A194472 -id:A194472 - cp's OEIS frontend

A064510 Numbers m such that the sum of the first k divisors of m is equal to m for some k.

1, 6, 24, 28, 496, 2016, 8128, 8190, 42336, 45864, 392448, 714240, 1571328, 33550336, 61900800, 91963648, 211891200, 1931236608, 2013143040, 4428914688, 8589869056, 10200236032, 137438691328, 214204956672

Offset: 1

Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Oct 06 2001

Obviously all perfect numbers are included in this sequence.
a(25) > 5*10^11. Other than perfect numbers, 104828758917120, 916858574438400, 967609154764800, 93076753068441600, 215131015678525440 and 1371332329173024768 are also terms. - Donovan Johnson, Dec 26 2012
a(25) > 10^12. - Giovanni Resta, Apr 15 2017

			Divisors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. 1+2+3+4+6+8 = 24.

Union of A000396 and A194472.

Cf. A109883, A109884, A109886, A185584, A185960, A190940, A307137, A318528, A378313, A378314.

subtract = If[ #1 < #2, Throw[ #1], #1 - #2]&; f[n_] := Catch @ Fold[subtract, n, Divisors @ n]; lst = {}; Do[ If[ f[n] == 0, AppendTo[lst, n]], {n, 10^8}]; lst (* Bobby R. Treat and Robert G. Wilson v, Jul 14 2005 *)
Select[Range[2000000],MemberQ[Accumulate[Divisors[#]],#]&] (* Harvey P. Dale, Mar 22 2012 *)

isok(n) = {my(d = divisors(n)); my(k = 1); while ((k <= #d) && ((sd = sum(j=1, k, d[j])) != n), k++;); (sd == n);} \\ Michel Marcus, Jan 16 2014

More terms from Don Reble, Dec 17 2001
a(19)-a(23) from Donovan Johnson, Aug 31 2008
a(24) from Donovan Johnson, Aug 11 2011

A240698 Partial sums of divisors of n, cf. A027750.

Original entry on oeis.org

1, 1, 3, 1, 4, 1, 3, 7, 1, 6, 1, 3, 6, 12, 1, 8, 1, 3, 7, 15, 1, 4, 13, 1, 3, 8, 18, 1, 12, 1, 3, 6, 10, 16, 28, 1, 14, 1, 3, 10, 24, 1, 4, 9, 24, 1, 3, 7, 15, 31, 1, 18, 1, 3, 6, 12, 21, 39, 1, 20, 1, 3, 7, 12, 22, 42, 1, 4, 11, 32, 1, 3, 14, 36, 1, 24, 1

Offset: 1

Reinhard Zumkeller, Apr 10 2014

Triangle read by rows in which row n lists the partial sums of divisors of n. - Omar E. Pol, Apr 12 2014

			.    n |  n-th row of A240698   |  n-th row of A027750
.  ----+------------------------+---------------------
.    1 |  1                     |  1
.    2 |  1, 3                  |  1, 2
.    3 |  1, 4                  |  1, 3
.    4 |  1, 3, 7               |  1, 2, 4
.    5 |  1, 6                  |  1, 5
.    6 |  1, 3, 6, 12           |  1, 2, 3, 6
.    7 |  1, 8                  |  1, 7
.    8 |  1, 3, 7, 15           |  1, 2, 4, 8
.    9 |  1, 4, 13              |  1, 3, 9
.   10 |  1, 3, 8, 18           |  1, 2, 5, 10
.   11 |  1, 12                 |  1, 11
.   12 |  1, 3, 6, 10, 16, 28   |  1, 2, 3, 4, 6, 12
.   13 |  1, 14                 |  1, 13 .

Reinhard Zumkeller, Rows n = 1..1000 of table, flattened

Cf. A000005 (row lengths), A240694.

Cf. A000203, A001065, A027750, A064510, A194472.

a240698 n k = a240698_tabf !! (n-1) !! (k-1)
a240698_row n = a240698_tabf !! (n-1)
a240698_tabf = map (scanl1 (+)) a027750_tabf

Table[Accumulate[Divisors[n]],{n,30}]//Flatten (* Harvey P. Dale, Dec 30 2019 *)

row(n) = my(d=divisors(n)); vector(#d, k, sum(i=1, k, d[i])); \\ Michel Marcus, Jan 24 2022

T(n,1) = 1, T(n,k) = T(n,k-1) + A027750(n,k), 1 < k <= n.
T(n,1) = 1;
T(n,A000005(n)) = A000203(n);
T(n,A000005(n)-1) = A001065(n), n > 1.

