A027687 -id:A027687 - cp's OEIS frontend

A259306 Numbers n such that there exists an x!=n that makes {n,n,x,x} an amicable multiset.

1441440, 1860768, 3205440, 3378240, 3423420, 3914820, 4029480, 4437720, 5738040, 6093360, 6807240, 7136640, 7239120, 7551360, 9402120, 10204740, 12270720, 12405120, 12942720, 13495680, 14627340, 14725620, 15233400, 17166240, 18097800, 19158048, 23023440

Offset: 1

Jeppe Stig Nielsen, Jun 23 2015

nonn

We call the multiset {n,n,x,x} amicable iff sigma(n)=sigma(x)=n+n+x+x.

If the condition x!=n were dropped, the terms from A027687 would also belong here.

John Cerkan, Table of n, a(n) for n = 1..1000

Cf. A027687, A259307, A259302, A259303, A259304, A259305, A036471.

isOK(n) = x=(sigma(n)-2*n)/2; x>0 && x!=n && denominator(x)==1 && sigma(x)==n+n+x+x

A023198 Numbers k such that sigma(k) >= 4*k.

Original entry on oeis.org

27720, 30240, 32760, 50400, 55440, 60480, 65520, 75600, 83160, 85680, 90720, 95760, 98280, 100800, 105840, 110880, 115920, 120120, 120960, 128520, 131040, 138600, 141120, 143640, 151200, 163800, 166320, 171360, 176400, 180180, 181440, 184800

Offset: 1

David W. Wilson

nonn

Called 4-abundant numbers. The first odd number is A119240(4) = 1853070540093840001956842537745897243375. - T. D. Noe, Mar 31 2011

Melvyn B. Nathanson, Elementary Methods in Number Theory, Springer, 2000, p 260.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Union of A027687 and A068404.

Select[Range[1000000], DivisorSigma[1, #] >= 4*#&] (* Vladimir Joseph Stephan Orlovsky, Apr 21 2010 *)

A216782 Numbers such that numerator(sigma(n)/n) is even and denominator(sigma(n)/n) is odd.

Original entry on oeis.org

3, 5, 6, 7, 11, 13, 14, 15, 17, 19, 21, 22, 23, 27, 28, 29, 30, 31, 33, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 51, 53, 54, 55, 57, 59, 60, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 75, 77, 78, 79, 83, 84, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97, 99, 101, 102

Offset: 1

Michel Marcus, Sep 16 2012

a(n) contains odd primes (A065091), odd squarefree semiprimes (A046388), perfect numbers (A000396), and 2n-multiperfect (A027687, A046061).

			sigma(3)/3 = 4/3 (even/odd).

Vincenzo Librandi, Table of n, a(n) for n = 1..8000

Cf. A216780, A216781, A324903 (characteristic function).

Subsequences: A000396, A027687, A043305 (without its initial 1), A046061, A046388, A065091, A336702 (without its initial 1).

Select[Range[1000], EvenQ[Numerator[DivisorSigma[1, #] / # ]] && OddQ[Denominator[DivisorSigma[1, #]/#]]&] (* Vincenzo Librandi, Jun 24 2014 *)
nedoQ[n_]:=Module[{ds=DivisorSigma[1,n]/n},EvenQ[Numerator[ds]]&&OddQ[ Denominator[ ds]]]; Select[Range[200],nedoQ] (* Harvey P. Dale, Feb 28 2015 *)

eoab(n) = {for (i=1, n, ab = sigma(i)/i; if ((numerator(ab) % 2 == 0) && (denominator(ab) % 2 == 1), print1(i, ", ")););}

A249670 a(n) = A017665(n)*A017666(n).

Original entry on oeis.org

1, 6, 12, 28, 30, 2, 56, 120, 117, 45, 132, 21, 182, 84, 40, 496, 306, 78, 380, 210, 672, 198, 552, 10, 775, 273, 1080, 2, 870, 60, 992, 2016, 176, 459, 1680, 3276, 1406, 570, 2184, 36, 1722, 112, 1892, 231, 390, 828, 2256, 372, 2793, 4650, 408, 1274, 2862

Offset: 1

Michel Marcus, Nov 03 2014

nonn

If n is a k-multiperfect, then a(n) = k.

