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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.
(Formerly N0416)
10
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
LINKS
G. E. Andrews, MacMahon's prime numbers of measurement, Amer. Math. Monthly, 82 (1975), 922-923.
EXAMPLE
From M. F. Hasler, Jun 26 2019:
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)
PROG
(MATLAB) 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
(PARI) 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
CROSSREFS
Differences give A002048, see also A048201.
See also A001856.
For n>2, a(n) equals A002049(n-1)+1 and A048204(n-2)+2.
Sequence in context: A308094 A292774 A026039 * A005282 A046185 A259964
KEYWORD
nonn
AUTHOR
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
EXTENSIONS
Definition corrected by Bryan S. Robinson, Mar 16 2006
Name edited by M. F. Hasler, Jun 26 2019
STATUS
approved

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Last modified May 27 02:08 EDT 2024. Contains 372847 sequences. (Running on oeis4.)