A002049 Prime numbers of measurement.

1, 3, 7, 12, 20, 30, 44, 59, 75, 96, 118, 143, 169, 197, 230, 264, 299, 335, 373, 413, 455, 501, 549, 598, 648, 701, 758, 818, 880, 944, 1009, 1079, 1156, 1236, 1317, 1400, 1485, 1571, 1661, 1752, 1844, 1944, 2048, 2155, 2263, 2379, 2498, 2622, 2749, 2881

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• changed

Comments

Partial sums of A002048. - Reinhard Zumkeller, May 23 2013

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..3000 from Reinhard Zumkeller)
• 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.
• 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.

Crossrefs

Cf. A002048.

a(n) = A004978(n+1)-1 = A048204(n-1)+1.

Programs

Haskell

import Data.List ((\\))
a002049 n = a002049_list !! (n-1)
a002049_list = g [1..] [] where
   g (x:xs) ys = (last zs) : g (xs \\ zs) (x : ys) where
     zs = scanl (+) x ys
-- Reinhard Zumkeller, May 23 2013

Mathematica

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. *)

Python

from itertools import count, accumulate, islice
from collections import deque
def A002049_gen(): # generator of terms
    aset, alist, c = set(), deque(), 0
    for k in count(1):
        if k in aset:
            aset.remove(k)
        else:
            yield (c:=c+k)
            aset |= set(k+d for d in accumulate(alist))
            alist.appendleft(k)
A002049_list = list(islice(A002049_gen(),20)) # Chai Wah Wu, Sep 01 2025

Formula

Andrews conjectures that a(n) ~ (1/2) n^2 log n / loglog n. - N. J. A. Sloane, Dec 01 2013