A019278 -id:A019278 - cp's OEIS frontend

A066961 Numbers k such that sigma(k) divides sigma(sigma(k)).

1, 5, 12, 54, 56, 87, 95, 276, 308, 427, 429, 446, 455, 501, 581, 611, 9120, 9180, 9504, 9720, 9960, 10296, 10620, 10740, 10824, 11070, 11310, 11480, 11484, 11556, 11628, 11748, 11934, 11960, 12024, 12036, 12072, 12084, 12376, 12460, 12510, 12570

Offset: 1

Benoit Cloitre, Jan 26 2002

nonn

Is this sequence finite?

These are numbers k such that sigma(k) is a multiply-perfect number (A007691). - Ivan N. Ianakiev, Sep 13 2016

			12 is in the sequence since sigma(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28 divides sigma(28) = 1 + 2 + 4 + 7 + 14 + 28 = 56. - _Michael B. Porter_, Sep 22 2016

Antti Karttunen, Table of n, a(n) for n = 1..3718 (first 1000 terms from Harry J. Smith)
Index entries for sequences where odd perfect numbers must occur, if they exist at all
Index entries for sequences related to sigma(n)

Cf. A000203, A051027.

Subsequences: A323653 (intersection with A007691, or equally, with A019278), A353365 (where the quotient is a power of 2).

[n: n in [1..13000] | (SumOfDivisors(SumOfDivisors(n)) mod SumOfDivisors(n) eq 0)]; // Vincenzo Librandi, Sep 13 2016

with(numtheory): A066961:=n->`if`(sigma(sigma(n)) mod sigma(n) = 0, n, NULL): seq(A066961(n), n=1..2*10^4); # Wesley Ivan Hurt, Sep 22 2016

Select[Range[30000], Divisible[DivisorSigma[1, DivisorSigma[1, #]], DivisorSigma[1, #]] &] (* Ivan N. Ianakiev, Sep 13 2016 *)

isok(n) = my(s=sigma(n)); s && ((sigma(s) % s) == 0); \\ Michel Marcus, Sep 17 2016

More terms from Lior Manor, Feb 06 2002

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.

A318060 a(n) is the denominator of sigma(sigma(n))/n.

Original entry on oeis.org

1, 1, 3, 1, 5, 3, 7, 1, 9, 10, 11, 3, 13, 7, 1, 1, 17, 9, 19, 5, 1, 22, 23, 1, 25, 13, 3, 7, 29, 2, 31, 4, 33, 17, 35, 9, 37, 19, 13, 20, 41, 1, 43, 11, 15, 46, 47, 3, 49, 25, 17, 52, 53, 3, 11, 7, 19, 29, 59, 1, 61, 31, 3, 1, 65, 66, 67, 17, 23, 70

Offset: 1

Michel Marcus, Aug 14 2018

			For n=2, sigma(sigma(n)) = 4, so a(2) = 1.

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

Cf. A000203 (sigma), A051027.

Cf. A019278 (where a(n) = 1), A318059 (numerator).

seq(denom((numtheory:-sigma@@2)(n)/n),n=1..200); # Robert Israel, Aug 15 2018

Table[(DivisorSigma[1,DivisorSigma[1,n]])/n,{n,70}]//Denominator (* Harvey P. Dale, Mar 30 2023 *)

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

A019292 Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (3,k)-perfect numbers.

Original entry on oeis.org

1, 12, 14, 24, 52, 98, 156, 294, 684, 910, 1368, 1440, 4480, 4788, 5460, 5840, 6882, 7616, 9114, 14592, 18288, 22848, 32704, 40880, 52416, 53760, 54864, 56448, 60960, 65472, 94860, 120960, 122640, 169164, 185535, 186368, 194432, 196137, 201872, 208026, 286160

Offset: 1

N. J. A. Sloane

nonn

Currently, up to k=50, the least integers to be (3,k)-perfect numbers are: 1, ?, ?, ?, 52, 98, ?, ?, ?, 12, ?, 14, ?, 5840, 7616, 294, ?, 201872, 169164, 24, 684, ?, ?, 910, ?, 40880, 60960, 4480, ?, 4788, 316160, 185535, 3138192, 1440, 186368, 5460, ?, 208026, 194432, 1454544, 481057305600, 26873600, 13225790247247872, 1937376, 10905024, ?, ?, 94860, ?, 683956224. - Michel Marcus, Jun 04 2017

