A027687 -id:A027687 - cp's OEIS frontend

A323653 Multiperfect numbers m such that sigma(m) is also multiperfect.

1, 459818240, 51001180160, 13188979363639752997731839211623940096, 5157152737616023231698245840143799191339008, 54530444405217553992377326508106948362108928, 133821156044600922812153118065015159487725568, 4989680372093758991515359988337845750507257510078971904

Offset: 1

Jaroslav Krizek, Jan 21 2019

nonn

Multiperfect numbers m such that sigma(m) divides sigma(sigma(m)).
Also k-multiperfect numbers m such that k*m is also multiperfect.
Corresponding values of numbers k(n) = sigma(a(n)) / a(n): 1, 3, 3, 5, 5, 5, 5, 5, ...
Corresponding values of numbers h(n) = sigma(k(n) * a(n)) / (k(n) * a(n)): 1, 4, 4, 6, 6, 6, 6, 6, ...
Number of k-multiperfect numbers m such that sigma(n) is also multiperfect for k = 3..6: 2, 0, 20, 0.
From Antti Karttunen, Mar 20 2021, Feb 18 2022: (Start)
Conjecture 1 (a): This sequence consists of those m for which sigma(m)/m is an integer (thus a term of A007691), and coprime with m. Or expressed in a slightly weaker form (b): {1} followed by those m for which sigma(m)/m is an integer, but not a divisor of m. In a slightly stronger form (c): For m > 1, sigma(m)/m is always the least prime not dividing m. This would imply both (a) and (b) forms.
Conjecture 2: This sequence is finite.
Conjecture 3: This sequence is the intersection of A007691 and A351458.
Conjecture 4: This is a subsequence of A349745, thus also of A351551 and of A351554.
Note that if there existed an odd perfect number x that were not a multiple of 3, then both x and 2*x would be terms in this sequence, as then we would have: sigma(x)/x = 2, sigma(2*x)/(2*x) = 3, sigma(6*x)/(6*x) = 4. See also the diagram in A347392 and A353365.
(End)
From Antti Karttunen, May 16 2022: (Start)
Apparently for all n > 1, A336546(a(n)) = 0. [At least for n=2..23], while A353633(a(n)) = 1, for n=1..23.
The terms a(1) .. a(23) are only cases present among the 5721 known and claimed multiperfect numbers with abundancy <> 2, as published 03 January 2022 under Flammenkamp's site, that satisfy the condition for inclusion in this sequence.
They are also the only 23 cases among that data such that gcd(n, sigma(n)/n) = 1, or in other words, for which the n and its abundancy are relatively prime, with abundancy in all cases being the least prime that does not divide n, A053669(n), which is a sufficient condition for inclusion in A351458.
(End)

			3-multiperfect number 459818240 is a term because number 3*459818240 = 1379454720 is a 4-multiperfect number.

Antti Karttunen, Table of n, a(n) for n = 1..23 (prepared from 1600 term b-file given in A007691).
Achim Flammenkamp, The Multiply Perfect Numbers Page
Index entries for sequences where any odd perfect numbers must occur
Index entries for sequences related to sigma(n)

Cf. A000396, A005820, A027687, A046060, A046061, A053669, A054030, A145551, A323652, A336546, A342659, A347392, A349745, A351458, A351551, A351554, A353365.

Intersection of any two of these three sequences: A007691, A019278, A066961.

[n: n in [1..10^6] | SumOfDivisors(n) mod n eq 0 and SumOfDivisors(SumOfDivisors(n)) mod SumOfDivisors(n) eq 0];

ismulti(n) = (sigma(n) % n) == 0;
isok(n) = ismulti(n) && ismulti(sigma(n)); \\ Michel Marcus, Jan 26 2019

A054030 Sigma(n)/n for n such that sigma(n) is divisible by n.

