A295296 - OEIS

%I #59 Dec 24 2020 02:28:28

%S 1,2,3,4,8,10,16,32,64,128,136,256,315,512,1024,2048,4096,8192,16384,

%T 32768,32896,65536,131072,262144,524288,1048576,2097152,4194304,

%U 8388608,16777216,33554432,67108864,134217728,268435456,536870912,1073741824,2147483648,2147516416

%N Numbers n such that the sum of their divisors + the number of ones in their binary expansion = 2n; numbers for which A000203(n) + A000120(n) = 2n.

%C Numbers n such that their binary weight is equal to their deficiency.

%C Numbers n such that A000120(n) = A033879(n), or equally A000203(n) = A005187(n), or equally A001065(n) = A011371(n).

%C 2^(2^k-1) * (2^(2^k) + 1) is in the sequence if and only if (2^(2^k) + 1) is a (Fermat) prime (A019434) which as of today is known to be the case for 0 <= k <= 4 giving the terms 3, 10, 136, 32896 and 2147516416. - _David A. Corneth_, Nov 26 2017

%C It would be nice to know whether 315 is the only term that is neither in A191363 nor a power of two.

%C Any term that is either a square or twice a square (in A028982) must be odious (in A000069), and vice versa.

%C If there's an odd term below 10^30 besides 315 then it must be divisible by a prime >= 23. - _David A. Corneth_, Nov 27 2017

%C 221753180448460815 is odd and also a term of this sequence. - _Alexander Violette_, Dec 24 2020

%H Giovanni Resta, <a href="/A295296/b295296.txt">Table of n, a(n) for n = 1..52</a>

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

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

%e A000203(315) = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 35 + 45 + 63 + 105 + 315 = 624. 2*315 - 624 = 6, and when 315 is written in binary, 100111011, we see that it has six 1-bits. Thus 315 is included in the sequence.

%t Select[Range[2^20], DivisorSigma[1, #] + DigitCount[#, 2, 1] == 2 # &] (* _Michael De Vlieger_, Nov 26 2017 *)

%o (PARI) for(n=1, oo, if(sigma(n)+hammingweight(n) == 2*n, print1(n, ", ")));

%Y Positions of zeros in A294898 and A294899.

%Y Subsequence of A005100 and also of A295298.

%Y Subsequences: A000079, A191363 (the five known terms).

%Y Cf. A000120, A000203, A001065, A005187, A011371, A019434, A033879, A141548.

%K nonn,base

%O 1,2

%A _Antti Karttunen_, Nov 26 2017

%E Terms a(35) and beyond from _Giovanni Resta_, Feb 27 2020