A048204 -id:A048204 - cp's OEIS frontend

A004978 a(n) = least integer m > a(n-1) such that m - a(n-1) != a(j) - a(k) for all j, k less than n; a(1) = 1, a(2) = 2.

1, 2, 4, 8, 13, 21, 31, 45, 60, 76, 97, 119, 144, 170, 198, 231, 265, 300, 336, 374, 414, 456, 502, 550, 599, 649, 702, 759, 819, 881, 945, 1010, 1080, 1157, 1237, 1318, 1401, 1486, 1572, 1662, 1753, 1845, 1945, 2049, 2156, 2264, 2380, 2499, 2623, 2750, 2882

Offset: 1

N. J. A. Sloane. This was in the 1973 "Handbook", but was then dropped from the database. Resubmitted by Clark Kimberling. Entry revised by N. J. A. Sloane, Jun 10 2012

Equivalently, if S(n) = { a(j) - a(k); n > j > k > 0 }, then a(n) = a(n-1) + M where M = min( {1, 2, 3, ...} \ S(n) ) is the smallest positive integer not in S(n). - M. F. Hasler, Jun 26 2019

			From _M. F. Hasler_, Jun 26 2019: (Start)
After a(1) = 1, a(2) = 2, we have a(3) = least m > a(2) such that m - a(2) = m - 2 is not in { a(j) - a(k); 1 <= k < j < 3 } = { a(2) - a(1) } = { 1 }. Thus we must have m - 2 = 2, whence m = 4.
The next term a(4) is the least m > a(3) such that m - a(3) = m - 4 is not in { a(j) - a(k); 1 <= k < j < 4 } = { 1, 4 - 2 = 2, 4 - 1 = 3 }, i.e., m = 4 + 4 = 8.
The next term a(5) is the least m > a(4) such that m - a(4) = m - 8 is not in { a(j) - a(k); 1 <= k < j < 5 } = { 1, 2, 3, 8 - 4 = 4, 8 - 2 = 6, 8 - 1 = 7 }, i.e., m = 5 + 8 = 13. (End)

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

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..2000 from Nathaniel Johnston)
G. E. Andrews, MacMahon's prime numbers of measurement, Amer. Math. Monthly, 82 (1975), 922-923.

Differences give A002048, see also A048201.
See also A001856.
For n>2, a(n) equals A002049(n-1)+1 and A048204(n-2)+2.

s=1:2000^2;d(1)=1;A004978(1)=1;A004978(2)=2;
for n=3:2000
  A004978(n)=A004978(n-1)+find([d,0]~=s(1:max(size(d))+1),1);
  d(end+1:end+n-1)=A004978(n)-A004978(1:n-1);
  d=sort(unique(d));
end
% Nathaniel Johnston, Feb 09 2011

A004978_vec(N,a=[1..N],S=[1])={for(n=3,N,a[n]=a[n-1]+S[1]+1;S=setunion(S,select(t->t>S[1],vector(n-1,k,a[n]-a[n-k])));for(k=1,#S-1, if(S[k+1]-S[k]>1, S=S[k..-1];next(2)));S[#S]==#S&&S=[#S]);a} \\ M. F. Hasler, Jun 26 2019

from itertools import count, accumulate, islice
from collections import deque
def A004978_gen(): # generator of terms
    aset, alist, c = set(), deque(), 1
    for k in count(1):
        if k in aset:
            aset.remove(k)
        else:
            yield c
            aset |= set(k+d for d in accumulate(alist))
            alist.appendleft(k)
            c += k
A004978_list = list(islice(A004978_gen(),20)) # Chai Wah Wu, Sep 01 2025

Definition corrected by Bryan S. Robinson, Mar 16 2006

Name edited by M. F. Hasler, Jun 26 2019

A002049 Prime numbers of measurement.

Original entry on oeis.org

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

N. J. A. Sloane

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

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).

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.

Cf. A002048.

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

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

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. *)

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

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

cp's OEIS Frontend

A004978 a(n) = least integer m > a(n-1) such that m - a(n-1) != a(j) - a(k) for all j, k less than n; a(1) = 1, a(2) = 2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

MATLAB

PARI

Python

Extensions