A094759 -id:A094759

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A005820

3-perfect (triply perfect, tri-perfect, triperfect or sous-double) numbers: numbers such that the sum of the divisors of n is 3n.
(Formerly M5376)

+10
93

120, 672, 523776, 459818240, 1476304896, 51001180160

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

These six terms are believed to comprise all 3-perfect numbers. - cf. the MathWorld link. - Daniel Forgues, May 11 2010

If there exists an odd perfect number m (a famous open problem) then 2m would be 3-perfect, since sigma(2m) = sigma(2)*sigma(m) = 3*2m. - Jens Kruse Andersen, Jul 30 2014

According to the previous comment from Jens Kruse Andersen, proving that this sequence is complete would imply that there are no odd perfect numbers. - Farideh Firoozbakht, Sep 09 2014

If 2 were prepended to this sequence, then it would be the sequence of integers k such that numerator(sigma(k)/k) = A017665(k) = 3. - Michel Marcus, Nov 22 2015

From Antti Karttunen, Mar 20 2021, Sep 18 2021, (Start):

Obviously, any odd triperfect numbers k, if they exist, have to be squares for the condition sigma(k) = 3*k to hold, as sigma(k) is odd only for k square or twice a square. The square root would then need to be a term of A097023, because in that case sigma(2*k) = 9*k. (See illustration in A347391).

Conversely to Jens Kruse Andersen's comment above, any 3-perfect number of the form 4k+2 would be twice an odd perfect number. See comment in A347870.

(End)

REFERENCES

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 120, p. 42, Ellipses, Paris 2008.

R. K. Guy, Unsolved Problems in Number Theory, B2.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

I. Stewart, L'univers des nombres, "Les nombres multiparfaits", Chap.15, pp 82-5, Belin/Pour la Science, Paris 2000.

David Wells, "The Penguin Book of Curious and Interesting Numbers," Penguin Books, London, 1986, pages 135, 159 and 185.

LINKS

Table of n, a(n) for n=1..6.

Abiodun E. Adeyemi, A Study of @-numbers, arXiv:1906.05798 [math.NT], 2019.

K. A. Broughan and Qizhi Zhou, Divisibility by 3 of even multiperfect numbers of abundancy 3 and 4, JIS 13 (2010) 10.1.5

A. Brousseau, Number Theory Tables, Fibonacci Association, San Jose, CA, 1973, p. 138.

S. Colbert-Pollack, J. Holdener, E. Rachfal, and Y. Xu A DIY Project: Construct Your Own Multiply Perfect Number!, Math Horizons, Vol. 28, pp. 20-23, February 2021.

F. Firoozbakht and M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.

Achim Flammenkamp, The Multiply Perfect Numbers Page [This page contains a lot of useful information, but be careful, not all the statements are correct. For example, it appears to claim that the six terms of this sequence are known to be complete, which is not the case. - N. J. A. Sloane, Sep 10 2014]

James Grime and Brady Haran, The Six Triperfect Numbers, Numberphile video (2018).

Fred Helenius, Link to Glossary and Lists

M. Kishore, Odd Triperfect Numbers, Mathematics of Computation, vol. 42, no. 165, 1984, pp. 231-233.

G. P. Michon, Multiperfect and hemiperfect numbers

Walter Nissen, Abundancy : Some Resources

N. J. A. Sloane & A. L. Brown, Correspondence, 1974

Eric Weisstein's World of Mathematics, Multiperfect Number

Eric Weisstein's World of Mathematics, Sous-Double

Wikipedia, Multiply perfect number, (section Triperfect numbers)

FORMULA

a(n) = 2*A326051(n). [provided no odd triperfect numbers exist] - Antti Karttunen, Jun 13 2019

EXAMPLE

120 = 2^3*3*5; sigma(120) = (2^4-1)/1*(3^2-1)/2*(5^2-1)/4 = (15)*(4)*(6) = (3*5)*(2^2)*(2*3) = 2^3*3^2*5 = (3) * (2^3*3*5) = 3 * 120. - Daniel Forgues, May 09 2010

MAPLE

A005820:=n->`if`(numtheory[sigma](n) = 3*n, n, NULL): seq(A005820(n), n=1..6*10^5); # Wesley Ivan Hurt, Oct 15 2017

MATHEMATICA

triPerfectQ[n_] := DivisorSigma[1, n] == 3n; A005820 = {}; Do[If[triPerfectQ[n], AppendTo[A005820, n]], {n, 10^6}]; A005820 (* Vladimir Joseph Stephan Orlovsky, Aug 07 2008 *)

Select[Range[10^6], DivisorSigma[1, #]==3#&] (* Harvey P. Dale, Jul 03 2023 *)

PROG

(PARI) isok(n) = sigma(n, -1) == 3; \\ Michel Marcus, Nov 22 2015

CROSSREFS

Cf. A000203, A000396, A007539, A017665, A019278, A027687, A046060, A046061, A068403, A075701, A097023, A171266, A259302, A259303, A306373, A326051, A326181, A329189, A335141, A335254, A347383, A347391.

