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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

%I N0416 #43 Jun 28 2019 08:51:58

%S 1,2,4,8,13,21,31,45,60,76,97,119,144,170,198,231,265,300,336,374,414,

%T 456,502,550,599,649,702,759,819,881,945,1010,1080,1157,1237,1318,

%U 1401,1486,1572,1662,1753,1845,1945,2049,2156,2264

%N 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.

%C 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

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

%H Nathaniel Johnston, <a href="/A004978/b004978.txt">Table of n, a(n) for n = 1..2000</a>

%H 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.

%e From _M. F. Hasler_, Jun 26 2019:

%e 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.

%e 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.

%e 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.

%e (End)

%o (MATLAB) s=1:2000^2;d(1)=1;A004978(1)=1;A004978(2)=2;

%o for n=3:2000

%o A004978(n)=A004978(n-1)+find([d,0]~=s(1:max(size(d))+1),1);

%o d(end+1:end+n-1)=A004978(n)-A004978(1:n-1);

%o d=sort(unique(d));

%o end

%o % _Nathaniel Johnston_, Feb 09 2011

%o (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

%Y Differences give A002048, see also A048201.

%Y See also A001856.

%Y For n>2, a(n) equals A002049(n-1)+1 and A048204(n-2)+2.

%K nonn

%O 1,2

%A _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

%E Definition corrected by Bryan S. Robinson, Mar 16 2006

%E Name edited by _M. F. Hasler_, Jun 26 2019