Original entry on oeis.org

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

Offset: 1

Asher Auel, Jan 19 2000

The graph supports the conjecture that all numbers except 2 appear only a finite number of times. Sequences A000396, A005820, A027687, A046060 and A046061 give the n for which the abundancy sigma(n)/n is 2, 3, 4, 5 and 6, respectively. See A134639 for the number of n having abundancy greater than 2. - T. D. Noe, Nov 04 2007

T. D. Noe, Table of n, a(n) for n = 1..1600 (using Flammenkamp's data)
Achim Flammenkamp, The Multiply Perfect Numbers Page
Eric Weisstein's World of Mathematics, Abundancy

Cf. A000203, A007691, A054024, A065997, A219545.

with(numtheory): for i while i < 33000 do
if sigma(i) mod i = 0 then print(sigma(i)/i) fi od;

for(n=1,1e7,if(denominator(k=sigma(n,-1))==1, print1(k", "))) \\ Charles R Greathouse IV, Mar 09 2014

a(n) = sigma(A007691(n))/A007691(n)

More terms from Jud McCranie, Jul 09 2000

More terms from David Wasserman, Jun 28 2004

A055153 Numbers k such that sigma(k) = 7k/2.

Original entry on oeis.org

4320, 4680, 26208, 20427264, 197064960, 21857648640, 57575890944, 88898072401645056, 301183421949935616, 9083288595228991885541376, 22290964134962716779872256, 230361837156847526055247872

Offset: 1

Jud McCranie, Jun 16 2000

nonn

			Sigma(4320)=15120=7*4320/2, so 4320 is in the sequence.

G. P. Michon, Table of n, a(n) for n = 1..21
G. P. Michon, Multiperfect and hemiperfect numbers
Walter Nissen, Abundancy : Some Resources
Walter Nissen, Abundancy : Some Resources
Walter Nissen, Primitive Friendly Pairs with friends < 2^34 with denom < 20000

Cf. A007539, A000396, A005820, A027687, A000203, A141643, A141645.

Do[If[DivisorSigma[1, m]==3.5*m, Print[m]], {m, 2*10^8}]

is(k)=sigma(k,-1)==7/2 \\ Charles R Greathouse IV, Mar 09 2014

Terms confirmed through a(5) by Ray Chandler, Sep 18 2008
a(6) and a(7) found by Yasutoshi Kohmoto and confirmed by Washington Bomfim, Oct 19 2008
Edited by N. J. A. Sloane, Sep 19 2008, Apr 18 2009
a(9) from Avinoam Kalma, a(12) from Yasutoshi Kohmoto, and a(8), a(10), a(13)-a(21) from Michel Marcus, added by Gerard P. Michon, Jun 04 2009

A348738 Numbers k for which A326042(k) < k, where A326042(n) = A064989(sigma(A003961(n))).

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97

Offset: 1

Antti Karttunen, Nov 02 2021

nonn

Claim: If there is an odd term y of A336702 larger than one, and it is the least one of such terms, then it should satisfy condition that for all nontrivial unitary divisor pairs d and x/d of x = A064989(y) [with gcd(d,x/d) = 1, 1 < d < x], the other divisor should reside in this sequence, and the other divisor in A348739. Proof: Applying A064989 to the odd terms of A336702 gives the fixed points of A326042. Suppose there are other odd terms in A336702 in addition to its initial 1, and let y be the least of these odd terms > 1 and x = A064989(y). Because A326042 (from here on indicated with f) is multiplicative, it follows that if we take any two nontrivial unitary divisors a and b of x, with x = a*b, gcd(a,b) = 1, 1 < a,b < x, then f(a)*f(b) = f(x) = x. Because f(x)/x = 1, we must have f(a)/a * f(b)/b = 1, as also the ratio f(n)/n is multiplicative. But f(a)/a and f(b)/b cannot be equal to 1, because then a and b would also be fixed by f, which contradicts our assumption that x were the least such fixed point larger than one. Therefore f(a) < a and f(b) > b, or vice versa. See also the comments in A348930, A348933.
Moreover, all odd perfect numbers (a subsequence of A336702), if such numbers exist, should also satisfy the same condition, regardless of whether they are the least of such numbers or not, because having a non-deficient proper divisor will push the abundancy index (ratio sigma(n)/n) of any number over 2. That is, for any such pair of nontrivial unitary divisors d and x/d, both A003961(d) and A003961(x/d) should be deficient, i.e., neither one should be in A337386. See also the condition given in A347383.
Terms that occur also in A337386 are: 120, 240, 360, 420, 480, 504, 540, 600, 630, ...

