%p a := remove(has,a, asum[i]);
%p od:
%p if printlist then
%p printf("%a %a\n",piv,a[piv]);
%p fi;
%p for i from 1 to piv do
%p asum := subsop(i=asum[i]+a[piv], asum);
%p od:
%p piv := piv+1;
%p od;
%p RETURN(a);
%p end:
%p A002048(40000,true);
%p # _R. J. Mathar_, Jun 04 2006
%t A002048[anmax_] := (a = {}; Do[AppendTo[a, i], {i, anmax}]; asum = {a[[1]] + a[[2]], a[[2]]}; Do[AppendTo[asum, 0], {i, 3, anmax}]; piv = 3; While[piv <= Length[a], Do[a = DeleteCases[a, asum[[i]]], {i, 1, piv - 2}]; Do[asum[[i]] += a[[piv]], {i, piv}]; piv = piv + 1;]; a); A002048[63] (* _Jean-François Alcover_, Jul 28 2011, converted from _R. J. Mathar_'s Maple prog. *)
%t searchMax = 200; segmNums = {1}; curr = 2; While[curr < searchMax, If[Not[MemberQ[Apply[Plus, Subsequences[segmNums], 1], curr]], AppendTo[segmNums, curr], ]; curr = curr + 1]; segmNums (* _Alonso del Arte_, Jan 25 2020 *)
%o (C++)
%o #include <iostream>
%o #include <vector>
%o #include <algorithm>
%o #define NMAX 400
%o using namespace std;
%o int main(int argc, char *argv[])
%o { vector<int> a; for(int i = 0; i < NMAX; i++) a.push_back(i+1); for(int piv = 2; piv < a.size(); piv++) for(int i = 0; i < piv - 1 && i < a.size() - 1; i++) { int su = a[i] + a[i + 1]; remove(a.begin(), a.end(), su); for(int j = i + 2; j < piv && j < a.size(); j++) { su += a[j]; remove(a.begin(), a.end(), su); if(su > NMAX) break; } } for(int i = 0; i < a.size() && a[i] < NMAX; i++) cout << a[i] << ","; return 0;
%o } /* _R. J. Mathar_, May 31 2006 */
%o (Haskell)
%o import Data.List ((\\))
%o a002048 n = a002048_list !! (n-1)
%o a002048_list = f [1..] [] where
%o f (x:xs) ys = x : f (xs \\ scanl (+) x ys) (x : ys)
%o -- _Reinhard Zumkeller_, May 23 2013
%o (C) // See Links section for C program by _Samuel B. Reid_, Jan 26 2020
%Y Cf. A002049 (partial sums), A004978, A005242, A033627.
%K nonn,nice
%O 1,2
%A _N. J. A. Sloane_
%E More terms from _R. J. Mathar_, May 31 2006
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