			14 is a term because applying sigma three times we see that 14 -> 24 -> 60 -> 168, and 168 = 12*14. So 14 is a (3,12)-perfect number. - _N. J. A. Sloane_, May 29 2017

Michel Marcus, Table of n, a(n) for n = 1..131
Graeme L. Cohen and Herman J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.
Michel Marcus, Unexhaustive list of terms

Cf. A019278 ((2,k)-perfect numbers), A019293.

isok(n) = denominator(sigma(sigma(sigma(n)))/n) == 1; \\ Michel Marcus, Jan 02 2017

More terms from Michel Marcus, Jan 02 2017

A019293 Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (4,k)-perfect numbers.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 15, 18, 21, 24, 26, 32, 39, 42, 60, 65, 72, 84, 96, 160, 182, 336, 455, 512, 672, 896, 960, 992, 1023, 1280, 1536, 1848, 2040, 2688, 4092, 5472, 5920, 7808, 7936, 10416, 11934, 16352, 16380, 18720, 20384, 21824, 23424, 24564, 29127, 30240, 33792, 36720, 41440

Offset: 1

N. J. A. Sloane

nonn

Similarly to A019278, 2 and sigma(2) are both terms, and this can happen also for further iterations of sigma on known terms, thus providing new terms outside the current known range. - Michel Marcus, May 01 2017

Also it occurs that many divisors of A019278 are terms of this sequence. - Michel Marcus, May 28 2017

			10 is a term because applying sigma four times we see that 10 -> 18 -> 39 -> 168 -> 120, and 120 = 12*10. So 10 is a (4,12)-perfect number.

Michel Marcus, Table of n, a(n) for n = 1..320
Graeme L. Cohen and Herman J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.
Michel Marcus, Unexhaustive list of terms

Cf. A019276, A019278, A019292, A129076.

isok(n) = denominator(sigma(sigma(sigma(sigma(n))))/n) == 1; \\ Michel Marcus, Apr 29 2017

Corrected by Michel Marcus, Apr 29 2017

A318175 Numbers m such that A188999(A188999(m)) = k*m for some k where A188999 is the bi-unitary sigma function.

Original entry on oeis.org

1, 2, 8, 9, 10, 15, 18, 21, 24, 30, 42, 60, 144, 160, 168, 240, 270, 288, 324, 480, 512, 630, 648, 960, 1023, 1200, 1404, 1428, 1536, 2046, 2400, 2808, 2856, 2880, 3276, 3570, 4092, 4320, 4608, 6552, 8925, 10080, 10368, 10752, 11550, 13824, 14280, 14976, 15345, 16368, 17850

Offset: 1

Michel Marcus, Aug 20 2018

nonn

As in A019278, here there are many instances where if x is a term, then A188999(x) is also a term.
Additionally, there exist longer chains of 3 or 4 elements; e.g.,
- 8 (3), 15 (4), 24 (5), 60 (6);
- 9 (2), 10 (3), 18 (4), 30 (5);
- 512 (3), 1023 (4), 1536 (5), 4092 (6);
- 8925 (4), 14976 (5), 35700 (6);
- 219969739395000 (16), 899826278400000 (17), 3519515830320000 (18).

			For m=2, A188999(2) = 3 and A188999(3) = 4, so 2 is a term with k=2.
For m=9, A188999(9) = 10 and A188999(10) = 18, so 9 is a term with k=2.

Giovanni Resta, Table of n, a(n) for n = 1..227 (terms < 10^12; first 185 terms from Tomohiro Yamada)
Tomohiro Yamada, 2 and 9 are the only biunitary superperfect numbers, arXiv:1705.00189 [math.NT], 2017. See Table 1.
Tomohiro Yamada, 2 and 9 are the only biunitary superperfect numbers, Annales Univ. Sci. Budapest., Sec. Comp., Volume 48 (2018). See Table 1.
Michel Marcus, Unexhaustive list of terms.

