A002048 Segmented numbers, or prime numbers of measurement.
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
Offset: 1
Examples
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.
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
- Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..7836 from R. J. Mathar)
- G. E. Andrews, MacMahon's prime numbers of measurement, Amer. Math. Monthly, 82 (1975), 922-923.
- Thomas Bloom, Problem 359, Erdős Problems.
- 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.
Programs
-
C
// See Links section for C program by Samuel B. Reid, Jan 26 2020
-
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
-
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 *)
-
Python
from itertools import count, accumulate, islice from collections import deque def A002048_gen(): # generator of terms aset, alist = set(), deque() for k in count(1): if k in aset: aset.remove(k) else: yield k aset |= set(k+d for d in accumulate(alist)) alist.appendleft(k) A002048_list = list(islice(A002048_gen(),20)) # Chai Wah Wu, Sep 01 2025
Formula
Andrews conjectures that lim_{n -> oo} n log n / (a(n) loglog n) = 1. - N. J. A. Sloane, Dec 01 2013
Extensions
More terms from R. J. Mathar, May 31 2006
Comments