A159907 -id:A159907 - cp's OEIS frontend

A218404 Numbers k for which sigma(k)/k - 1/4 is an integer.

40, 224, 360, 2016, 174592, 524160, 1571328, 492101632, 4428914688, 83350779494400, 2163091537152000, 2251247910912000, 2931032952207360, 7487050986455040, 155086041146982400, 369127262068899840, 12682035945406464000, 12848564519239680000

Offset: 1

Zdenek Cervenka, Oct 28 2012

a(10) > 10^11. - Donovan Johnson, Oct 31 2012
a(10) > 10^12. - Giovanni Resta, Nov 04 2012
Note that there are no terms here with abundancy 5/4. - Michel Marcus, Jun 26 2013

Cf. A159907, A218405.

a(8)-a(9) from Donovan Johnson, Oct 31 2012

a(10)-a(18) from Michel Marcus, Jun 26 2013

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

A088912 a(n) = smallest m such that sigma(m) = (n+1/2)*m.

Original entry on oeis.org

2, 24, 4320, 8910720, 17116004505600, 170974031122008628879954060917200710847692800, 12749472205565550032020636281352368036406720997031277595140988449695952806020854579200000

Offset: 1

Farideh Firoozbakht, Nov 29 2003

2 is the only number m such that sigma(m)=1.5*m.
A direct consequence of Robin's theorem is that a(6)>5E16, a(7)>1.898E29, a(8)>2.144E51, a(9)>9.877E89 and a(10)>6.023E157. - Washington Bomfim, Oct 30 2008
If the Riemann hypothesis (RH) is true then Robin's theorem (Guy Robin, 1984) implies that the n-th term of this sequence is greater than exp(exp((n+1/2)/exp(gamma))) where gamma=0.5772156649... is the Euler-Mascheroni constant (A001620). For the 6th term (which is actually 1.7*10^44) this lower bound is 5.0*10^16. Similarly, if RH is true, the next term (7th term) is at least 1.9*10^29 (and is probably more than 10^90 or so). - Gerard P. Michon, Jun 10 2009
From Gerard P. Michon, Jul 04 2009: (Start)
An upper bound for a(7) is provided by a 97-digit integer of abundancy 15/2 (5.71379...10^96) discovered by Michel Marcus on July 4, 2009. The factorization of that number is: 2^53 3^15 5^6 7^6 11^3 13 17 19^3 23 29 31 37 41 43 61 73 79 97 181 193 199 257 263 4733 11939 19531 21803 87211 262657.
Similarly, an upper bound for a(8) is provided by a 286-digit integer of abundancy 17/2 (3.30181...10^285) equal to x/17, where x is the smallest known number of abundancy 9 (a 287-digit integer discovered by Fred W. Helenius in 1995). This is so because 17 happen to occur with multiplicity 1 in the factorization of x. (End)
A new upper bound for a(7) was found on Aug 15 2009 by Michel Marcus, who broke his own record by finding two "small" multiples of 2^35*3^20*5^5*7^6*11^2*13^2*17 that are of abundancy 15/2. The lower one (1.27494722...10^88) has only 89 digits. - Gerard P. Michon, Aug 15 2009
These are the least hemiperfects of abundancy n + 1/2. - Walter Nissen, Aug 17 2010
On Jul 24 2010, Michel Marcus found a 191-digit integer of abundancy 17/2 (2.7172904...10^190) whose factorization starts with 2^81 3^29 5^9 7^10 11^4 13^3 17^2 19 23^2... This is the best upper bound to a(8) known so far. - Gerard P. Michon, Aug 22 2010

			a(2)=24 because 1+2+3+4+6+8+12+24=2.5*24 and 24 is the earliest m such that sigma(m)=2.5*m.

Guy Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, J. Math. Pures Appl. 63 (1984), 187-213.

G. P. Michon, Multiplicative functions: Abundancy = sigma(n)/n
G. P. Michon, Multiperfect and hemiperfect integers
G. P. Michon and M. Marcus, Hemiperfect numbers of abundancy 11/2
G. P. Michon and M. Marcus, Hemiperfect numbers of abundancy 13/2
G. P. Michon and M. Marcus, Hemiperfect numbers of abundancy 15/2
G. P. Michon and M. Marcus, Hemiperfect numbers of abundancy 17/2
Walter Nissen, Abundancy : Some Resources
Wikipedia, Hemiperfect number
Wikipedia, Riemann hypothesis

Cf. A007539, A000396, A005820, A027687, A046060, A046061.

Cf. A159907 (hemiperfect numbers: half-integral abundancy), A141643 (abundancy = 5/2), A055153 (abundancy = 7/2), A141645 (abundancy = 9/2), A159271 (abundancy = 11/2), A160678 (abundancy = 13/2).

a[n_] := (For[m=1, DivisorSigma[1, m]!=(n+1/2)m, m++ ];m); Do[Print[a[n]], {n, 4}]

a(5)-a(6) from Robert Gerbicz, Apr 19 2009
Cross-references from Gerard P. Michon, Jun 10 2009
Edited by M. F. Hasler, Mar 17 2013
a(7) from Michel Marcus confirmed and added by Max Alekseyev, Jun 05 2025

A246454 Numbers k that divide 2*sigma(k).