Allan C. Wechsler, Table of n, a(n) for n = 1..10000

Cf. A000203 (sigma(n)).
Cf. A017665/A017666 (abundancy of n).
Cf. A009194 (gcd(n, sigma(n))), A064987 (n*sigma(n)).
Cf. A000396, A005820, A027687, A046060, A046061.

a249670 n = div (n * s) (gcd n s ^ 2)
 where s = sum (filter (\k -> mod n k == 0) [1..n])
-- Allan C. Wechsler, Mar 31 2023

a249670[n_Integer] := Numerator[DivisorSigma[-1, n]]*Denominator[DivisorSigma[-1, n]]; a249670 /@ Range[80] (* Michael De Vlieger, Nov 10 2014 *)

a(n) = my(ab = sigma(n)/n); numerator(ab)*denominator(ab);

a(n) = A064987(n)/A009194(n)^2.
a(A000396(n)) = 2 (perfect).
a(A005820(n)) = 3 (tri-perfect).
For p prime, a(p) = p*(p+1).

A357494 Numbers k such that s(k) = 4*k, where s(k) is the sum of divisors of k that have a square factor (A162296).

Original entry on oeis.org

902880, 1534680, 361674720, 767685600, 4530770640, 4941414720, 5405788800, 5517818880, 16993944000, 20429240832, 94820077440

Offset: 1

Amiram Eldar, Oct 01 2022

Analogous to 4-perfect numbers (A027687) with nonsquarefree divisors.
Equivalently, numbers k such that A325314(k) = -3*k.
There are no numbers k below 10^11 such that A162296(k) = m*k for integers m > 4.

			902880 is a term since A162296(902880) = 3611520 = 4*902880.

Cf. A162296, A325314.
Subsequence of A013929 and A023198.
Numbers k such that A162296(k) = m*k: A005117 (m=0), A001248 (m=1), A322609 (m=2), A357493 (m=3), this sequence (m=4).
Similar sequence: A027687.

q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1) == 4*n]; Select[Range[2, 2*10^6], q]

A380848 Numbers k such that A380845(k) = 4*k.

Original entry on oeis.org

123832800, 247695840, 268337160, 495421920, 536707080, 990874080, 1073446920, 1981778400, 2146926600, 3963587040, 4293885960, 7927204320, 8587804680, 15854438880, 17175642120, 31708908000, 34351317000, 63417846240, 68702666760, 124884879840, 126713795040, 126835722720

Offset: 1

Amiram Eldar, Feb 05 2025

Analogous to 4-perfect numbers (A027687) with A380845 instead of A000203.
All the terms are 4-abundant numbers (A068404), because A380845(k) <= A000203(k) with equality only when k is a power of 2, and powers of 2 are deficient numbers (A005100).
Are there numbers k such that A380845(k) = m*k for integers m >= 5? There are none below 1.6*10^11.

Cf. A000203, A005100, A027687, A380845, A380846, A380847.

Subsequence of A068404.

q[k_] := Module[{h = DigitCount[k, 2, 1]}, DivisorSum[k, # &, DigitCount[#, 2, 1] == h &] == 4*k]; Select[Range[3*10^8], q]

isok(k) = {my(h = hammingweight(k)); sumdiv(k, d, d*(hammingweight(d) == h)) == 4*k;}

a(19)-a(22) from Jinyuan Wang, Feb 12 2025

A384841 Numbers k for which sigma(k - x) + sigma(k + x) = 8*k has at least one nonnegative solution.

