A294898 - OEIS

%I #36 Dec 16 2024 13:29:29

%S 0,0,0,0,2,-2,3,0,3,0,7,-6,9,1,2,0,14,-5,15,-4,7,5,18,-14,16,7,10,-3,

%T 24,-16,25,0,16,12,19,-21,33,13,18,-12,37,-15,38,1,8,16,41,-30,38,4,

%U 26,3,48,-16,33,-11,30,22,53,-52,55,23,16,0,44,-14,63,8,39,-7,66,-53,69,31,22,9,54,-16,73,-28,38,35,78,-59,58

%N Deficiency minus binary weight: a(n) = A033879(n) - A000120(n) = A005187(n) - A000203(n).

%C "Least deficient numbers" or "almost perfect numbers" are those k for which A033879(k) = 1, or equally, for which a(k) = -A048881(k-1). The only known solutions are powers of 2 (A000079), all present also in A295296. See also A235796 and A378988. - _Antti Karttunen_, Dec 16 2024

%H Antti Karttunen, <a href="/A294898/b294898.txt">Table of n, a(n) for n = 1..16384</a>

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

%F a(n) = A005187(n) - A000203(n).

%F a(n) = A011371(n) - A001065(n).

%F a(n) = A033879(n) - A000120(n).

%F Sum_{k=1..n} a(k) ~ c * n^2, where c = 1 - zeta(2)/2 = 0.177532... . - _Amiram Eldar_, Feb 22 2024

%t Array[IntegerExponent[(2 #)!, 2] - DivisorSigma[1, #] &, 85] (* _Michael De Vlieger_, Nov 26 2017 *)

%o (Scheme) (define (A294898 n) (- (A005187 n) (A000203 n)))

%o (PARI) A294898(n) = ((2*n-sigma(n))-hammingweight(n)); \\ _Antti Karttunen_, Dec 16 2024

%Y Cf. A000120, A000203, A001065, A005187, A011371, A013661, A033879, A048881, A235796, A294896, A294899, A297114 (Möbius transform), A317844 (difference from a(n)), A326133, A326138, A324348 (a(n) applied to Doudna sequence), A379008 (a(n) applied to prime shift array), A378988.

%Y Cf. A295296 (positions of zeros), A295297 (parity of a(n)).

%K sign

%O 1,5

%A _Antti Karttunen_, Nov 25 2017

%E Name edited by _Antti Karttunen_, Dec 16 2024