A064597 -id:A064597 - cp's OEIS frontend

A327942 Numbers k such that both k and k+1 are nonunitary abundant numbers (A064597).

165375, 893024, 1047375, 1576575, 2282175, 2304224, 2858624, 3614624, 4068224, 4096575, 4597424, 4975424, 6591375, 7574175, 8555624, 9511424, 10446975, 10749375, 10872224, 11477024, 12535424, 13773375, 13946624, 14277375, 15926624, 16041375, 16505775, 16769024

Offset: 1

Amiram Eldar, Sep 30 2019

nonn

			165375 is in the sequence since both 165375 and 165376 are nonunitary abundant: nusigma(165375) = 179280 > 165375, and nusigma(165376) = 183600 > 165376 (nusigma is the sum of nonunitary divisors, A048146).

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

Cf. A048146, A064597, A094889, A307823.

Cf. A096399, A292704, A318167, A327635.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); nuabQ[n_] := Times @@ (f @@@ FactorInteger[n]) - Times @@ (1 + Power @@@ FactorInteger[n]) > n; s = {}; q1 = False; Do[q2 = nuabQ[n]; If[q1 && q2, AppendTo[s, n - 1]]; q1 = q2, {n, 2, 10^7}]; s

A064591 Nonunitary perfect numbers: k is the sum of its nonunitary divisors; i.e., k = sigma(k) - usigma(k).

Original entry on oeis.org

24, 112, 1984, 32512, 134201344, 34359476224, 549754765312

Offset: 1

Dean Hickerson, Sep 25 2001

There are no other terms up to 1.2*10^14.
If P (A000396) is an even perfect number, then 4*P is in the sequence. Are there any others?
If there are no terms of another form, the sequence goes on with 9223372032559808512 = 2^32 * A000668(8), 10633823966279326978618770463815368704 = 2^62 * A000668(9), 766247770432944429179173512337214552523989286192676864 = 2^90 * A000668(10), ... - Michel Lagneau, Jan 21 2015
Conjecture: let s0 be the sum of the inverses of the even divisors of a number n and s1 the sum of the inverses of the odd divisors of n; then n is in the sequence iff 2*s0-s1 = 1. - Michel Lagneau, Jan 21 2015
Ligh & Wall proved that 2^(p+1)*(2^p-1) is a term if p and 2^p-1 are primes, and that all the nonunitary perfect numbers below 10^6 are of this form. - Amiram Eldar, Sep 27 2018
If k is in the sequence and k = 2^m*p^a then k is of the form 4*P for an even perfect P. See the link to MathOverflow. - Joshua Zelinsky, Mar 07 2024

			The sum of the nonunitary divisors of 24 is 2 + 4 + 6 + 12 = 24.

Steve Ligh and Charles R. Wall, Functions of Nonunitary Divisors, Fibonacci Quarterly, Vol. 25 (1987), pp. 333-338.
MathOverflow, Are all nonunitary perfect numbers in the form 4k where k is an even perfect number?

Cf. A000203 (sigma), A034448 (usigma), A048146, A063870.

Cf. A064592, A064593, A064594, A064595, A064596, A064597, A064598, A064599.

nusigma[ n_ ] := DivisorSigma[ 1, n ]-Times@@(1+Power@@#&/@FactorInteger[ n ]); For[ n=1, True, n++, If[ nusigma[ n ]==n, Print[ n ] ] ]
Do[s0=0;s1=0;Do[d=Divisors[n][[i]];If[Mod[d,2]==0,s0=s0+1/d,s1=s1+1/d],{i,1,Length[Divisors[n]]}];If[2*s0-s1==1,Print[n]],{n,2,10^9,2}] (* Michel Lagneau, Jan 21 2015 *)

A348274 Noninfinitary abundant numbers: numbers k such that A348271(k) > k.

