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A002048 Segmented numbers, or prime numbers of measurement.
(Formerly M0972 N0363)
7
1, 2, 4, 5, 8, 10, 14, 15, 16, 21, 22, 25, 26, 28, 33, 34, 35, 36, 38, 40, 42, 46, 48, 49, 50, 53, 57, 60, 62, 64, 65, 70, 77, 80, 81, 83, 85, 86, 90, 91, 92, 100, 104, 107, 108, 116, 119, 124, 127, 132, 133, 137, 141, 144, 145, 148, 150, 151, 154, 158, 159, 163, 165 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

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

REFERENCES

R. K. Guy, Unsolved Problems in Number Theory, E30.

Š. Porubský, On MacMahon's segmented numbers and related sequences. Nieuw Arch. Wisk. (3) 25 (1977), no. 3, 403--408. MR0485763 (58 #5575)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

R. J. Mathar, Table of n, a(n) for n = 1..7836

G. E. Andrews, MacMahon's prime numbers of measurement, Amer. Math. Monthly, 82 (1975), 922-923.

R. L. Graham and C. B. A. Peck, Problem E1910, Amer. Math. Monthly, 75 (1968), 80-81.

R. K. Guy, Letter to G. E. Andrews, Apr 14 1975

P. A. MacMahon, The prime numbers of measurement on a scale, Proc. Camb. Phil. Soc. 21 (1923), 651-654; reprinted in Coll. Papers I, pp. 797-800.

Samuel B. Reid, C program for A002048

Eric Weisstein's World of Mathematics, Prime Number of Measurement.

FORMULA

Andrews conjectures that lim_{n -> oo} n log n / (a(n) loglog n) = 1. - N. J. A. Sloane, Dec 01 2013

EXAMPLE

Although 5 is the sum of the terms 1 and 4, those prior terms are not consecutive, and therefore 5 is in the sequence.

6 is not in the sequence because it is the sum of consecutive prior terms 2 and 4.

7 is not in the sequence either because it is also the sum of consecutive prior terms, in this case 1, 2, 4.

8 is in the sequence because no sum whatsoever of distinct prior terms adds up to 8.

MAPLE

A002048 := proc(anmax::integer, printlist::boolean)

local a, asum, su, i, piv, j;

a := [];

for i from 1 to anmax do

a := [op(a), i];

od:

if printlist then

printf("%d %d\n", 1, a[1]);

printf("%d %d\n", 2, a[2]);

fi;

asum := [a[1]+a[2], a[2]];

for i from 3 to anmax do

asum := [op(asum), 0];

od:

piv := 3;

while piv <= nops(a) do

for i from 1 to piv-2 do

a := remove(has, a, asum[i]);

od:

if printlist then

printf("%a %a\n", piv, a[piv]);

fi;

for i from 1 to piv do

asum := subsop(i=asum[i]+a[piv], asum);

od:

piv := piv+1;

od;

RETURN(a);

end:

A002048(40000, true);

# R. J. Mathar, Jun 04 2006

MATHEMATICA

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. *)

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 *)

PROG

(C++)

#include <iostream>

#include <vector>

#include <algorithm>

#define NMAX 400

using namespace std;

int main(int argc, char *argv[])

{ 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;

} /* R. J. Mathar, May 31 2006 */

(Haskell)

import Data.List ((\\))

a002048 n = a002048_list !! (n-1)

a002048_list = f [1..] [] where

   f (x:xs) ys = x : f (xs \\ scanl (+) x ys) (x : ys)

-- Reinhard Zumkeller, May 23 2013

(C) // See Links section for C program by Samuel B. Reid, Jan 26 2020

CROSSREFS

Cf. A002049 (partial sums), A004978, A005242, A033627.

Sequence in context: A036404 A186077 A018498 * A174989 A190809 A067941

Adjacent sequences:  A002045 A002046 A002047 * A002049 A002050 A002051

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from R. J. Mathar, May 31 2006

STATUS

approved

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Last modified February 24 09:21 EST 2020. Contains 332209 sequences. (Running on oeis4.)