login
Integers which when multiplied by 3 have an odd number of 1's in their binary expansion (cf. A000069).
6

%I #25 Aug 06 2023 03:05:57

%S 7,14,23,27,28,29,31,39,46,54,56,57,58,62,71,78,87,91,92,93,95,103,

%T 107,108,109,111,112,113,114,115,116,117,119,123,124,125,127,135,142,

%U 151,155,156,157,159,167,174,182,184,185,186,190,199,206,214

%N Integers which when multiplied by 3 have an odd number of 1's in their binary expansion (cf. A000069).

%C In other words, numbers n such that 3n is odious.

%C Numbers n such that valuation(binomial(3n,n),2) is odd. - _Benoit Cloitre_, Jun 06 2004

%C Intersection of A000069 and A008585 (multiples of 3), divided by 3.

%H G. C. Greubel, <a href="/A036556/b036556.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A180963(n)/3. - _Amiram Eldar_, Aug 06 2023

%t Select[ Range[ 214 ], OddQ[ Plus@@IntegerDigits[ 3#, 2 ]]& ]

%o (PARI) for(n=1,214,if(valuation(binomial(3*n,n),2)%2==1,print1(n,","))) \\ _Benoit Cloitre_, Jun 06 2004

%o (Magma) [ n : n in [0..150] | IsOdd(&+Intseq(3*n, 2))]; // _Vincenzo Librandi_, Apr 13 2011

%Y Cf. A000069, A036555, A036557, A180963.

%K nonn,base

%O 1,1

%A _Dan Hoey_

%E Definition corrected by _N. J. A. Sloane_, Jan 09 2007