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Numbers j such that sigma(sigma(j)) = k*j for some k.
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%I #186 Nov 04 2023 08:42:52

%S 1,2,4,8,15,16,21,24,42,60,64,84,160,168,240,336,480,504,512,960,1023,

%T 1344,1536,4092,4096,10752,13824,16368,29127,32256,32736,47360,57120,

%U 58254,61440,65472,65536,86016,116508,217728,262144,331520,343976,466032,550095

%N Numbers j such that sigma(sigma(j)) = k*j for some k.

%C Let sigma^m (j) be the result of applying the sum-of-divisors function (A000203) m times to j; call j (m,k)-perfect if sigma^m (j) = k*j; then this is the sequence of (2,k)-perfect numbers.

%C From _Michel Marcus_, May 14 2016: (Start)

%C For these numbers, the quotient k = sigma(sigma(j))/j is an integer (see A098223). Then also k = (sigma(s)/s)*(sigma(j)/j) with s = sigma(j). That is, k = abundancy(s)*abundancy(j).

%C So looking at the abundancy of these terms may be interesting. Indeed we see that 459818240 and 51001180160 are actually 3-perfect numbers (A005820), and the reason they are here is that they are coprime to 3. So their sums of divisors are 4-perfect numbers (A027687), yielding q=12.

%C In a similar way, we can see that the 5-perfect numbers (A046060) that are coprime to 5 will be terms of this sequence with q=30. There are 20 such numbers, the smallest being 13188979363639752997731839211623940096. (End)

%C From _Michel Marcus_, May 15 2016: (Start)

%C It is also interesting to note that for a(2)=8, s=sigma(8)=15 is also a term. This happens to be the case for chains of several terms in a row:

%C 8, 15, 24, 60, 168, 480 with k = 3,4,7,8,9,10;

%C 512, 1023, 1536, 4092, 10752, 32736 with k = 3,4,7,8,9,10;

%C 29127, 47360, 116508, 331520, 932064, 2983680 with k = 4,7,8,9,13,14;

%C 1556480, 3932040, 14008320 with k = 9,13,14;

%C 106151936, 251650560, 955367424 with k = 9,13,14;

%C 312792480, 1505806848 with k = 19,20;

%C 6604416000, 30834059256 with k = 19,20;

%C 9623577600, 46566269568 with k = 19,20.

%C When j is a term, we can test if s=sigma(j) is also a term; this way we get 6 more terms: 572941926400, 845734196736, 1422976331052, 4010593484800, 11383810648416, 36095341363200.

%C And the corresponding chains are:

%C 173238912000, 845734196736 with k = 19,20;

%C 355744082763, 572941926400, 1422976331052, 4010593484800, 11383810648416, 36095341363200 with k = 4,7,8,9,13,14. (End)

%C From _Altug Alkan_, May 17 2016: (Start)

%C Here are additional chains for the above list:

%C 57120, 217728 with k = 13,14;

%C 343976, 710400 with k = 7,8;

%C 1980342, 5621760 with k = 10,14;

%C 4404480, 14913024 with k = 11,12;

%C 238608384, 775898880 with k = 11,12. (End)

%C Currently, the coefficient pairs are [1, 1], [3, 4], [4, 7], [7, 8], [8, 9], [9, 10], [9, 13], [10, 14], [11, 12], [13, 14], [16, 17], [16, 21], [17, 18], [19, 20], [23, 24], [25, 26], [25, 31], [27, 28], [29, 30], [31, 32], [32, 33], [37, 38]. It is interesting to note that for some of them, the pair (s,t) also satisfies t=sigma(s). - _Michel Marcus_, Jul 03 2016; Sep 06 2016

%C Using these empirical pairs of coefficients in conjunction with the first comment allows us to determine whether some term is the sum of divisors of another yet unknown smaller term. - _Michel Marcus_, Jul 04 2016

%C For m in A090748 = A000043 - 1 and c in A205597 (= odd a(n)), c*2^m is in the sequence, unless 2^(m+1)-1 | sigma(c). Indeed, from sigma(x*y) = sigma(x)*sigma(y) for gcd(x,y) = 1, we get sigma(sigma(c*2^m)) = sigma(sigma(c))*2^(m+1), so c*2^m is in the sequence if sigma(sigma(c))/c = k/2 (where k can't be odd: A330598 has no odd c). - _M. F. Hasler_, Jan 06 2020

%H Giovanni Resta, <a href="/A019278/b019278.txt">Table of n, a(n) for n = 1..145</a> (first 130 terms from Jud McCranie)

%H G. L. Cohen and H. J. J. te Riele, <a href="http://projecteuclid.org/euclid.em/1047565640">Iterating the sum-of-divisors function</a>, Experimental Mathematics, 5 (1996), pp. 93-100.

%H Michel Marcus, <a href="/A019278/a019278_22.txt">List of terms, grouped by quotient, not exhaustive</a>.

%t Select[Range[100000], Mod[DivisorSigma[1, DivisorSigma[1, #]], #] == 0 &] (* _Carl Najafi_, Aug 22 2011 *)

%o (PARI) is_A019278(n)=sigma(sigma(n))%n==0 \\ _M. F. Hasler_, Jul 02 2016

%o (Python)

%o from sympy.ntheory import divisor_sigma as D

%o print([i for i in range(1, 10000) if D(D(i, 1), 1)%i==0]) # _Indranil Ghosh_, Mar 17 2017

%o (Magma) [m: m in [1..560000]| IsIntegral(DivisorSigma(1,DivisorSigma(1,m))/m)]; // _Marius A. Burtea_, Nov 16 2019

%Y Cf. A098219, A098220, A098221, A098222, A098223, A008333, A051027, A019276.

%Y For sigma see A000203 and A007691.

%Y Cf. A205597 (odd terms), A323653 (those terms that are in A007691, i.e., for which sigma(n)/n is also an integer), A330598 (half-integer ratio).

%K nonn

%O 1,2

%A _N. J. A. Sloane_

%E Simpler definition from _M. F. Hasler_, Jul 02 2016