Positions of positive terms in A348736, positions of 1's in A348737 (characteristic function).
Almost complement of A348739.
Cf. A000203, A003961, A003973, A006039, A027687, A064989, A326042, A336702, A337386, A347383, A348741, A348748 (corresponding odd numbers).
Subsequences: A000040, A374464 (after its initial 1).
Cf. also A348930, A348933.

f1[2, e_] := 1; f1[p_, e_] := NextPrime[p, -1]^e; s1[1] = 1; s1[n_] := Times @@ f1 @@@ FactorInteger[n]; f2[p_, e_] := NextPrime[p]^e; s2[1] = 1; s2[n_] := Times @@ f2 @@@ FactorInteger[n]; Select[Range[100], s1[DivisorSigma[1, s2[#]]] < # &] (* Amiram Eldar, Nov 04 2021 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A326042(n) = A064989(sigma(A003961(n)));
isA348738(n) = (A326042(n)

A068404 Numbers k such that sigma(k) > 4*k.

Original entry on oeis.org

27720, 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, 191520

Offset: 1

Benoit Cloitre, Mar 02 2002

nonn

This sequence is of positive density, see for example Davenport. The density is between 0.000176 and 0.004521 according to the McDaniel College link. - Charles R Greathouse IV, Sep 07 2012
From Amiram Eldar, Feb 13 2021: (Start)
Behrend (1933) found the bounds (0.00003, 0.025) for the asymptotic density.
Wall et al. (1972) found the bounds (0.0001, 0.0147).
Using Deléglise's method the upper bound for the density found by McDaniel College is 0.000679406. (End)

Harold Davenport, Über numeri abundantes, Sitzungsber. Preuss. Akad. Wiss., Phys.-Math. Kl., No. 6 (1933), pp. 830-837.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Felix Behrend, Über numeri abundantes II, Preuss. Akad. Wiss. Sitzungsber., Vol. 6 (1933), pp. 280-293; alternative link.
Marc Deléglise, Bounds for the Density of Abundant Integers, Experimental Mathematics, Vol. 7, No. 2 (1998), pp. 137-143.
Richard Laatsch, Measuring the Abundancy of Integers, Mathematics Magazine, Vol. 59, No. 2 (1986), pp. 84-92, alternative link.
Gordon L. Miller and Mary T. Whalen, Multiply Abundant Numbers, School Science and Mathematics, Volume 95, Issue 5 (May 1995), pp. 256-259.
Summer 2010 research group on Abundancy, Abundancy Bounds 2010, McDaniel College, 2010.
Charles R. Wall, Phillip L. Crews and Donald B. Johnson, Density Bounds for the Sum of Divisors Function, Mathematics of Computation, Vol. 26, No. 119 (1972), pp. 773-777; Errata, Vol. 31, No. 138 (1977), p. 616.

Cf. A068403, A001221, A215264.

Cf. A027687 (4-perfect numbers).

Select[Range[27720,9!,60], 4*#Vladimir Joseph Stephan Orlovsky, Apr 21 2010 *)

A001221(a(n)) >= 4 (Laatsch, 1986). - Amiram Eldar, Nov 07 2020

A332223 a(1) = 1, and for n > 1, a(n) = A005940(1+sigma(A156552(n))).

Original entry on oeis.org

1, 2, 4, 5, 8, 9, 16, 7, 25, 18, 32, 25, 64, 21, 21, 49, 128, 27, 256, 35, 40, 121, 512, 49, 125, 385, 49, 121, 1024, 13, 2048, 13, 225, 1573, 105, 77, 4096, 57, 187, 343, 8192, 63, 16384, 65, 55, 4693, 32768, 121, 625, 32, 15625, 85, 65536, 81, 180, 91, 253, 9945, 131072, 175, 262144, 508079, 625, 847, 729, 169, 524288, 2057, 2601, 105, 1048576