Original entry on oeis.org

1, 2, 6, 24, 28, 120, 496, 672, 4320, 4680, 8128, 26208, 30240, 32760, 523776, 2178540, 8910720, 17428320, 20427264, 23569920, 33550336, 45532800, 91963648, 142990848, 197064960, 459818240, 1379454720, 1476304896, 8583644160, 8589869056, 10200236032

Offset: 1

Jaroslav Krizek, Aug 26 2014

nonn

Numbers k such that 2*A000203(k) / k is an integer.

Union of A007691 (multiply-perfect numbers) and A159907 (numbers with half-integral abundancy index).

			Number 24 is in the sequence because 24 divides 2*sigma(24); 24 divides 2*60.

Jens Kruse Andersen, Table of n, a(n) for n = 1..55

Cf. A000203, A007691, A159907.

[n: n in [1..1000000] | Denominator(2*(SumOfDivisors(n))/n) eq 1];

for(n=1,10^8,if((2*sigma(n))%n==0,print1(n,", "))) \\ Derek Orr, Aug 26 2014

A330746 Number of values of k, 1 <= k <= n, with A017666(k) = A017666(n), where A017666(n) = n/gcd(n, sigma(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 3, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 4, 1, 1, 3, 2, 1, 1, 1, 2, 1, 2, 1, 3, 1, 4, 1, 2, 1, 1, 1, 1, 3, 1, 1, 2, 1, 4, 1, 2, 1, 5, 1, 2, 1, 1, 1, 5, 1, 1, 3, 2, 1, 1, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 3, 1, 2, 3, 1, 1, 6, 4, 4, 1, 2, 4, 2, 1, 1, 1, 1, 1, 4, 1, 1, 3

Offset: 1

Antti Karttunen, Jan 11 2020

nonn

Ordinal transform of A017666.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A009194, A014567, A017666.
A left inverse of following sequences: A007691, A159907, A245775.
Cf. also A331175.

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A017666(n) = (n/gcd(n, sigma(n)));
v330746 = ordinal_transform(vector(up_to, n, A017666(n)));
A330746(n) = v330746[n];

For all n >= 1, a(A014567(n)) = 1.

For all n >= 1, a(A007691(n)) = a(A159907(n)) = a(A245775(n)) = n.

A067237 Numbers k such that gcd(sigma(k),k) = k/5.

Original entry on oeis.org

5, 10, 15, 30, 60, 90, 140, 420, 1170, 2480, 3360, 6200, 7440, 8190, 18600, 40640, 114660, 121920, 131040, 297600, 997920, 2618880, 5059200, 64995840, 72602880, 95472000, 102136320, 167751680, 197308800, 433305600, 503255040, 668304000, 714954240, 1307124000, 1381161600, 1502582400

Offset: 1

Benoit Cloitre, Feb 20 2002

nonn

Also numbers k such that denominator(sigma(k)/k) = 5. - David A. Corneth, Oct 15 2023

			30 is in the sequence as gcd(sigma(30), 30) = gcd(72, 30) = 6 = 30/5. - _David A. Corneth_, Oct 15 2023

David A. Corneth, Table of n, a(n) for n = 1..166 (first 61 terms from Jud McCranie, terms <= 10^28)

Cf. A000203.

Cf. similar sequences with A017666(n)=k: A159907 (k=2), A245775 (k=3), A229088 (k=4), A262359 (k=6).

is(n) = gcd(sigma(n), n) == n/5 - David A. Corneth, Oct 15 2023

More terms from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Nov 15 2004

A218405 Numbers k for which sigma(k)/k - 3/4 is an integer.

Original entry on oeis.org

4, 47616, 293760, 1782144, 3485664, 134209536, 282977280, 746444160, 1459956960, 1684126080, 1716728832, 27298252800, 41233360896, 376591138560, 719045268480, 1622308746240, 38965526046720, 41769966551040, 985261739212800, 3067660098994176

Offset: 1

Zdenek Cervenka, Oct 28 2012

nonn

a(14) > 10^11. - Donovan Johnson, Oct 31 2012

a(16) > 10^12. - Giovanni Resta, Nov 04 2012

Cf. A218404, A159907.

a(6)-a(13) from Donovan Johnson, Oct 31 2012
a(14)-a(15) from Giovanni Resta, Nov 04 2012
More terms from Michel Marcus, Jun 26 2013

A239876 Partial sums of A229110, where A229110(n) = antisigma(n) mod n = A024816(n) mod n.

