A391729 - OEIS

%I #13 Dec 20 2025 17:24:35

%S 1,3,1,7,1,3,1,15,1,3,4,6,3,3,1,31,1,3,4,6,1,8,4,10,1,7,1,6,1,3,1,63,

%T 1,3,4,6,5,7,8,10,1,3,12,14,1,7,4,16,3,3,1,15,1,3,5,10,7,3,1,6,3,3,1,

%U 127,1,3,4,6,5,7,8,10,1,9,4,12,5,12,8,15,3,3,6,6,1,21,4,21,1,3,6,12,1,7,4,26,1,7,1

%N a(n) = Sum_{k=1..A003817(n)} [A048720(A065621(n),k) == n*k], where [ ] is the Iverson bracket.

%C a(n) gives the index (column number) of A003817(n) on the row n of array A115872.

%H Antti Karttunen, <a href="/A391729/b391729.txt">Table of n, a(n) for n = 1..16383</a>

%H <a href="/index/Con#CongruCrossDomain">Index entries for sequences defined by congruent products between domains N and GF(2)[X]</a>

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>.

%H <a href="/index/Ca#CARRYLESS">Index entries for sequences related to carryless arithmetic</a>.

%F For all n >= 1, A115872(n, a(n)) = A003817(n).

%o (PARI)

%o A003817(n) = (1<<(log(2*n+1)\log(2)))-1;

%o A006068(n) = { my(s=1, ns); while(1, ns = n >> s; if(0==ns, return(n)); n = bitxor(n, ns); s <<= 1); };

%o A048720(b, c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);

%o A391729(n) = sum(k=1,A003817(n),my(t=bitxor(k,k*n)); (!(hammingweight(t)%2) && A006068(t) == 2*A048720(k,n-1))); \\ Why this works? See comments in A115873.

%Y Cf. A003817, A048720, A065621, A115873, A391590 (positions of 1's).

%Y Cf. array A115872.

%K nonn,base

%O 1,2

%A _Antti Karttunen_, Dec 20 2025