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Below n=5x10^11, q=5 and 17 quotients do not appear; smallest numbers providing integer quotients = 1, 2, 3, 4,..., 16,... are as follows: 1, 2, 8, 15, ?, 42, 24, 60, 168, 480, 57669920, 2200380, 57120, 217278, 1058148, 7526400, ... - updated by Jud McCranie, Feb 08 2012

The above sequence is now A272930. - Franklin T. Adams-Watters, May 11 2016

See A019278 for the actual numbers x such that x | sigma(sigma(x)). - M. F. Hasler, Jul 03 2016

See the subsequence A274685 of odd terms for a remark on frequent pairs of the form (30k-3, 30k-1).

If m is in the sequence and gcd(k,m)=1, then k*m is also in the sequence. One might call "primitive" those terms which are not of this form, i.e., not a "coprime" multiple of an earlier term. The primitive terms are the primes and powers of primes within the sequence, cf. below.

Integers m > 0 where an integer k exists such that A000203(m) = A008587(k). - Felix Fröhlich, Jul 02 2016

For any prime p <> 5 there is an exponent k in {1, 3, 4} (depending on whether p is in A030433, A003631 or A030430) such that p^k is in this sequence. Given these p^k, the sequence consists of all numbers of the form n*p^(q*(k+1)-1) where n is coprime to p and q >= 1. Otherwise said, all numbers m which have some prime factor p with multiplicity q*(k+1)-1, where k = k(p) in {1, 3, 4} as introduced before. - M. F. Hasler, Jul 10 2016

The subsequence of odd terms in A274397.

If n is in the sequence and gcd(n,m)=1 for some odd m, then n*m is also in the sequence. One might call "primitive" those terms which are not of this form, i.e., not a "coprime" multiple of an earlier term. The list of these primitive terms is (19, 27, 29, 59, 79, 89, 109, 139, 149, 179, 199, 229, 239, 269, 343, 349, 359, 379, 389, 409, 419, 439, 449, 479, 499, ...). The primitive terms are the primes and powers of primes within the sequence. If a prime power p^k (k >= 1) is in the sequence, then p^(m(k+1)-1) is in the sequence for any m >= 1, since 1+p+...+p^(m(k+1)-1) = (1+p+...+p^k)(1+p^(k+1)+...+p^((m-1)*(k+1))). For example, with the prime p=19 we also have all odd powers 19^3, 19^5, ..., and with 27 = 3^3, we also have 27^5, 27^9, ... in the sequence.

On the other hand, for any prime p <> 5 there is an exponent k in {1, 3, 4} such that p^k is in this sequence (and therewith all higher powers of the form given above).

One may notice that there are many pairs of the form (30k-3, 30k-1), e.g., 27,29; 57,59; 87,89; 177,179; 237,239; 295,299; ... Indeed, it is likely that 30k-1 is prime and in this case, if 10k-1 is also prime, then sigma(30k-3) = 40k is divisible by 5 and sigma(30k-1) = 30k is also divisible by 5.

It is also known that a(20) = 11975040.

Then for higher indices n, we have:

a(19) <= 5235707393280;

a(21) <= 49110437376000;

a(22) <= 106780561395056640;

a(24) <= 1099525819392000;

a(25) <= 41252767395840;

a(26) <= 202768780032000.

It is also known that a(12) = 71400.

Then for higher indices n, we have:

a(11) <= 414230544000;

a(13) <= 2667897127526400;

a(14) <= 446464417259520;

a(15) <= 23613006957281280;

a(16) <= 22227004800;

a(17) <= 3134896756992000;

a(18) <= 15414516736819200;

a(20) <= 53129265653882880.

a(16) = 8420630400. - Giovanni Resta, Aug 25 2018

From Daniel Suteu, Oct 10 2019: (Start)

a(32) <= 14814918518400,

a(33) <= 64464915715200,

a(34) <= 709114072867200,

a(35) <= 9881550651772800,

a(36) <= 76648784785372800,

a(37) <= 2376112328346556800. (End)

For n <= 5, the values k <= 10^6 such that sigma(sigma(k)) = sigma(sigma(k) - k) + n*k are:

For n = 1; 4, 144, 16384.

For n = 2; 3, 7, 8, 31, 84, 127, 8191, 35743, 42048, 131071, 524287.

For n = 3; 14, 20, 372, 1081, 16246, 98292.

For n = 4; 54, 168, 308, 504, 994, 3870, 4994, 7800, 16488, 17360, 51995, 475664.

For n = 5; 1496, 1704, 14976, 21552, 379938, 854168.

a(10) = 16708685994, a(9) > 6*10^11. - Giovanni Resta, Sep 18 2017