Original entry on oeis.org

0, 0, 2, 5, 9, 12, 18, 23, 28, 35, 45, 47, 59, 70, 76, 85, 101, 107, 125, 133, 143, 162, 184, 184, 203, 226, 240, 254, 282, 285, 315, 332, 350, 381, 403, 438, 474, 509, 531, 541, 581, 590, 632, 658, 670, 713, 759, 803, 844, 876, 906, 938, 990, 1005, 1043, 1063

Offset: 1

Jaroslav Krizek, Mar 29 2014

nonn

Antisigma(n) = A024816(n) = sum of non-divisors of n.

See A239877 - values of n for which a(n)/n is an integer.

Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Cf. A024816, A229110, A239877.

[&+[(k*(k+1)div 2 - SumOfDivisors (k)) mod k: k in [1..n]]: n in [1..1000]]

Accumulate[Table[Mod[Total[Complement[Range[n],Divisors[n]]],n],{n,60}]] (* Harvey P. Dale, Jul 05 2024 *)

a(n) = Sum_{k = 1...n} antisigma(k) mod k = Sum_{k = 1...n} A229110(k).

a(n) = a(n-1) for numbers n from A159907 if there are no odd multiply perfect numbers (A007691).

A262359 Numbers k such that denominator(sigma(k)/k) = 6.

Original entry on oeis.org

18, 22932, 14520576, 1610563584, 1907020800, 2836487808, 6399679104, 70912195200, 82819376640, 159991977600, 2732372020224, 6164773235712, 68309300505600, 148068998977536, 154119330892800, 264727305267840, 649657533767040, 3701724974438400, 42503412523106304, 220312341220608000

Offset: 1

Michel Marcus, Sep 19 2015

nonn

			sigma(18)/18 = 13/6, hence 18 is a term.

David A. Corneth, Table of n, a(n) for n = 1..54 (terms <= 10^28)

Cf. similar sequences with A017666(n)=k: A159907 (k=2), A245775 (k=3), A229088 (k=4), A067237 (k=5).

Cf. A000203.

isok(n) = denominator(sigma(n, -1)) == 6;

lista(nn) = {k = 6; nb = 0; while (nb != nn, if (denominator(sigma(k,-1)) == 6, print1(k, ", "); nb++); k += 6;);}

a(10)-a(12) from Jud McCranie, Oct 14 2023

More terms from David A. Corneth, Oct 15 2023

A348411 Numbers whose divisors have a harmonic mean with a denominator 2.

Original entry on oeis.org

3, 15, 42, 84, 135, 308, 420, 546, 1428, 1488, 1890, 2295, 2660, 3780, 6210, 7440, 9424, 12180, 13392, 18018, 20832, 24384, 24570, 43152, 43400, 64260, 66960, 77490, 90090, 98420, 121920, 127710, 155610, 200340, 204600, 227664, 316992, 348688, 353400, 461776, 483210

Offset: 1

Amiram Eldar, Oct 17 2021

nonn

Numbers k such that A099378(k) = 2.
The odd terms seem to be relatively rare: 3, 15, 135, 2295, 544635, 9258795, 22330035, 39118408875, ...
If k is in this sequence, then 2*k is in A348412.

			3 is a term since the harmonic mean of its divisors, {1, 3}, is 3/2.
15 is a term since the harmonic mean of its divisors, {1, 3, 5, 15}, is 5/2.

Amiram Eldar, Table of n, a(n) for n = 1..310

Cf. A001599, A099377, A099378, A348412.

Similar sequences: A159907, A330598.

filter:= proc(n) local L,h;
  L:= map(t->1/t,numtheory:-divisors(n));
  denom(nops(L)/convert(L,`+`))=2;
end proc:
select(filter, [$1..10^6]); # Robert Israel, Oct 17 2021

Select[Range[10^5], Denominator[DivisorSigma[0, #]/DivisorSigma[-1, #]] == 2 &]
Select[Range[500000],Denominator[HarmonicMean[Divisors[#]]]==2&] (* Harvey P. Dale, Apr 06 2023 *)

isok(m) = my(d=divisors(m)); denominator(#d/sum(k=1, #d, 1/d[k])) == 2; \\ Michel Marcus, Oct 18 2021

from sympy import gcd, divisor_sigma
A348411_list = [n for n in range(1,10**3) if (lambda x, y: 2*gcd(x,y*n)==x)(divisor_sigma(n),divisor_sigma(n,0))] # Chai Wah Wu, Oct 20 2021

cp's OEIS Frontend

A218404 Numbers k for which sigma(k)/k - 1/4 is an integer.

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

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A088912 a(n) = smallest m such that sigma(m) = (n+1/2)*m.

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A246454 Numbers k that divide 2*sigma(k).

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A330746 Number of values of k, 1 <= k <= n, with A017666(k) = A017666(n), where A017666(n) = n/gcd(n, sigma(n)).

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A067237 Numbers k such that gcd(sigma(k),k) = k/5.

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A218405 Numbers k for which sigma(k)/k - 3/4 is an integer.

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A239876 Partial sums of A229110, where A229110(n) = antisigma(n) mod n = A024816(n) mod n.

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