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A019278 Numbers j such that sigma(sigma(j)) = k*j for some k.

%I A019278 #186 Nov 04 2023 08:42:52
%S A019278 1,2,4,8,15,16,21,24,42,60,64,84,160,168,240,336,480,504,512,960,1023,
%T A019278 1344,1536,4092,4096,10752,13824,16368,29127,32256,32736,47360,57120,
%U A019278 58254,61440,65472,65536,86016,116508,217728,262144,331520,343976,466032,550095
%N A019278 Numbers j such that sigma(sigma(j)) = k*j for some k.
%C A019278 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 A019278 From _Michel Marcus_, May 14 2016: (Start)
%C A019278 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 A019278 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 A019278 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 A019278 From _Michel Marcus_, May 15 2016: (Start)
%C A019278 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 A019278 8, 15, 24, 60, 168, 480 with k = 3,4,7,8,9,10;
%C A019278 512, 1023, 1536, 4092, 10752, 32736 with k = 3,4,7,8,9,10;
%C A019278 29127, 47360, 116508, 331520, 932064, 2983680 with k = 4,7,8,9,13,14;
%C A019278 1556480, 3932040, 14008320 with k = 9,13,14;
%C A019278 106151936, 251650560, 955367424 with k = 9,13,14;
%C A019278 312792480, 1505806848 with k = 19,20;
%C A019278 6604416000, 30834059256 with k = 19,20;
%C A019278 9623577600, 46566269568 with k = 19,20.
%C A019278 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 A019278 And the corresponding chains are:
%C A019278 173238912000, 845734196736 with k = 19,20;
%C A019278 355744082763, 572941926400, 1422976331052, 4010593484800, 11383810648416, 36095341363200 with k = 4,7,8,9,13,14. (End)
%C A019278 From _Altug Alkan_, May 17 2016: (Start)
%C A019278 Here are additional chains for the above list:
%C A019278 57120, 217728 with k = 13,14;
%C A019278 343976, 710400 with k = 7,8;
%C A019278 1980342, 5621760 with k = 10,14;
%C A019278 4404480, 14913024 with k = 11,12;
%C A019278 238608384, 775898880 with k = 11,12. (End)
%C A019278 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 A019278 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 A019278 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 A019278 Giovanni Resta, <a href="/A019278/b019278.txt">Table of n, a(n) for n = 1..145</a> (first 130 terms from Jud McCranie)
%H A019278 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 A019278 Michel Marcus, <a href="/A019278/a019278_22.txt">List of terms, grouped by quotient, not exhaustive</a>.
%t A019278 Select[Range[100000], Mod[DivisorSigma[1, DivisorSigma[1, #]], #] == 0 &] (* _Carl Najafi_, Aug 22 2011 *)
%o A019278 (PARI) is_A019278(n)=sigma(sigma(n))%n==0 \\ _M. F. Hasler_, Jul 02 2016
%o A019278 (Python)
%o A019278 from sympy.ntheory import divisor_sigma as D
%o A019278 print([i for i in range(1, 10000) if D(D(i, 1), 1)%i==0]) # _Indranil Ghosh_, Mar 17 2017
%o A019278 (Magma) [m: m in [1..560000]| IsIntegral(DivisorSigma(1,DivisorSigma(1,m))/m)]; // _Marius A. Burtea_, Nov 16 2019
%Y A019278 Cf. A098219, A098220, A098221, A098222, A098223, A008333, A051027, A019276.
%Y A019278 For sigma see A000203 and A007691.
%Y A019278 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 A019278 nonn
%O A019278 1,2
%A A019278 _N. J. A. Sloane_
%E A019278 Simpler definition from _M. F. Hasler_, Jul 02 2016