Offset: 1

Antti Karttunen, Feb 12 2020

nonn

From Antti Karttunen, Jul 31 - Aug 06 2020: (Start)
As a curiosity, like with sigma, also here a(14) = a(15). [Cf. also A003973 and A341512]
Question: is it possible that a(k) = 2*k for any k? If not, then the deficiency (A033879) cannot be -1, and there are no quasiperfect numbers. If there were such cases, then A156552(k) = q would be an instance of quasiperfect number, which should also be an odd square, thus k would need to be of the form 4u+2.
In range n <= 10000, a(n) is a nontrivial multiple of n only at n = [25, 35, 343, 539, 847, 3315] with a(n) = [125, 105, 2401, 2695, 2541, 9945]. The quotients are thus also odd: 5, 3, 7, 5, 3, 3.
This rather meager empirical evidence motivates a conjecture that no quotient a(n)/n may be an even integer, and particularly, never a power of 2 larger than one, which (when translated back to the ordinary, unconjugated sigma) claims that it is not possible that sigma(n) = 2^k * n + 2^k - 1, for any n > 1, k > 0. See also A336700 and A336701, where this leads to a rather surprising empirical observation.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..10000 (computed using Hans Havermann's factorization of A156552)
P. Hagis and G. L. Cohen, Some Results Concerning Quasiperfect Numbers, J. Austral. Math. Soc. Ser. A 33, 275-286, 1982.
V. Siva Rama Prasad and C. Sunitha, On quasiperfect numbers, Notes on Number Theory and Discrete Mathematics, Vol. 23, 2017, No. 3, 73-78.
Eric Weisstein's World of Mathematics, Quasiperfect Number
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000203, A005940, A027687, A156552, A246277, A323243, A324201, A324899, A324909, A332224, A332225, A332463 (Möbius transform).
Cf. A003961, A332449, A332450, A332451, A332460 (for other functions similarly conjugated).
Cf. also A003973, A033879, A326042, A332222, A336700, A336701, A341512.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A332223(n) = if(1==n,n,A005940(1+sigma(A156552(n))));

A332223(n) = if(1==n,n,A005940(1+sumdiv(A156552(n),d,d))); \\ Antti Karttunen, Aug 04 2020

For n > 1, a(n) = A005940(1+A000203(A156552(n))) = A005940(1+A323243(n)).
a(A324201(n)) = A003961(A324201(n)). [It's an open problem whether A324201 gives all such solutions]
For n > 1, a(n) = A005940(1 + (Sum_{d|A156552(n)} d)). - Antti Karttunen, Aug 04 2020

A347392 Numbers k such that nearest common ancestor of k and sigma(k) in Doudna tree (A347879) is the grandparent of k.

Original entry on oeis.org

8, 9, 12, 13, 24, 35, 160, 455, 42550, 127650, 8041950, 22469750, 58506250, 67409250, 175518750, 394055550, 4246782750

Offset: 1

Antti Karttunen, Aug 30 2021

Note how 13 * 35 = 455.
If there exists any odd perfect numbers x, with sigma(x) = 2x, then 2*x would be a term of this sequence, as then sigma(2*x) = 6*x would be situated as a descendant under the other branch of the grandparent of 2*x (a parent of x), which is m = A064989(x), with m in A005101. Opn x itself would be a term of A336702. Furthermore, if such x is not a multiple of 3 (in which case m is odd and in A005231), then also 3x would be a term of this sequence as sigma(3*x) = 4*sigma(x) = 8*x would be situated as a grandchild of 2x, with 2x being a first cousin of 3x. Also, in that case, 6*x would be located in A336702 (particularly, in A027687) because then sigma(6*x) = 12*sigma(x) = 24*x = 4*(6*x).
.
                <--A003961--  m  ---(*2)--->
              .............../ \...............
             /                                 \
            /                                   \
           /                                     \
          x                                       2m
  etc..../ \......2x = sigma(x)          3x....../ \......4m
                  / \                     / \            / \
               etc.  \                 etc.  \        etc.  etc.
                      \                       \
                       4x          sigma(2x) = 6x
                      / \                     / \
                  etc    \                 etc.  \
                          \                       \
                          8x = sigma(3x)          12x
                             if m odd               \
                                                     \
                                                    24x = sigma(6x) if m odd.
.
Furthermore, if there were any hypothetical odd terms y in A005820 (triperfect numbers), then 2y would be a term of this sequence. See the diagram in A347391.
If it exists, a(18) > 2^33.

			455 is included in the sequence as sigma(455) = 672, and the nearest common ancestor of 455 and 672 in Doudna tree is 42, which is the grandparent of 455 [as 455 = A003961(A003961(42))] and the grand-grand-grand-parent of 672 [as 672 = (2^4)*42].

Positions of 2's in A347381.
Cf. A000203, A005101, A005231, A005820, A005940, A027687, A336702, A347391, A347393, A347394, A347879, A348041.
Cf. also A336834, A353365.