Original entry on oeis.org

36, 48, 80, 144, 180, 240, 252, 288, 300, 324, 336, 396, 400, 432, 468, 528, 560, 576, 588, 612, 624, 684, 720, 768, 784, 816, 828, 880, 900, 912, 960, 1008, 1040, 1044, 1104, 1116, 1200, 1232, 1260, 1280, 1296, 1332, 1360, 1392, 1440, 1456, 1476, 1488, 1520, 1548

Offset: 1

Amiram Eldar, Oct 09 2021

nonn

The first odd term is a(3577) = 99225.

The number of terms not exceeding 10^k, for k = 1, 2, ..., are 0, 3, 31, 360, 3605, 36160, 360840, 3618980, 36144059, ... Apparently this sequence has an asymptotic density 0.0361...

			36 is a term since A348271(36) = 41 > 36.

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

Cf. A348271, A327633.
Subsequence of A005101.
Similar sequences: A034683, A064597, A129575, A129656, A292982.

f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ f @@@ FactorInteger[n]; s[n_] := DivisorSigma[1,n] - isigma[n]; Select[Range[1500], s[#] > # &]

A348604 Nonexponential abundant numbers: numbers k such that A160135(k) > k.

Original entry on oeis.org

24, 30, 42, 48, 54, 60, 66, 70, 72, 78, 84, 90, 96, 102, 114, 120, 126, 132, 138, 150, 156, 160, 162, 168, 174, 180, 186, 192, 198, 210, 216, 222, 224, 240, 246, 258, 264, 270, 280, 282, 288, 294, 300, 312, 318, 320, 330, 336, 352, 354, 360, 366, 378, 384, 390

Offset: 1

Amiram Eldar, Oct 25 2021

nonn

The smallest odd term is a(1357) = 8505.

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 0, 13, 148, 1595, 15688, 158068, 1578957, 15762209, 157745113, 1577808429, ... Apparently this sequence has an asymptotic density 0.157...

			24 is a term since A160135(24) = 30 > 24.

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

Cf. A160135, A348601.
Subsequence of A005101.
Similar sequences: A034683, A064597, A129575, A129656, A292982, A348274.

esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; q[n_] := DivisorSigma[1, n] - esigma[n] > n; Select[Range[400], q]

A094889 Odd nonunitary abundant numbers.

Original entry on oeis.org

33075, 99225, 165375, 231525, 259875, 297675, 363825, 429975, 467775, 496125, 552825, 562275, 571725, 606375, 628425, 675675, 694575, 716625, 760725, 779625, 798525, 826875, 848925, 883575, 893025, 921375, 937125, 959175, 1003275

Offset: 1

Lekraj Beedassy, Jun 16 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
J. A. Sellers, A Note On Infinitely Many Odd Nonunitary Abundant Numbers

Cf. A064597.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); nuabQ[n_] := Times @@ (f @@@ FactorInteger[n]) - Times @@ (1 + Power @@@ FactorInteger[n]) > n; Select[Range[3, 10^6, 2], nuabQ] (* Amiram Eldar, May 12 2019 *)

{ usigma(n)=local(s=1,fac,i); fac=factor(n); for(i=1,matsize(fac)[1], s=s*(1+fac[i,1]^fac[i,2])); return(s); } nusigma(n)=sigma(n)-usigma(n); forstep(n=1,2^24,2,if(nusigma(n)>n,print1(n",")))

Corrected and extended by Jason Earls, Jun 18 2004

A380929 Numbers k such that A380845(k) > 2*k.

Original entry on oeis.org

36, 72, 84, 140, 144, 168, 180, 264, 270, 280, 288, 300, 336, 360, 372, 392, 450, 520, 528, 532, 540, 558, 560, 576, 594, 600, 612, 620, 672, 720, 744, 756, 780, 784, 840, 900, 930, 1036, 1040, 1050, 1056, 1064, 1068, 1080, 1092, 1116, 1120, 1134, 1152, 1170, 1180, 1188, 1200

Offset: 1

Amiram Eldar, Feb 08 2025

Analogous to abundant numbers (A005101) with A380845 instead of A000203.

