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

1, 2, 4, 5, 8, 9, 16, 7, 25, 18, 32, 25, 64, 21, 21, 49, 128, 27, 256, 35, 40, 121, 512, 49, 125, 385, 49, 121, 1024, 13, 2048, 13, 225, 1573, 105, 77, 4096, 57, 187, 343, 8192, 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

Offset

1,2

Comments

From Antti Karttunen, Jul 31 - Aug 06 2020: (Start)

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

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.

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.

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.

(End)

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000 (computed using Hans Havermann's factorization of A156552)

P. Hagis and G. L. Cohen, Some Results Concerning Quasiperfect Numbers, J. Austral. Math. Soc. Ser. A 33, 275-286, 1982.

V. Siva Rama Prasad and C. Sunitha, On quasiperfect numbers, Notes on Number Theory and Discrete Mathematics, Vol. 23, 2017, No. 3, 73-78.

Eric Weisstein's World of Mathematics, Quasiperfect Number

Index entries for sequences computed from indices in prime factorization

Index entries for sequences related to sigma(n)

Formula

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

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

For n > 1, a(n) = A005940(1 + (Sum_{d|A156552(n)} d)). - Antti Karttunen, Aug 04 2020

Prog

(PARI)
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
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
A332223(n) = if(1==n, n, A005940(1+sigma(A156552(n))));
(PARI) A332223(n) = if(1==n, n, A005940(1+sumdiv(A156552(n), d, d))); \\ Antti Karttunen, Aug 04 2020

Crossrefs

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

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

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

Sequence in context: A195364 A279793 A305076 * A269361 A200947 A350356

Adjacent sequences: A332220 A332221 A332222 * A332224 A332225 A332226

Keyword

nonn

Author

Antti Karttunen, Feb 12 2020

Status

approved