Cf. A188999 (bi-unitary sigma).

Cf. A019278 (analog for sigma), A318182 (analog for infinitary sigma).

bsigma[n_] := If[n==1, 1, Product[{p, e} = pe; If[OddQ[e], (p^(e+1)-1)/(p-1), ((p^(e+1)-1)/(p-1)-p^(e/2))], {pe, FactorInteger[n]}]];
Reap[For[m = 1, m < 20000, m++, If[Divisible[bsigma @ bsigma @ m, m], Sow[m]]]][[2, 1]] (* Jean-François Alcover, Sep 22 2018 *)

a188999(n) = {f = factor(n); for (i=1, #f~, p = f[i, 1]; e = f[i, 2]; f[i, 1] = if (e % 2, (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1) -p^(e/2)); f[i, 2] = 1; ); factorback(f); }
isok(n) = frac(a188999(a188999(n))/n) == 0;

A098221 a(n) is the smallest number x such that floor(sigma(sigma(x))/x) = n or the A098219(x) quotient equals n.

Original entry on oeis.org

1, 2, 8, 6, 40, 30, 24, 60, 120, 480, 540, 1560, 2520, 10920, 27720, 30240, 191520, 524160, 360360, 3243240, 5765760, 28828800, 109549440, 438197760, 766846080, 3834230400, 9081072000, 32974381440, 147516969600, 880887047040, 2802822422400

Offset: 1

Labos Elemer, Oct 25 2004

a(30) <= 880887047040. a(31) <= 2802822422400. - Donovan Johnson, Feb 16 2013

			n = 10: a(10) = 480 because floor(sigma(sigma(480))/480) = floor(sigma(1512)/480) = floor(4800/480) = 4800/480 = n = 10.

Cf. A000203, A051027, A008333, A019278, A098219, A098220, A098222, A327630.

t=Table[0, {100}];Do[s=g[n];If[s<101&&t[[s]]==0, t[[s]]=n], {n, 1, 1000000}];t

a(n) = Min{x;floor(A051027(x)/x)=n}.

a(20)-a(26) from Donovan Johnson, Dec 29 2008
a(27)-a(29) from Donovan Johnson, Feb 16 2013
a(30)-a(31) from Giovanni Resta, Feb 27 2020

A098223 Integer quotients when sigma(sigma(x))/x is an integer.

Original entry on oeis.org

1, 2, 2, 3, 4, 2, 3, 7, 6, 8, 2, 6, 6, 9, 8, 6, 10, 10, 3, 8, 4, 6, 7, 8, 2, 9, 10, 8, 4, 10, 10, 7, 13, 8, 8, 8, 2, 6, 8, 14, 2, 9, 7, 8, 6, 9, 8, 13, 8, 15, 14, 6, 9, 9, 8, 10, 12, 14, 13, 8, 8, 11, 6, 14, 16, 12, 14, 12, 16, 15, 12, 18, 16, 11, 8, 22

Offset: 1

Labos Elemer, Oct 25 2004

nonn

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

Jud McCranie and Giovanni Resta, Table of n, a(n) for n = 1..145 (first 130 terms from Jud McCranie)

Cf. A000203, A098219-A098222, A019278, A008333, A051027, A272930.

with(numtheory): A098223:=n->`if`(sigma(sigma(n)) mod n = 0, sigma(sigma(n))/n, NULL): seq(A098223(n), n=1..10^5); # Wesley Ivan Hurt, Oct 10 2014

Select[DivisorSigma[1, DivisorSigma[1, #]]/# &@ Range[10^6], IntegerQ] (* Michael De Vlieger, May 11 2016 *)

for(n=1,1e7, sigma(sigma(n))%n||print1(sigma(sigma(n))/n",")) \\ M. F. Hasler, Jul 03 2016

In order of appearance the sigma(sigma(A019278(n)))/A019278(n) quotients which are by definition integers.

A330598 Numbers k such that the denominator of sigma(sigma(k))/k is equal to 2.

