A064591 -id:A064591 - cp's OEIS frontend

A064597 Nonunitary abundant numbers: the sum of the nonunitary divisors of n is larger than n; i.e., sigma(n) - usigma(n) > n.

36, 48, 72, 80, 96, 108, 120, 144, 160, 168, 180, 192, 200, 216, 224, 240, 252, 264, 280, 288, 300, 312, 320, 324, 336, 352, 360, 384, 392, 396, 400, 408, 416, 432, 448, 456, 468, 480, 504, 528, 540, 552, 560, 576, 588, 600, 612, 624, 640, 648, 672, 684

Offset: 1

Dean Hickerson, Sep 25 2001

			The sum of the nonunitary divisors of 36 is 2 + 3 + 6 + 12 + 18 = 41.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A048146, A063846, A064591, A064598.

nusigma[ n_ ] := DivisorSigma[ 1, n ]-Times@@(1+Power@@#&/@FactorInteger[ n ]); For[ n=1, True, n++, If[ nusigma[ n ]>n, Print[ n ] ] ]

usigma(n)= { local(f,s=1); f=factor(n); for(i=1, matsize(f)[1], s*=1 + f[i, 1]^f[i, 2]); return(s) } { n=0; for (m=1, 10^9, if (sigma(m) - usigma(m) > m, write("b064597.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 19 2009

A325814 a(n) = n - A048146(n), where A048146 is the sum of non-unitary divisors of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 2, 6, 10, 11, 4, 13, 14, 15, 2, 17, 9, 19, 8, 21, 22, 23, 0, 20, 26, 15, 12, 29, 30, 31, 2, 33, 34, 35, -5, 37, 38, 39, 4, 41, 42, 43, 20, 27, 46, 47, -8, 42, 35, 51, 24, 53, 18, 55, 8, 57, 58, 59, 12, 61, 62, 39, 2, 65, 66, 67, 32, 69, 70, 71, -33, 73, 74, 55, 36, 77, 78, 79, -4, 42, 82, 83, 20

Offset: 1

Antti Karttunen, May 23 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000203, A033879, A034448, A034460, A048146, A064591 (positions of zeros), A228058, A325812, A325813, A325824.
Cf. also A325314.
Cf. A002117, A013661.

Table[n - DivisorSum[n, # &, ! CoprimeQ[#, n/#] &], {n, 84}] (* Michael De Vlieger, May 27 2019 *)

A048146(n) = (sigma(n)-A034448(n));
A325814(n) = (n-A048146(n));

a(n) = n - A048146(n).
a(n) = A033879(n) + A034460(n).
a(A228058(n)) = A325824(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = 1/2 - zeta(2) * (1 - 1/zeta(3)) / 2 = 0.3617493553... . - Amiram Eldar, Feb 22 2024

A319745 Nonunitary harmonic numbers: numbers such that the harmonic mean of their nonunitary divisors is an integer.

Original entry on oeis.org

4, 9, 12, 18, 24, 25, 45, 49, 54, 60, 112, 121, 126, 150, 168, 169, 270, 289, 294, 336, 361, 529, 560, 594, 637, 726, 841, 961, 1014, 1232, 1369, 1638, 1680, 1681, 1734, 1849, 1984, 2166, 2184, 2209, 2430, 2520, 2688, 2700, 2809, 2850, 3174, 3481, 3721, 3780

Offset: 1

Amiram Eldar, Sep 27 2018

nonn

Includes all the numbers with a single nonunitary divisor. Those with more than one: 12, 18, 24, 45, 54, 60, 112, ...
Supersequence of A064591 (nonunitary perfect numbers).
Ligh & Wall showed that if p, 2p-1 and 2^p-1 are distinct primes (A172461, except for 2), then the following numbers are in the sequence: 6*p^2, p^2*(2p-1), 6*p^2*(2p-1), 2^(p+1)*3*(2^p-1), 2^(p+1)*15*(2^p-1) and 2^(p+1)*(2p-1)*(2^p-1).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Steve Ligh and Charles R. Wall, Functions of Nonunitary Divisors, Fibonacci Quarterly, Vol. 25 (1987), pp. 333-338.

