A064591 -id:A064591 - cp's OEIS frontend

A327946 Nonunitary pseudoperfect numbers (A327945) that equal to the sum of a subset of their nonunitary divisors in a single way.

24, 36, 80, 112, 200, 312, 352, 392, 408, 416, 456, 552, 588, 684, 696, 744, 888, 984, 1032, 1088, 1116, 1128, 1216, 1272, 1332, 1416, 1464, 1472, 1548, 1608, 1692, 1704, 1752, 1856, 1896, 1908, 1936, 1984, 1992, 2124, 2136, 2196, 2288, 2328, 2412, 2424, 2472

Offset: 1

Amiram Eldar, Sep 30 2019

nonn

The nonunitary version of A064771.

			The nonunitary divisors of 36 are {2, 3, 6, 12, 18}, and {6, 12, 18} is the only subset that sums to 36.

Cf. A064771, A064591, A064597, A295829, A327945.

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 == 1, AppendTo[s, n]], {n, 1, 700}]; s

A247825 Numbers which are the difference between the sum of their bi-unitary divisors and the sum of their unitary divisors.

Original entry on oeis.org

24, 240, 360

Offset: 1

Paolo P. Lava, Sep 29 2014

No further terms up to 10^8. Is there a relation with 6, 60 and 90, the 3 only bi-unitary perfects? - Michel Marcus, Oct 05 2014

a(4), if it exists, is larger than 10^11. - Giovanni Resta, Apr 15 2017

			Divisors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. Unitary divisors are 1, 3, 8, 24 and their sum is 36. Bi-unitary divisors are 1, 2, 3, 4, 6, 8, 12, 24 and their sum is 60. Then 60 - 36 = 24.
Divisors of 240 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240. Unitary divisors are 1, 3, 5, 15, 16, 48, 80, 240 and their sum is 408. Bi-unitary divisors are 1, 2, 3, 5, 6, 8, 10, 15, 16, 24, 30, 40, 48, 80, 120, 240 and their sum is 648. Then 648 - 408 = 240.

C. R. Wall, Bi-unitary perfect numbers, Proc. Am. Math. Soc. 33 (1) (1972) 39-42.
Wikipedia, Unitary divisor

Cf. A034448, A064591, A188999.

Q:=proc(n) local a, e, p, f; a:=1 ;for f in ifactors(n)[2] do e:=op(2,f); p:=op(1,f);
if type(e,odd) then a:=a*(p^(e+1)-1)/(p-1); else a:=a*((p^(e+1)-1)/(p-1)-p^(e/2)); fi; od: a ; end:
P:=proc(h) local a,b,k,n;
for n from 1 to h do a:=divisors(n); b:=0;
for k from 1 to nops(a) do if gcd(a[k],n/a[k])=1 then b:=b+a[k]; fi; od;
if Q(n)-b=n then print(n); fi; od; end: P(10^6);

up(p, e) = p^e+1;
bup(p, e) = my(ret = (p^(e+1) - 1)/(p-1)); if ((e % 2) == 0, ret -= p^(e/2)); ret;
isok(n) = f = factor(n); n == (prod(k=1, #f~, bup(f[k,1], f[k,2])) - prod(k=1, #f~, up(f[k,1], f[k,2]))); \\ Michel Marcus, Oct 05 2014

A257996 Let s0 and s1 be the sums of the reciprocals of the even and odd divisors of n, respectively. The sequence lists the numbers n such that 3s0 - 2s1 = 1.