Original entry on oeis.org

14412, 17640, 25581, 25623, 25659, 26208, 30240, 31110, 31380, 31500, 32340, 32736, 32760, 34958, 35112, 44211, 44343, 45048, 45324, 45444, 46578, 48090, 49368, 51674, 52045, 52290, 53103, 53127, 53460, 54000, 54180, 59400, 59940, 60228, 60903, 60914, 60987, 61920, 62340, 62370

Offset: 1

Michel Marcus, Jun 10 2025

nonn

Michel Marcus, Table of n, a(n) for n = 1..100

Cf. A000203 (sigma), A027687 (a subsequence).
Cf. A383268, A384838, A384839, A384840.
Cf. A383758.

isok(k) = for (x=0, k-1, if (sigma(k - x) + sigma(k + x) == 8*k, return(1)));

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

A091443 Multiperfect numbers n which are divisible by sopfr(n) (multiperfect number: sigma(n) = k*n with k integer, sopfr: Sum of prime factors with repetition).

Original entry on oeis.org

1379454720, 14182439040, 212517062615531520, 27099073228001299660800, 680489641226538823680000, 15229814702070563916152832000, 34111227434420791224041472000, 59023729003862626557345792000

Offset: 1

Sven Simon, Jan 10 2004

nonn

The sequence contains multiperfect numbers with multiplicity k from 3..8. They are extracted from a list with about 5000 multiperfect numbers with multiplicity from 2..11. Because of the size of these numbers, no numbers with multiplicity k > 8 were found, even though there were about 3000 of them in the list. 95% of the multiperfect numbers with multiplicity from 3..8 are known.
Conjecture: the sequence is finite.
There are 5255 multiperfect numbers known with multiplicity 3 to 11. No more findings for A091443 so we still have 33 multiperfect numbers divisible by their sopfr (without the trivial case 1). With multiplicity 3..8 quite surely all are found (only very few - if any - missing). It is estimated that there are about 2200 with multiplicity 9 and 2091 of them are already found. With multiplicity 10 of estimated 4500 1161 are known. So far no multiperfect number with multiplicity 9 or 10 is divisible by its sopfr (with repetition). Using sopfr without repetition (A114887), there is one number with multiplicity 9 (or more). - Sven Simon, Feb 12 2012

			a(1): 1379454720 = 2^8*3*5*7*19*37*73, sopfr(n)= 2^5*5.

Sven Simon, Table of n, a(n) for n = 1..33 [Conjectured to be complete]
Achim Flammenkamp, The Multiply Perfect Numbers Page (See here for the latest information about the search)
Achim Flammenkamp, he Multiply Perfect Numbers Page [Local copy in pdf format of the page as of Jun 04 2020, made with permission of the author]
Achim Flammenkamp, Datafile_2013_12_31 (with 5311 multiperfect numbers). [_Sven Simon_, Feb 15 2014]
Eric Weisstein's World of Mathematics, Multiperfect numbers

Cf. A000203, A001414, A005820, A027687, A046060, A046061.

Intersection of A007691 and A036844. - Michel Marcus, Oct 08 2017

A171263 Primes of the form 1 + 4-multiperfect numbers.

Original entry on oeis.org

30241, 2178541, 23569921, 66433720321, 630532357710420079508428362350593, 1928622300236318049928258133164033, 651350717502447739281012140234441171379683329

Offset: 1

M. F. Hasler and Farideh Firoozbakht, Apr 15 2010

A subsequence of A093034. If p is a term of this sequence then for each positive integer k, x=p^k is a solution for the equation sigma(phi(x))=4(x-1). See comment lines of the sequence A093034.

			p=30241 is prime and sigma(p-1)/(p-1)=4, so p is in the sequence.

Cf. A027687, A093034, A171264, A171265.

cp's OEIS Frontend

A259306 Numbers n such that there exists an x!=n that makes {n,n,x,x} an amicable multiset.

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A023198 Numbers k such that sigma(k) >= 4*k.

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A216782 Numbers such that numerator(sigma(n)/n) is even and denominator(sigma(n)/n) is odd.

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A249670 a(n) = A017665(n)*A017666(n).

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A357494 Numbers k such that s(k) = 4*k, where s(k) is the sum of divisors of k that have a square factor (A162296).

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A380848 Numbers k such that A380845(k) = 4*k.

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A384841 Numbers k for which sigma(k - x) + sigma(k + x) = 8*k has at least one nonnegative solution.

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

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