A098223 -id:A098223 - cp's OEIS frontend

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

1, 2, 4, 8, 15, 16, 21, 24, 42, 60, 64, 84, 160, 168, 240, 336, 480, 504, 512, 960, 1023, 1344, 1536, 4092, 4096, 10752, 13824, 16368, 29127, 32256, 32736, 47360, 57120, 58254, 61440, 65472, 65536, 86016, 116508, 217728, 262144, 331520, 343976, 466032, 550095

Offset: 1

N. J. A. Sloane

nonn

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.
From Michel Marcus, May 14 2016: (Start)
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).
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.
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)
From Michel Marcus, May 15 2016: (Start)
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:
8, 15, 24, 60, 168, 480 with k = 3,4,7,8,9,10;
512, 1023, 1536, 4092, 10752, 32736 with k = 3,4,7,8,9,10;
29127, 47360, 116508, 331520, 932064, 2983680 with k = 4,7,8,9,13,14;
1556480, 3932040, 14008320 with k = 9,13,14;
106151936, 251650560, 955367424 with k = 9,13,14;
312792480, 1505806848 with k = 19,20;
6604416000, 30834059256 with k = 19,20;
9623577600, 46566269568 with k = 19,20.
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.
And the corresponding chains are:
173238912000, 845734196736 with k = 19,20;
355744082763, 572941926400, 1422976331052, 4010593484800, 11383810648416, 36095341363200 with k = 4,7,8,9,13,14. (End)
From Altug Alkan, May 17 2016: (Start)
Here are additional chains for the above list:
57120, 217728 with k = 13,14;
343976, 710400 with k = 7,8;
1980342, 5621760 with k = 10,14;
4404480, 14913024 with k = 11,12;
238608384, 775898880 with k = 11,12. (End)
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
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
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

Giovanni Resta, Table of n, a(n) for n = 1..145 (first 130 terms from Jud McCranie)
G. L. Cohen and H. J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.
Michel Marcus, List of terms, grouped by quotient, not exhaustive.

Cf. A098219, A098220, A098221, A098222, A098223, A008333, A051027, A019276.
For sigma see A000203 and A007691.
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).

[m: m in [1..560000]| IsIntegral(DivisorSigma(1,DivisorSigma(1,m))/m)]; // Marius A. Burtea, Nov 16 2019

Select[Range[100000], Mod[DivisorSigma[1, DivisorSigma[1, #]], #] == 0 &] (* Carl Najafi, Aug 22 2011 *)

is_A019278(n)=sigma(sigma(n))%n==0 \\ M. F. Hasler, Jul 02 2016

from sympy.ntheory import divisor_sigma as D
print([i for i in range(1, 10000) if D(D(i, 1), 1)%i==0]) # Indranil Ghosh, Mar 17 2017

Simpler definition from M. F. Hasler, Jul 02 2016

A272930 a(n) is the least k such that sigma(sigma(k)) = n*k, where sigma(n) is the sum of the divisors of n, or 0 if no such k exists.

Original entry on oeis.org

1, 2, 8, 15

Offset: 1

Franklin T. Adams-Watters, May 11 2016

If a(5) is not zero, it exceeds 5*10^11 (see A098223). Likewise for a(17).
a(6) to a(16) are 42, 24, 60, 168, 480, 4404480, 2200380, 57120, 217728, 1058148, 7526400. a(18) is 39352320.
Is a(n) in fact nonzero for every positive n? - Franklin T. Adams-Watters, Jan 22 2019 [who previously conjectured that it is]
a(19) to a(26) are 312792480, 1505806848, 341543854080, 83825280, 13460388480, 8530704000, 58350015360, 284430182400. - Michel Marcus, May 18 2016
From Michel Marcus, May 18 2016; Jul 19 2016, Aug 23 2016, Sep 06 2016: (Start)
a(17) <= 336421458837032140800;
a(27) <= 4641476998878720;
a(28) <= 23479734980782080;
a(29) <= 4670834235654671884800;
a(30) <= 7526652811748265000960;
a(31) <= 45781120625942782080;
a(32) <= 242094947364010540800;
a(33) <= 216462850095065333760000;
a(34) <= 2366077977040955880819916800;
a(35) <= 8076837429313362044375040000;
a(36) <= 2634106558176405916291008921600;
a(37) <= 299500004890186577026355605378405509365760000000;
a(38) <= 45103591381041833364829469933568000. (End)

			sigma(8) = 15. sigma(15) = 24 = 3*8. Since this does not work for any value smaller than 8, a(3) = 8.

See the links in A019278. - _Altug Alkan_, May 31 2016 and May 18 2016

Cf. A000203 (sigma), A098223, A051027, A019279, A007539, A019278.

with(numtheory):
a:=proc(n) local k :
for k while sigma(sigma(k))<>n*k do od : k end: # Robert FERREOL, Apr 11 2018

Table[SelectFirst[Range[10^2], Nest[DivisorSigma[1, #] &, #, 2] == n # &], {n, 4}] (* Michael De Vlieger, May 11 2016, Version 10 *)

a(n)=my(r=1);while(sigma(sigma(r))!=n*r,r++);r \\ works only if a(n) is not zero.

A318059 a(n) is the numerator of sigma(sigma(n))/n.

Original entry on oeis.org

1, 2, 7, 2, 12, 14, 15, 3, 14, 39, 28, 14, 24, 30, 4, 2, 39, 28, 42, 24, 3, 91, 60, 7, 32, 48, 10, 30, 72, 13, 63, 13, 124, 60, 124, 28, 60, 84, 40, 117, 96, 6, 84, 56, 56, 195, 124, 14, 80, 64, 65, 171, 120, 20, 39, 45, 62, 117, 168, 8, 96, 126, 10, 2, 224, 403, 126, 78, 84, 403

Offset: 1

Michel Marcus, Aug 14 2018

			Fractions begin with 1, 2, 7/3, 2, 12/5, 14/3, 15/7, 3, 14/9, 39/10, 28/11, 14/3, ...

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000203 (sigma), A051027.

Cf. A098223 (when denominator is 1), A318060 (denominator).

seq(numer((numtheory:-sigma@@2)(n)/n),n=1..100); # Robert Israel, Aug 14 2018

a[n_] := Numerator[DivisorSigma[1, DivisorSigma[1, n]]/n]; Array[a, 100] (* Amiram Eldar, Apr 04 2024 *)

PARI
```
a(n) = numerator(sigma(sigma(n))/n);
```

For n=2, sigma(sigma(n)) = 4, so a(2) = 2.

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A205643 (2,k)-perfect numbers (A019278) such that the next (2,k)-perfect number has the same value of k (in A098223).

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

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A272930 a(n) is the least k such that sigma(sigma(k)) = n*k, where sigma(n) is the sum of the divisors of n, or 0 if no such k exists.

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A318059 a(n) is the numerator of sigma(sigma(n))/n.

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