Original entry on oeis.org

120, 1456, 121024, 2198352216064, 576458003527499776

Offset: 1

Michel Lagneau, May 16 2015

nonn

Let D0 = {d0(i)}, i = 1..p, the set of the p even divisors of a number n and D1 = {d1(n)}, j = 1..q the set of the q odd divisors of n. Then a(n) is the number such that 3*Sum_{i=1..p} 1/d0(i)- 2*Sum_{j=1..q} 1/d1(j) = 1.
Property of the sequence:
We observe that a(n) = 2^(k+1)*(2^k-1)*(2^(k+1) - 3) = (2*A000668(m) + 2)*A000668(m)*(2*A000668(m) - 1) where A000668(m) = 2^k - 1 is a Mersenne prime and (2*A000668(m)-1) = 2^(k+1)- 3 is also a prime number.
The corresponding values of k are 2, 3, 5, 13, 19, ... and the corresponding values of m are 1, 2, 3, 5, 7, ...
Generalization:
It is possible to introduce general sequences of numbers such that a*s0 + b*s1 = c with very interesting properties for some integers a, b, c.
Example 1: with (a, b, c) = (2, -1, 1) we find the sequence A064591 = 24, 112, 1984, 32512, ... (non-unitary perfect numbers).
Example 2: with (a, b, c) = (2, -1, 0) we find the sequence A016825(n) = 2, 6, 10, 14, 18, 22, ...
Example 3: with (a, b, c) = (1, 1, 2) we find the sequence A000396(n) = 6, 28, 496, 8128,... (perfect numbers).
Example 4: with (a, b, c) = (4, -3, 1) we find the sequence 48, 224, 3968, 65024, ... = 2*A064591(n) = A000668(n)*2^p for some p where A000668 lists the Mersenne primes.
Example 5: with (a, b, c) = (6, -5, 1) we find the sequence 240, 2912, 242048, ... which equals twice the sequence obtained with (a, b, c) = (3, -2, 1).
Example 6: with (a, b, c) = (7, -6, 1) we find the sequence 2150, 13104, 24800, ...

			120 = 2^3*3*5 = (2*A000668(1)+2)* A000668(1)*(2*A000668(1)-1);
1456 = 2^4*7*13 = (2*A000668(2)+2)* A000668(2)*(2*A000668(2)-1);
121024 = 2^6*31*61 =(2*A000668(3)+2)* A000668(3)*(2*A000668(3)-1);
2198352216064 = 2^14*8191*16381= (2*A000668(5)+2)*A000668(5)*(2*A000668(5)-1);
576458003527499776 = 2^20*524287*1048573 = (2*A000668(7)+2)* A000668(7)*(2*A000668(7)-1).

Cf. A000668.

with(numtheory):nn:=100000:
for n from 2 by 2 to nn do :
   x:=divisors(n):n0:=nops(x):s:=sum('x[i]', 'i'=1..n0):
    s0:=0:s1:=0:
    for k from 1 to n0 do:
     if irem(x[k],2)=0
     then
     s0:=s0+1/x[k]
     else
     s1:=s1+1/x[k]:
     fi:
    od:
    if 3*s0-2*s1=1 then print(n):else fi:od:

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[3*s0-2*s1==1,Print[n]],{n,2,10^9,2}]

siod(n) = sumdiv(n, d, (d%2)/d);
seod(n) = sumdiv(n, d, (1-d%2)/d);
isok(n) = 3*seod(n)-2*siod(n) == 1; \\ Michel Marcus, May 16 2015

A273813 Composite numbers whose sum of unitary divisors is a multiple of the sum of their aliquot parts.

Original entry on oeis.org

6, 24, 112, 1984, 32512, 171197, 667879, 780625, 56513539, 134201344, 488265625, 5203009849, 9130639447, 34359476224, 47390685029, 96595448129

Offset: 1

Paolo P. Lava, May 31 2016

A064591 is a subsequence of this sequence.
The ratios are 2, 1, 1, 1, 1, 12, 16, 4, 40, 1, 4, 100, 112, 1, 156, 180, ...
Up to 3*10^11 all the terms are of the form p^e*q. In particular, if 2^k-1 is prime, then 2^(k+1)(2^k-1) is a term. Similarly, if 2*5^k-1 is prime, then 5^k*(2*5^k-1) is a term. By solving appropriate Diophantine equations it is also possible to obtain large terms of the form p^2*q, like 1300253^2*1099140634496715133. - Giovanni Resta, Jun 01 2016

			Unitary divisors of 6 are 1, 2, 3, 6 and their sum is 12. Aliquot parts are 1, 2, 3 and their sum is 6.  Then, 12 / 6 = 2.
Unitary divisors of 24 are 1, 3, 8, 24 and their sum is 36. Aliquot parts are 1, 2, 3, 4, 6, 8, 12 and their sum is 36.  Then, 36 / 36 = 1.
Unitary divisors of 171197 are 1, 169, 1013, 171197 and their sum is 172380. Aliquot parts are 1, 13, 169, 1013, 13169 and their sum is 14365.  Then, 172380 / 14365 = 12.