Cf. A001599, A006086, A063947, A064591, A172461, A286325.

nudiv[n_] := Block[{d = Divisors[n]}, Select[d, GCD[#, n/#] > 1 &]]; nhQ[n_]:= Module[ {divs=nudiv[n]}, Length[divs] > 0 && IntegerQ[HarmonicMean[divs]]]; Select[Range[30000], nhQ]

hm(v) = #v/sum(k=1, #v, 1/v[k]);
vnud(n) = select(x->(gcd(x, n/x)!=1), divisors(n));
isok(n) = iferr(denominator(hm(vnud(n))) == 1, E, 0); \\ Michel Marcus, Oct 28 2018

A064594 Nonunitary multiply perfect numbers: the sum of the nonunitary divisors of n is a multiple of n; i.e., n divides sigma(n) - usigma(n).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 24, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 109

Offset: 1

Dean Hickerson, Sep 25 2001

Trivially includes all squarefree numbers (A005117). See A064595 for the others.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A048146, A064591, A064592, A064593, A064595, A064596.

nusigma[ n_ ] := DivisorSigma[ 1, n ]-Times@@(1+Power@@#&/@FactorInteger[ n ]); For[ n=1, True, n++, If[ Mod[ nusigma[ n ], n ]==0, Print[ n ] ] ]

usigma(n)= { local(f,s=1); f=factor(n); for(i=1, matsize(f)[1], s*=1 + f[i, 1]^f[i, 2]); return(s) }
{ n=0; for (m=1, 10^9, if ((sigma(m) - usigma(m)) % m == 0, write("b064594.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 19 2009

A064595 Nontrivial nonunitary multiply perfect numbers: the sum of the nonunitary divisors of n is a positive multiple of n; i.e., (sigma(n) - usigma(n))/n is a positive integer.

Original entry on oeis.org

24, 112, 1984, 2520, 31320, 32512, 1163160, 2208384, 3053232, 6535296, 13472928, 123165600, 134201344, 186076800, 365959296, 3456573120, 4401782352, 5517818880, 18924364800, 34359476224, 325377803520

Offset: 1

Dean Hickerson, Sep 25 2001

Cf. A048146, A064591, A064592, A064593, A064594, A064596.

nusigma[ n_ ] := DivisorSigma[ 1, n ]-Times@@(1+Power@@#&/@FactorInteger[ n ]); For[ n=1, True, n++, v=nusigma[ n ]; If[ v>0&&Mod[ v, n ]==0, Print[ n ] ] ]

A322609 Numbers k such that s(k) = 2*k, where s(k) is the sum of divisors of k that have a square factor (A162296).

Original entry on oeis.org

24, 54, 112, 150, 294, 726, 1014, 1734, 1984, 2166, 3174, 5046, 5766, 8214, 10086, 11094, 13254, 16854, 19900, 20886, 22326, 26934, 30246, 31974, 32512, 37446, 41334, 47526, 56454, 61206, 63654, 68694, 71286, 76614, 96774, 102966, 112614, 115926, 133206

Offset: 1

Amiram Eldar, Dec 20 2018

nonn

This sequence is infinite since 6*p^2 is included for all primes p. Terms that are not of the form 6*p^2: 112, 1984, 19900, 32512, 134201344, ...
Includes 4*k if k is an even perfect number: see A000396. - Robert Israel, Jan 06 2019
From Amiram Eldar, Oct 01 2022: (Start)
24 = 6*prime(1)^2 = 4*A000396(1) is the only term that is common to the two forms that are mentioned above.
19900 is the only term below 10^11 which is not of any of these two forms. Are there any other such terms?
All the known nonunitary perfect numbers (A064591) are also of the form 4*k, where k is an even perfect number.
Equivalently, numbers k such that A325314(k) = -k. (End)

			24 is a term since A162296(24) = 48 = 2*24.

Amiram Eldar, Table of n, a(n) for n = 1..12091 (terms below 10^11; terms 1..300 from Robert Israel)

Cf. A162296, A325314.
Subsequence of A005101 and A013929.
Numbers k such that A162296(k) = m*k: A005117 (m=0), A001248 (m=1), this sequence (m=2), A357493 (m=3), A357494 (m=4).
Similar sequences: A000396, A002827, A007357, A054979, A064591, A322486, A323757, A324707, A327633.

filter:= proc(n) convert(remove(numtheory:-issqrfree,numtheory:-divisors(n)),`+`)=2*n end proc:
select(filter, [$1..200000]); # Robert Israel, Jan 06 2019

s[1]=0; s[n_] := DivisorSigma[1,n] - Times@@(1+FactorInteger[n][[;;,1]]); Select[Range[10000], s[#] == 2# &]

s(n) = sumdiv(n, d, d*(1-moebius(d)^2)); \\ A162296
isok(n) = s(n) == 2*n; \\ Michel Marcus, Dec 20 2018

from sympy import divisors, factorint
A322609_list = [k for k in range(1,10**3) if sum(d for d in divisors(k,generator=True) if max(factorint(d).values(),default=1) >= 2) == 2*k] # Chai Wah Wu, Sep 19 2021

A064592 Nonunitary doubly perfect numbers: the sum of the nonunitary divisors of n is 2n; i.e., sigma(n) - usigma(n) = 2n.

Original entry on oeis.org

2520, 31320, 1163160, 2208384, 3053232, 6535296, 13472928, 123165600, 365959296, 4401782352, 5517818880, 612014161920

Offset: 1

Dean Hickerson, Sep 25 2001

There are no others up to 7.9*10^12, but here's a larger one: 2846972800010800140288.