Subsequence of the following sequences: A007691, A069085, A153501, A216780, A292365, A336458, A336461, A336745, and if there are no odd terms, then also of A334410.

Positions of 120's in A094759, 119's in A326200.

KEYWORD

nonn,nice,more

AUTHOR

N. J. A. Sloane

EXTENSIONS

Wells gives the 6th term as 31001180160, but this is an error.

Edited by Farideh Firoozbakht and N. J. A. Sloane, Sep 09 2014 to remove some incorrect statements.

STATUS

approved

A095301

Numbers n such that there is some k < n with n*sigma(k) = k*sigma(n).

+10
8

28, 140, 200, 224, 234, 270, 308, 364, 476, 496, 532, 600, 644, 672, 700, 812, 819, 868, 936, 1036, 1148, 1170, 1204, 1316, 1400, 1484, 1488, 1540, 1638, 1652, 1708, 1800, 1820, 1876, 1988, 2016, 2044, 2200, 2212, 2324, 2380, 2464, 2480, 2492, 2574, 2600

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Original name: Numbers n such that A094759(n) < n.

Agrees with A050973 without duplicates.

Also numbers n such that the value sigma(n)/n has already been reached before n. If n belongs to the sequence then A214701(n) = A214701(n-1). - Michel Marcus, Aug 19 2012

REFERENCES

B. Spearman and K. S. Williams, Handbook of Estimates in the Theory of Numbers, Carleton Math. Lecture Note Series No. 14, 1975; see p. 3.2, Eq. (3.9).

LINKS

Michel Marcus and Donovan Johnson, Table of n, a(n) for n = 1..10000 (first 180 terms from Michel Marcus)

P. Erdős, Remarks on number theory II: Some problems on the sigma function, Acta Arith., 5 (1959), 171-177.

EXAMPLE

A094759(28) = 6 < 28, hence 28 is in the sequence.

PROG

(PARI) for(n=1, 2600, s=sigma(n); k=1; while(n*sigma(k)!=k*s, k++); if(k<n, print1(n, ", ")));

(PARI) allab = []; nb = 0; for (i=1, n, ab = sigma(i)/i; already = 0; if (length(allab) > 0, for (j=1, length(allab), if (ab == allab[j], already = 1; break); ); ); if (already == 1, nb++; print1(i, ", "), allab = concat(allab, ab); ); )

/* Michel Marcus, Aug 19 2012 */

CROSSREFS

Cf. A000203, A094759, A050973.

KEYWORD

nonn

AUTHOR

Klaus Brockhaus, Jun 01 2004

EXTENSIONS

New name from Charles R Greathouse IV, Jul 21 2015

STATUS

approved

A094757

Least positive k <= n such that n*pi(k) = k*pi(n), where pi(n) is the number of primes <= n (A000720).

+10
4

1, 2, 3, 2, 5, 2, 7, 2, 9, 10, 11, 12, 13, 14, 10, 16, 17, 18, 19, 10, 21, 22, 23, 16, 25, 26, 27, 28, 29, 27, 31, 32, 27, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 40, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 56, 64, 65, 66, 67, 68, 69, 70, 71, 72

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Conjecture: For every n there exists a k different from n (possibly k > n) such that n*pi(k) = k*pi(n).

From David A. Corneth, Nov 15 2019: (Start)

If n*pi(k) = k*pi(n) then n/pi(n) = k/pi(k). So to find terms, one can make a list of pairs (k/pi(k), k) and sort them.

Then if for two such pairs (m/pi(m), m) and (k/pi(k), k), m > k have the same first element, i.e., m/pi(m) = k/pi(k) then a(m) = k for the least k with that ratio.