Cf. A001065, A034448, A064591.

with(numtheory): P:=proc(q) local a,b,k,n;
for n from 2 to q do if not isprime(n) then a:=ifactors(n)[2]; b:=mul(a[k][1]^a[k][2]+1,k=1..nops(a));
if type(b/(sigma(n)-n),integer) then print(n); fi; fi; od; end: P(10^9);

Select[Range[10^6], Function[n, CompositeQ@ n && Mod[Total@ Select[Divisors@ n, GCD[#, n/#] == 1 &], DivisorSigma[1, n] - n] == 0]] (* Michael De Vlieger, Jun 01 2016 *)

a(9)-a(16) from Giovanni Resta, Jun 01 2016

A327944 Numbers m that are equal to the sum of their first k consecutive nonunitary divisors, but not all of them (i.e k < A048105(m)).

Original entry on oeis.org

480, 2688, 17640, 131712, 2095104, 3576000, 4248288, 16854816, 41055200, 400162032, 637787520, 788259840, 1839272960, 2423592576

Offset: 1

Amiram Eldar, Sep 30 2019

The nonunitary version of Erdős-Nicolas numbers (A194472).

If all the nonunitary divisors are permitted (i.e. k <= A048105(n)), then the nonunitary perfect numbers (A064591) are included.

			480 is in the sequence since its nonunitary divisors are 2, 4, 6, 8, 10, 12, 16, 20, 24, 30, 40, 48, 60, 80, 120 and 240 and 2 + 4 + 6 + 8 + 10 + 12 + 16 + 20 + 24 + 30 + 40 + 48 + 60 + 80 + 120 = 480.~

Cf. A048105, A064591, A064597, A064599.

Cf. A194472, A293618, A309843.

ndivs[n_] := Block[{d = Divisors[n]}, Select[d, GCD[ #, n/# ] > 1 &]]; ndivs2[n_] := Module[{d=ndivs[n]},If[Length[d]<2,{},Drop[d, -1] ]]; subtr = If[#1 < #2, Throw[#1], #1 - #2] &; selDivs[n_] := Catch@Fold[subtr, n,ndivs2[n]]; a = {}; Do[ If[selDivs[n] == 0, AppendTo[a, n]; Print[n]], {n, 2, 10^6}]; a (* after Alonso del Arte at A194472 *)

A327947 Nonunitary pseudoperfect numbers (A327945) that equal to the sum of a subset of their nonunitary divisors in more ways than any smaller nonunitary pseudoperfect number.

Original entry on oeis.org

24, 48, 72, 96, 144, 216, 240, 288, 360, 480, 720, 1080, 1440, 2160, 2880, 3600, 4320, 5040, 7200, 7560, 10080, 15120, 20160, 25200, 30240

Offset: 1

Amiram Eldar, Sep 30 2019

The nonunitary version of A065218.

The corresponding numbers of ways are 1, 2, 4, 5, 15, 28, 34, 63, 211, 279, 6025, 17436, 187794, 2035726, 5965563, 36449982, 250420995, 3426156924, 8991176276, 37016127059, 6770551810345, 1095548357870254, 13524344273940115, 604532928571438678, 33370817837127087825, ...

			24 is the least number which is the sum of its nonunitary divisor, thus a(1) = 24.
48 is the least number which is the sum of a subset of its nonunitary divisor in two ways: 24 + 12 + 8 + 4 and 24 + 12 + 8 + 4 + 2, thus a(2) = 48.

Cf. A064591, A064597, A065218, A327945, A327946.

nudiv[n_] := Module[{d = Divisors[n]}, Select[d, GCD[#, n/#] > 1 &]]; s = {}; cm = 0; 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 > cm, cm = c; AppendTo[s, n]], {n, 1, 1000}]; s

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A327946 Nonunitary pseudoperfect numbers (A327945) that equal to the sum of a subset of their nonunitary divisors in a single way.

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A247825 Numbers which are the difference between the sum of their bi-unitary divisors and the sum of their unitary divisors.

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A257996 Let s0 and s1 be the sums of the reciprocals of the even and odd divisors of n, respectively. The sequence lists the numbers n such that 3*s0 - 2*s1 = 1.

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A273813 Composite numbers whose sum of unitary divisors is a multiple of the sum of their aliquot parts.

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A327944 Numbers m that are equal to the sum of their first k consecutive nonunitary divisors, but not all of them (i.e k < A048105(m)).

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A327947 Nonunitary pseudoperfect numbers (A327945) that equal to the sum of a subset of their nonunitary divisors in more ways than any smaller nonunitary pseudoperfect number.

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A257996 Let s0 and s1 be the sums of the reciprocals of the even and odd divisors of n, respectively. The sequence lists the numbers n such that 3s0 - 2s1 = 1.