Ligh & Wall found the first 11 terms of this sequence. - Amiram Eldar, Sep 27 2018

Steve Ligh and Charles R. Wall, Functions of Nonunitary Divisors, Fibonacci Quarterly, Vol. 25 (1987), pp. 333-338.

Cf. A048146, A064591, A064593, A064594, A064595, A064596.

nusigma[ n_ ] := DivisorSigma[ 1, n ]-Times@@(1+Power@@#&/@FactorInteger[ n ]); For[ n=1, True, n++, If[ nusigma[ n ]==2n, Print[ n ] ] ]

isok(n) = sumdiv(n, d, if (gcd(d, n/d)!=1, d)) == 2*n; \\ Michel Marcus, Jul 31 2017

A335142 Nonunitary Zumkeller numbers: numbers whose set of nonunitary divisors is nonempty and can be partitioned into two disjoint sets of equal sum.

Original entry on oeis.org

24, 48, 54, 80, 96, 112, 120, 150, 160, 168, 180, 192, 216, 224, 240, 252, 264, 270, 280, 294, 312, 320, 336, 352, 360, 378, 384, 396, 408, 416, 432, 448, 456, 468, 480, 486, 504, 528, 540, 552, 560, 594, 600, 612, 624, 630, 640, 672, 684, 696, 702, 704, 720, 726

Offset: 1

Amiram Eldar, May 25 2020

nonn

Apparently, most of the terms are nonunitary abundant (A064597). Term that are nonunitary deficient (A064598) are 54, 150, 270, 294, 378, ...

			24 is a term since its set of nonunitary divisors, {2, 4, 6, 12}, can be partitioned into the two disjoint sets, {2, 4, 6} and {12}, whose sum is equal: 2 + 4 + 6 = 12.

Cf. A064591, A064597, A064598, A083207.

nuzQ[n_] := Module[{d = Select[Divisors[n], GCD[#, n/#] > 1 &], sum, x}, sum = Plus @@ d; sum > 0 && EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] > 0]; Select[Range[1000], nuzQ]

A064593 Nonunitary triply perfect numbers: the sum of the nonunitary divisors of n is 3n; i.e., sigma(n) - usigma(n) = 3n.

Original entry on oeis.org

186076800, 3456573120, 18924364800, 325377803520, 2666567816640, 8538872307840, 11169839116800

Offset: 1

Dean Hickerson, Sep 25 2001

There are no others up to 1.6*10^13, but there are at least 20 larger ones, between 87916260096000 and 19424258892617980729341453189120.

Ligh & Wall found a(1), a(4) and a(5). - Amiram Eldar, Sep 27 2018

Steve Ligh and Charles R. Wall, Functions of Nonunitary Divisors, Fibonacci Quarterly, Vol. 25 (1987), pp. 333-338.

Cf. A048146, A064591, A064592, A064594, A064595, A064596.

A064596 Smallest nonunitary multiperfect number of index n; i.e., smallest k such that sigma(k) - usigma(k) = nk.

Original entry on oeis.org

1, 24, 2520, 186076800

Offset: 0

Dean Hickerson, Sep 25 2001

4.5*10^14 < a(4) <= 1081895452249824576000.

Cf. A048146, A064591, A064592, A064593, A064594, A064595.

nusigma[ n_ ] := DivisorSigma[ 1, n ]-Times@@(1+Power@@#&/@FactorInteger[ n ]); a[ n_ ] := For[ k=1, True, k++, If[ nusigma[ k ]==n k, Return[ k ] ] ]

cp's OEIS Frontend

A064597 Nonunitary abundant numbers: the sum of the nonunitary divisors of n is larger than n; i.e., sigma(n) - usigma(n) > n.

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A325814 a(n) = n - A048146(n), where A048146 is the sum of non-unitary divisors of n.

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A319745 Nonunitary harmonic numbers: numbers such that the harmonic mean of their nonunitary divisors is an integer.

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A064594 Nonunitary multiply perfect numbers: the sum of the nonunitary divisors of n is a multiple of n; i.e., n divides sigma(n) - usigma(n).

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A064595 Nontrivial nonunitary multiply perfect numbers: the sum of the nonunitary divisors of n is a positive multiple of n; i.e., (sigma(n) - usigma(n))/n is a positive integer.

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A322609 Numbers k such that s(k) = 2*k, where s(k) is the sum of divisors of k that have a square factor (A162296).

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A064592 Nonunitary doubly perfect numbers: the sum of the nonunitary divisors of n is 2n; i.e., sigma(n) - usigma(n) = 2n.

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A335142 Nonunitary Zumkeller numbers: numbers whose set of nonunitary divisors is nonempty and can be partitioned into two disjoint sets of equal sum.

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