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A278967 a(n) = least multiple of the n-th prime that has the minimum Hamming weight (=A278966(n)).

%I A278967 #16 May 23 2025 01:13:37
%S A278967 2,3,5,7,33,65,17,513,69,16385,31,262145,1025,129,517,67108865,
%T A278967 536870913,1073741825,8589934593,8449,73,1027,2199023255553,89,
%U A278967 16777217,1125899906842625,515,9007199254740993,262145,16385,127,36893488147419103233,17179869185,590295810358705651713,18889465931478580854785
%N A278967 a(n) = least multiple of the n-th prime that has the minimum Hamming weight (=A278966(n)).
%C A278967 Apart from the first term, all terms are odd.
%F A278967 a(n) = 2^(A014664(n)/2) + 1 whenever A014664(n) is even. Also, a(n) = prime(n) whenever prime(n) is in A143027. - _Max Alekseyev_, May 22 2025
%e A278967 2 = 2^1 has Hamming weight 1 and so a(1) = 2.
%e A278967 3 = 2^1 + 2^0 has Hamming weight 2, and any multiple of 3 has a Hamming weight at least as high, so a(2) = 3.
%e A278967 5 = 2^2 + 2^0 has Hamming weight 2 and so similarly a(3) = 5.
%e A278967 7 = 2^2 + 2^1 + 2^0 has Hamming weight 3, and all powers of 2 are 1, 2, or 4 mod 7, and so all multiples of 7 have Hamming weight at least 3, so a(4) = 7.
%e A278967 11 = 2^3 + 2^1 + 2^0 has Hamming weight 3 but 33 = 2^5 + 2^0 has Hamming weight 2 so a(5) = 33.
%o A278967 (PARI) min1s(p)=my(o=znorder(Mod(2,p)), v1=Set(powers(Mod(2,p),o)), v=v1, s=1); while(!setsearch(v,Mod(0,p)), v=setbinop((x,y)->x+y,v,v1); s++); s
%o A278967 a(n,p=prime(n))=my(m=min1s(p),t=p,k=2*p); while(hammingweight(t)>m, t+=k); t
%Y A278967 Contains A143027 as subsequence.
%Y A278967 Cf. A278966, A278968, A086342.
%K A278967 nonn
%O A278967 1,1
%A A278967 _Charles R Greathouse IV_, Dec 02 2016
%E A278967 a(23)-a(25) from _Charles R Greathouse IV_, Dec 09 2016
%E A278967 Name clarified and terms a(26) onward added by _Max Alekseyev_, May 22 2025