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A002049 Prime numbers of measurement.

%I A002049 M2633 N1044 #45 Sep 02 2025 09:22:38
%S A002049 1,3,7,12,20,30,44,59,75,96,118,143,169,197,230,264,299,335,373,413,
%T A002049 455,501,549,598,648,701,758,818,880,944,1009,1079,1156,1236,1317,
%U A002049 1400,1485,1571,1661,1752,1844,1944,2048,2155,2263,2379,2498,2622,2749,2881
%N A002049 Prime numbers of measurement.
%C A002049 Partial sums of A002048. - _Reinhard Zumkeller_, May 23 2013
%D A002049 R. K. Guy, Unsolved Problems in Number Theory, E30.
%D A002049 Š. Porubský, On MacMahon's segmented numbers and related sequences. Nieuw Arch. Wisk. (3) 25 (1977), no. 3, 403--408. MR0485763 (58 #5575)
%D A002049 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002049 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002049 Chai Wah Wu, <a href="/A002049/b002049.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..3000 from Reinhard Zumkeller)
%H A002049 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 A002049 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 A002049 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.
%F A002049 Andrews conjectures that a(n) ~ (1/2) n^2 log n / loglog n. - _N. J. A. Sloane_, Dec 01 2013
%t A002049 A002048[anmax_] := (a = {}; Do[AppendTo[a, i], {i, 1, 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, 1, piv}]; piv = piv + 1;]; a); A002048[200] // Accumulate (* _Jean-François Alcover_, Oct 05 2016, adapted from _R. J. Mathar_'s Maple code in A002048. *)
%o A002049 (Haskell)
%o A002049 import Data.List ((\\))
%o A002049 a002049 n = a002049_list !! (n-1)
%o A002049 a002049_list = g [1..] [] where
%o A002049    g (x:xs) ys = (last zs) : g (xs \\ zs) (x : ys) where
%o A002049      zs = scanl (+) x ys
%o A002049 -- _Reinhard Zumkeller_, May 23 2013
%o A002049 (Python)
%o A002049 from itertools import count, accumulate, islice
%o A002049 from collections import deque
%o A002049 def A002049_gen(): # generator of terms
%o A002049     aset, alist, c = set(), deque(), 0
%o A002049     for k in count(1):
%o A002049         if k in aset:
%o A002049             aset.remove(k)
%o A002049         else:
%o A002049             yield (c:=c+k)
%o A002049             aset |= set(k+d for d in accumulate(alist))
%o A002049             alist.appendleft(k)
%o A002049 A002049_list = list(islice(A002049_gen(),20)) # _Chai Wah Wu_, Sep 01 2025
%Y A002049 Cf. A002048.
%Y A002049 a(n) = A004978(n+1)-1 = A048204(n-1)+1.
%K A002049 nonn,nice,changed
%O A002049 1,2
%A A002049 _N. J. A. Sloane_