

A295296


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.


11



1, 2, 3, 4, 8, 10, 16, 32, 64, 128, 136, 256, 315, 512, 1024, 2048, 4096, 8192, 16384, 32768, 32896, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 2147516416
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OFFSET

1,2


COMMENTS

Numbers n such that their binary weight is equal to their deficiency.
Numbers n such that A000120(n) = A033879(n), or equally A000203(n) = A005187(n), or equally A001065(n) = A011371(n).
2^(2^k1) * (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
It would be nice to know whether 315 is the only term that is neither in A191363 nor a power of two.
Any term that is either a square or twice a square (in A028982) must be odious (in A000069), and vice versa.
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


LINKS

Giovanni Resta, Table of n, a(n) for n = 1..52
Index entries for sequences related to binary expansion of n
Index entries for sequences related to sigma(n)


EXAMPLE

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 1bits. Thus 315 is included in the sequence.


MATHEMATICA

Select[Range[2^20], DivisorSigma[1, #] + DigitCount[#, 2, 1] == 2 # &] (* Michael De Vlieger, Nov 26 2017 *)


PROG

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


CROSSREFS

Positions of zeros in A294898 and A294899.
Subsequence of A005100 and also of A295298.
Subsequences: A000079, A191363 (the five known terms).
Cf. A000120, A000203, A001065, A005187, A011371, A019434, A033879, A141548.
Sequence in context: A005542 A037171 A308811 * A186417 A207644 A114854
Adjacent sequences: A295293 A295294 A295295 * A295297 A295298 A295299


KEYWORD

nonn,base,changed


AUTHOR

Antti Karttunen, Nov 26 2017


EXTENSIONS

Terms a(35) and beyond from Giovanni Resta, Feb 27 2020


STATUS

approved



