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A332223 a(1) = 1, and for n > 1, a(n) = A005940(1+sigma(A156552(n))). 10

%I #70 Aug 06 2020 22:03:11

%S 1,2,4,5,8,9,16,7,25,18,32,25,64,21,21,49,128,27,256,35,40,121,512,49,

%T 125,385,49,121,1024,13,2048,13,225,1573,105,77,4096,57,187,343,8192,

%U 63,16384,65,55,4693,32768,121,625,32,15625,85,65536,81,180,91,253,9945,131072,175,262144,508079,625,847,729,169,524288,2057,2601,105,1048576

%N a(1) = 1, and for n > 1, a(n) = A005940(1+sigma(A156552(n))).

%C From _Antti Karttunen_, Jul 31 - Aug 06 2020: (Start)

%C As a curiosity, like with sigma, also here a(14) = a(15).

%C Question: is it possible that a(k) = 2*k for any k? If not, then the deficiency (A033879) cannot be -1, and there are no quasiperfect numbers. If there were such cases, then A156552(k) = q would be an instance of quasiperfect number, which should also be an odd square, thus k would need to be of the form 4u+2.

%C In range n <= 10000, a(n) is a nontrivial multiple of n only at n = [25, 35, 343, 539, 847, 3315] with a(n) = [125, 105, 2401, 2695, 2541, 9945]. The quotients are thus also odd: 5, 3, 7, 5, 3, 3.

%C This rather meager empirical evidence motivates a conjecture that no quotient a(n)/n may be an even integer, and particularly, never a power of 2 larger than one, which (when translated back to the ordinary, unconjugated sigma) claims that it is not possible that sigma(n) = 2^k * n + 2^k - 1, for any n > 1, k > 0. See also A336700 and A336701, where this leads to a rather surprising empirical observation.

%C (End)

%H Antti Karttunen, <a href="/A332223/b332223.txt">Table of n, a(n) for n = 1..10000</a> (computed using Hans Havermann's factorization of A156552)

%H P. Hagis and G. L. Cohen, <a href="http://dx.doi.org/10.1017/S1446788700018401">Some Results Concerning Quasiperfect Numbers</a>, J. Austral. Math. Soc. Ser. A 33, 275-286, 1982.

%H V. Siva Rama Prasad and C. Sunitha, <a href="http://nntdm.net/papers/nntdm-23/NNTDM-23-3-073-078.pdf">On quasiperfect numbers</a>, Notes on Number Theory and Discrete Mathematics, Vol. 23, 2017, No. 3, 73-78.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/QuasiperfectNumber.html">Quasiperfect Number</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

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

%F For n > 1, a(n) = A005940(1+A000203(A156552(n))) = A005940(1+A323243(n)).

%F a(A324201(n)) = A003961(A324201(n)). [It's an open problem whether A324201 gives all such solutions]

%F For n > 1, a(n) = A005940(1 + (Sum_{d|A156552(n)} d)). - _Antti Karttunen_, Aug 04 2020

%o (PARI)

%o A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940

%o A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552

%o A332223(n) = if(1==n,n,A005940(1+sigma(A156552(n))));

%o (PARI) A332223(n) = if(1==n,n,A005940(1+sumdiv(A156552(n),d,d))); \\ _Antti Karttunen_, Aug 04 2020

%Y Cf. A000203, A005940, A027687, A156552, A246277, A323243, A324201, A324899, A324909, A332224, A332225, A332463 (Möbius transform).

%Y Cf. A003961, A332449, A332450, A332451, A332460 (for other functions similarly conjugated).

%Y Cf. also A033879, A326042, A332222, A336700, A336701.

%K nonn

%O 1,2

%A _Antti Karttunen_, Feb 12 2020

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