PARI
```
isA347392(n) = (2==A347381(n));
```

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A252463(n) = if(!(n%2),n/2,A064989(n));
isA347391(n) = if(1==n,0,my(m=A252463(n), s=sigma(n)); while(s>m, if(s==n, return(0)); s = A252463(s)); (s==m));
isA347391_or_A347392(n) = if(1==n,0,my(m=A252463(A252463(n)), s=sigma(n)); while(s>m, if(s==n, return(0)); s = A252463(s)); (s==m));
isA347392(n) = (isA347391_or_A347392(n) && !isA347391(n));

A069070 Numbers n such that n*sigma(n) is a perfect square.

Original entry on oeis.org

1, 40, 81, 135, 216, 224, 400, 819, 1372, 3240, 3744, 4650, 6318, 18144, 21700, 27930, 30240, 32400, 32760, 69312, 71148, 91694, 111132, 174592, 175500, 185220, 215472, 241395, 278318, 293907, 327600, 336675, 362700, 386232, 515450, 958737

Offset: 1

Benoit Cloitre, Apr 05 2002

Also n such that the squarefree part of n (A007913) equals the squarefree part of sigma(n), A355928.
Also n such that abundancy of n, sigma(n)/n is a rational square. - Michel Marcus, Oct 06 2013
See A230043, resp. A230538, for n whose abundancy is a rational cube, resp. fourth power. - M. F. Hasler, Nov 02 2013

Giovanni Resta, Table of n, a(n) for n = 1..1896 (terms < 4*10^12, first 500 terms from Donovan Johnson)

Cf. A000203, A000290, A007913, A064987, A355928.
Cf. A008848, A027687 (subsequences).
Cf. also A230043, A230538.
Positions of 0's in A355929.

Select[Range[1000000],IntegerQ[Sqrt[# DivisorSigma[1,#]]]&] (* Harvey P. Dale, Dec 24 2012 *)

for(n=1,1000000,if(issquare(n*sigma(n)),print1(n,",")))

isok(n) = issquare(sigma(n)/n); \\ Michel Marcus, Oct 06 2013

More terms from Rick L. Shepherd, Apr 07 2002

A259304 Numbers n such that there exists an x!=n that makes {n,n,n,x} an amicable multiset.

Original entry on oeis.org

796320, 1205820, 1277640, 1963080, 5154240, 8599500, 8694000, 9022860, 10820880, 12149280, 15845760, 17173800, 18023040, 19323360, 21360240, 22720320, 25494480, 27052200, 27335880, 28365480, 28574280, 28704060, 29211000, 30329640, 30373200, 30450420, 31433220

Offset: 1

Jeppe Stig Nielsen, Jun 23 2015

nonn

We call the multiset {n,n,n,x} amicable iff sigma(n)=sigma(x)=n+n+n+x.
For the x values, see A259305.
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, A259305, A259302, A259303, A259306, A036471.

isOK(n) = x=sigma(n)-3*n; x>0&x!=n&sigma(x)==3*n+x

A259305 Numbers n such that there exists an x!=n that makes {x,x,x,n} an amicable multiset.

Original entry on oeis.org

756000, 1220940, 1437480, 1973160, 5019840, 9058140, 9703260, 11058480, 11641680, 13085280, 17089920, 18117000, 19370880, 21282912, 21628080, 25295760, 25585560, 26936640, 27763560, 28191240, 29104200, 29203020, 30450420, 32839560, 33279120, 33398820

Offset: 1

Jeppe Stig Nielsen, Jun 23 2015

nonn

We call the multiset {x,x,x,n} amicable iff sigma(x)=sigma(n)=x+x+x+n.
For the x values, see A259304.
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, A259304, A259302, A259303, A259306, A036471.

isOK(n) = x=(sigma(n)-n)/3; x>0&x!=n&denominator(x)==1&sigma(x)==3*x+n

cp's OEIS Frontend

A323653 Multiperfect numbers m such that sigma(m) is also multiperfect.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

PARI

A054030 Sigma(n)/n for n such that sigma(n) is divisible by n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

PARI

Formula

Extensions

A055153 Numbers k such that sigma(k) = 7k/2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A348738 Numbers k for which A326042(k) < k, where A326042(n) = A064989(sigma(A003961(n))).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A068404 Numbers k such that sigma(k) > 4*k.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

A332223 a(1) = 1, and for n > 1, a(n) = A005940(1+sigma(A156552(n))).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

Formula

A347392 Numbers k such that nearest common ancestor of k and sigma(k) in Doudna tree (A347879) is the grandparent of k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

A069070 Numbers n such that n*sigma(n) is a perfect square.

Views