Amarnath Murthy's conjecture above is false. For n = 3 we have pi(n)/n = 2/3. For no other k we have pi(k)/k = 2/3. Therefore the conjecture is false. (End)

LINKS

David A. Corneth, Table of n, a(n) for n = 1..10007

EXAMPLE

a(15) = 10 as 15*pi(10) = 15*4 = 60 = 10*pi(15) = 10*6.

For k in {2, 4, 6, 8} we have pi(k)/k = 1/2 and for no k < 2 this holds. So for all these values a(k) = 2. - David A. Corneth, Nov 15 2019

MATHEMATICA

Table[SelectFirst[Range[n], n PrimePi[#] == # PrimePi[n] &], {n, 72}] (* Michael De Vlieger, Dec 14 2019 *)

PROG

(PARI) {m=72; pi=vector(m, n, omega(n!)); for(n=1, m, k=1; while(n*pi[k]!=k*pi[n], k++); print1(k, ", "))}

(PARI) first(n) = {n = nextprime(n); my(v = vector(n), t = -1, q = 1, res = vector(n), m); v[1] = [0, 1]; v[2] = [1/2, 2]; forprime(p = 2, n, t++; for(c = q, p - 1, v[c] = [t/c, c]; ); q = p ); v[n] = [t/n, n]; v = vecsort(v); res[1] = 1; for(i = 2, #v, if(v[i-1][1] != v[i][1], m = v[i][2]; ); res[v[i][2]] = m ); res } \\ David A. Corneth, Nov 15 2019

CROSSREFS

Cf. A095299 for n such that a(n) < n.

Cf. A000720, A094758, A094759.

KEYWORD

nonn,easy

AUTHOR

Amarnath Murthy, May 30 2004

EXTENSIONS

Edited and extended by Klaus Brockhaus, Jun 01 2004

STATUS

approved

A094758

Least k <= n such that n*tau(k) = k*tau(n), where tau(n) is the number of divisors of n (A000005).

+10
4

1, 1, 3, 4, 5, 3, 7, 8, 9, 5, 11, 8, 13, 7, 15, 16, 17, 9, 19, 20, 21, 11, 23, 9, 25, 13, 27, 28, 29, 15, 31, 32, 33, 17, 35, 36, 37, 19, 39, 40, 41, 21, 43, 44, 45, 23, 47, 48, 49, 25, 51, 52, 53, 27, 55, 56, 57, 29, 59, 40, 61, 31, 63, 64, 65, 33, 67, 68, 69, 35, 71, 72, 73, 37

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

EXAMPLE

6*tau(3) = 6*2 = 3*4 = 3*tau(6), hence a(6) = 3.

MAPLE

A094758 := proc(n)

for k from 1 to n do

if n*numtheory[tau](k) = k*numtheory[tau](n) then

return k;

end if;

end do:

end proc: # R. J. Mathar, Nov 15 2019

MATHEMATICA

a[n_] := Module[{k = 1, r = DivisorSigma[0, n]/n}, While[DivisorSigma[0, k]/k != r, k++]; k]; Array[a, 100] (* Amiram Eldar, Aug 19 2019 *)

PROG

(PARI) for(n=1, 75, s=numdiv(n); k=1; while(n*numdiv(k)!=k*s, k++); print1(k, ", "));

CROSSREFS

Cf. A095300 for n such that a(n) < n.

Cf. A000005, A094757, A094759.

KEYWORD

nonn

AUTHOR

Amarnath Murthy, May 30 2004

EXTENSIONS

Edited and extended by Klaus Brockhaus, Jun 01 2004

STATUS

approved

A326200

Lexicographically earliest sequence such that a(i) = a(j) => sigma(i)/i = sigma(j)/j for all i, j.

+10
4

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 6, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Restricted growth sequence transform of the abundancy index of n.

For all i, j:

a(i) = a(j) <=> A094759(i) = A094759(j),

a(i) = a(j) => A017665(i) = A017665(j),

a(i) = a(j) => A017666(i) = A017666(j).

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..105664

Achim Flammenkamp, The Multiply Perfect Numbers Page

PROG

(PARI)

up_to = 105664; \\ (In the same equivalence class as 78, 364 and 6448).

rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om, invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om, invec[i], i); outvec[i] = u; u++ )); outvec; };

v326200 = rgs_transform(vector(up_to, n, sigma(n)/n));

A326200(n) = v326200[n];

CROSSREFS

Cf. A000396 (positions of 6's), A005820 (positions of 119's).

Cf. A017665, A017666, A074902, A094759.

Cf. A218406, A218407, A218408, A218427, A326202.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 13 2019

STATUS

approved

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