Original entry on oeis.org

30, 2046, 245760, 301056, 450560, 1171456, 1351680, 3514368, 14515200, 16760832, 19611648, 77220864, 159373824, 357291648, 391444480, 477216768, 555714432, 754928640, 765414240, 1006602240, 1761500160, 2330913312, 4314834944, 8369053056, 20449394784, 37949317120

Offset: 1

Michel Marcus, Dec 19 2019

nonn

Although the definition here is similar to the one in A019278, it appears that this sequence does not have the same nice features as A019278.

Otherwise said: sigma(sigma(k))/k is half-integer, or: sigma(sigma(k)) is an odd multiple of k/2. This also implies that all terms are even. - M. F. Hasler, Jan 06 2020

			sigma(sigma(30))/30 = sigma(72)/30 = 195/30 = 13/2 so 30 is a term.

Giovanni Resta, Table of n, a(n) for n = 1..38 (terms < 10^13)
Michel Marcus, Unexhaustive list of terms

Cf. A019278 (denominator is 1), A051027 (sigma(sigma)).

Cf. A000203 (sigma), A159907 (hemiperfect numbers).

isok(n) = denominator(sigma(sigma(n))/n) == 2;

a(22)-a(26) from Giovanni Resta, Dec 20 2019

A318182 Numbers m such that A049417(A049417(m)) = k*m for some k where A049417 is the infinitary sigma function.

Original entry on oeis.org

1, 2, 8, 9, 10, 15, 18, 24, 30, 60, 720, 1020, 4080, 8925, 14688, 14976, 16728, 17850, 35700, 36720, 37440, 66912, 71400, 285600, 308448, 381888, 428400, 602208, 636480, 763776, 856800, 1321920, 1505520, 3011040, 3084480, 21679488, 22276800, 30844800

Offset: 1

Michel Marcus, Aug 20 2018

nonn

a(86) > 3*10^11. All the prime factors of the first 85 terms belong to the set {2, 3, 5, 7, 11, 13, 17, 41, 43, 257}. - Giovanni Resta, Aug 25 2018
Like in A019278, here there are many instances where if x is a term, then A049417(x) is also a term.
Additionally, there exist longer chains of 3 or 4 elements like:
- 8 (3), 15 (4), 24 (5), 60 (6);
- 9 (2), 10 (3), 18 (4), 30 (5);
- 31615920 (4), 50585472 (5), 126463680 (6), 252927360 (12);
- 963407296051200 (16), 3134896756992000 (17), 15414516736819200 (18);
- 3541951043592192 (5), 8854877608980480 (6), 17709755217960960 (12), 53129265653882880 (20);
- 4829933241262080 (11), 17709755217960960 (12), 53129265653882880 (20);
  7871002319093760 (9), 26564632826941440 (10), 70839020871843840 (13), 265646328269414400 (14).

Giovanni Resta, Table of n, a(n) for n = 1..85 (first 58 terms from Tomohiro Yamada)
Tomohiro Yamada, Infinitary superperfect numbers, Annales Mathematicae et Informaticae, 47 (2017) pp. 211-218. See Table 1.
Michel Marcus, Unexhaustive list of terms

Cf. A049417 (infinitary sigma).

Cf. A019278 (analog for sigma), A318175 (analog for bi-unitary sigma).

a049417(n) = {my(b, f=factorint(n)); prod(k=1, #f[, 2], b = binary(f[k, 2]); prod(j=1, #b, if(b[j], 1+f[k, 1]^(2^(#b-j)), 1)));}
isok(n) = frac(a049417(a049417(n))/n) == 0;

More terms from Giovanni Resta, Aug 25 2018

cp's OEIS Frontend

A066961 Numbers k such that sigma(k) divides sigma(sigma(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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A318060 a(n) is the denominator of sigma(sigma(n))/n.

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A019292 Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (3,k)-perfect numbers.

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A019293 Let sigma_m (n) be result of applying sum-of-divisors function m times to n; call n (m,k)-perfect if sigma_m (n) = k*n; sequence gives the (4,k)-perfect numbers.

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A318175 Numbers m such that A188999(A188999(m)) = k*m for some k where A188999 is the bi-unitary sigma function.

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A098221 a(n) is the smallest number x such that floor(sigma(sigma(x))/x) = n or the A098219(x) quotient equals n.

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