A278967 a(n) = least multiple of the n-th prime that has the minimum Hamming weight (=A278966(n)).
2, 3, 5, 7, 33, 65, 17, 513, 69, 16385, 31, 262145, 1025, 129, 517, 67108865, 536870913, 1073741825, 8589934593, 8449, 73, 1027, 2199023255553, 89, 16777217, 1125899906842625, 515, 9007199254740993, 262145, 16385, 127, 36893488147419103233, 17179869185, 590295810358705651713, 18889465931478580854785
Offset: 1
Keywords
Examples
2 = 2^1 has Hamming weight 1 and so a(1) = 2. 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. 5 = 2^2 + 2^0 has Hamming weight 2 and so similarly a(3) = 5. 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. 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.
Programs
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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 a(n,p=prime(n))=my(m=min1s(p),t=p,k=2*p); while(hammingweight(t)>m, t+=k); t
Formula
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
Extensions
a(23)-a(25) from Charles R Greathouse IV, Dec 09 2016
Name clarified and terms a(26) onward added by Max Alekseyev, May 22 2025
Comments