			36 is a term since A380845(36) = 84 > 2 * 36 = 72.

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

Cf. A000203, A380845, A380846.
Subsequence of A005101.
Subsequences: A380847, A380848, A380930, A380931.
Similar sequences: A034683, A064597, A087248, A129575, A129656, A292982, A348274, A348604, A379029.

q[k_] := Module[{h = DigitCount[k, 2, 1]}, DivisorSum[k, # &, DigitCount[#, 2, 1] == h &] > 2*k]; Select[Range[1200], q]

isok(k) = {my(h = hammingweight(k)); sumdiv(k, d, d*(hammingweight(d) == h)) > 2*k;}

A327948 Nonunitary weird numbers: numbers that are nonunitary abundant but not nonunitary pseudoperfect.

Original entry on oeis.org

280, 3344, 16120, 23320, 28768, 31648, 37088, 41720, 42280, 43168, 43960, 45640, 46760, 48440, 50120, 50680, 53480, 54040, 55160, 55720, 59080, 62440, 63560, 64120, 65240, 66920, 67480, 69088, 70280, 71960, 73640, 75320, 75880, 77560, 78680, 79240, 82040

Offset: 1

Amiram Eldar, Sep 30 2019

nonn

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

The nonunitary version of A006037.
Cf. A064597, A327945.
Cf. A064114, A292986, A306984, A321146.

nudiv[n_] := Module[{d = Divisors[n]}, Select[d, GCD[#, n/#] > 1 &]]; s = {}; Do[d = nudiv[n]; If[Total[d] <= n, Continue[]]; c = SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n]; If[c == 0, AppendTo[s, n]], {n, 1, 10^5}]; s

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]

A063846 Numbers k such that sigma(k) - usigma(k) > 2k.

Original entry on oeis.org

1440, 1800, 2160, 2880, 3024, 3600, 4320, 5040, 5400, 5760, 6048, 6480, 7056, 7200, 7560, 7920, 8064, 8640, 9000, 9072, 9360, 9504, 9720, 10080, 10584, 10800, 11088, 11520, 11880, 12096, 12240, 12600, 12960, 13680, 14040, 14112, 14400, 15120

Offset: 1

Jason Earls, Aug 25 2001

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)

Cf. A034448, A048146, A034683, A064597.

u(n) = sumdiv(n,d, if(gcd(d,n/d)==1,d));
j=[]; for(n=1,20000, if(sigma(n)-u(n)>2*n,j=concat(j,n))); j

u(n) = sumdiv(n, d, if(gcd(d, n/d)==1, d))
{ n=0; for (m=1, 10^9, if(sigma(m) - u(m) > 2*m, write("b063846.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 01 2009

A064598 Nonunitary deficient numbers: the sum of the nonunitary divisors of n is less than n; i.e., sigma(n) - usigma(n) < n.

Original entry on oeis.org

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

Offset: 1

Dean Hickerson, Sep 25 2001

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

Cf. A048146, A064591, A064597.

nusigma[ n_ ] := DivisorSigma[ 1, n ]-Times@@(1+Power@@#&/@FactorInteger[ n ]); For[ n=1, True, n++, If[ nusigma[ 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("b064598.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 19 2009

cp's OEIS Frontend

A327942 Numbers k such that both k and k+1 are nonunitary abundant numbers (A064597).

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A064591 Nonunitary perfect numbers: k is the sum of its nonunitary divisors; i.e., k = sigma(k) - usigma(k).

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A348274 Noninfinitary abundant numbers: numbers k such that A348271(k) > k.

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A348604 Nonexponential abundant numbers: numbers k such that A160135(k) > k.

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A094889 Odd nonunitary abundant numbers.

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A380929 Numbers k such that A380845(k) > 2*k.

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A327948 Nonunitary weird numbers: numbers that are nonunitary abundant but not nonunitary pseudoperfect.

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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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A063846 Numbers k such that sigma(k) - usigma(k) > 2k.

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