A293618 Numbers n that equal the sum of their first k consecutive aliquot bi-unitary divisors, but not all of them (i.e k < A286324(n)-1).

Original entry on oeis.org

24, 360, 432, 1344, 2016, 19440, 45360, 68544, 714240, 864000, 1468800, 1571328, 1900800, 2391120, 2888704, 3057600, 4586400, 5241600, 103194000

Offset: 1

Amiram Eldar, Oct 13 2017

The bi-unitary version of Erdős-Nicolas numbers (A194472).

If all the aliquot bi-unitary divisors are permitted (i.e. k <= A286324(n)-1), then the 3 bi-unitary perfect numbers, 6, 60 and 90, are included.

			24 is in the sequence since its aliquot bi-unitary divisors are 1, 2, 3, 4, 6, 8, 12 and 24 and 1 + 2 + 3 + 4 + 6 + 8 = 24.

Cf. A188999, A194472, A222266, A286324.

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bdiv[m_] := Select[Divisors[m], Last@Intersection[f@#, f[m/#]] == 1 &]; subtr = If[#1 < #2, Throw[#1], #1 - #2] &; selDivs[n_] := Catch@Fold[subtr, n, Drop[bdiv[n], -2]]; a = {}; Do[ If[selDivs[n] == 0, AppendTo[a, n]; Print[n]], {n, 2, 10^6}]; a (* after Alonso del Arte at A194472 *)

A318168 Reverse Erdős-Nicolas numbers: abundant numbers m such that the sum of the last k proper divisors of m is equal to m for some k.

Original entry on oeis.org

18, 42, 54, 66, 78, 102, 114, 126, 138, 162, 174, 186, 196, 198, 222, 234, 246, 258, 282, 294, 306, 318, 342, 354, 366, 378, 402, 414, 426, 438, 462, 474, 486, 498, 522, 534, 546, 558, 582, 594, 606, 618, 642, 654, 666, 678, 702, 714, 726, 738, 762, 774, 786

Offset: 1

Amiram Eldar, Aug 20 2018

nonn

Apparently most of the terms are sum of their 3 largest proper divisors and are included in A074837. Terms that are not there are 196, 812, 868, 1036, 1148, 1204, 1316, 1372, 1484, 1652, 1708, 1876, 1998, 2044, ...
The possible values of k seem to be rather sparse. Up to 2*10^10, such values are: 3 (minimal m = 18), 5 (196), 9 (15376), 13 (1032256), 15 (34155), 16 (20482), 17 (33345), 19 (8415), 21 (407715), 23 (1273725), 26 (89245784), 32 (479198624), 36 (125226568), 40 (12499150), 45 (5905148248), 46 (1375270384), 68 (13968326788), and 91 (159030135). - Giovanni Resta, Aug 21 2018
If 2^p - 1 is prime then ((2^p - 1)^n)*2^(p-1) is in the sequence for n > 1. - Davide Rotondo, Oct 02 2021
From Mauro Fiorentini, Jan 08 2024: (Start)
More generally, if n is an even perfect number, with odd prime factor p, all prime factors of m are greater than n, k >= 0 and p^k*m > 1, n*p^k*m is in the sequence.
Also, if n is in the sequence and all prime factors of m are greater than n, n*m is in the sequence (note that n is not necessarily a multiple of an even perfect number).
It follows that there are infinitely many odd terms in the sequence, that the asymptotic density of the sequence is greater than 0.073482 and that the difference between consecutive terms is at most 24. (End)
The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 0, 5, 68, 737, 7352, 73704, 737142, 7370307, 73699222, 737011233, 7370145824, ... . Apparently, the asymptotic density of this sequence exists and equals 0.073701... . - Amiram Eldar, Apr 18 2024

			196 is in the sequence since its proper divisors are 1, 2, 4, 7, 14, 28, 49, 98, and 7 + 14 + 28 + 49 + 98 = 196.

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

Cf. A000043, A000668, A074837, A194472.

subtr = If[#1 < #2, Throw[#1], #1 - #2] &; selDivs[n_] := Catch@Fold[subtr, n, Reverse[Rest[Most[Divisors[n]]]]]; s={}; Do[If[selDivs[n] == 0, AppendTo[s, n]], {n, 2, 1000}]; s

isok(n) = {my(d = Vecrev(divisors(n))); if (vecsum(d) > 2*n, my(s = 0); for (i=2, #d, s += d[i]; if (s == n, return(n)););); return (0);} \\ Michel Marcus, Aug 21 2018

A306373 Integers m such that the sum of the first k divisors is equal to 2*m for some k less than the number of divisors of m.

Original entry on oeis.org

120, 672, 4320, 4680, 26208, 523776, 20427264, 29795040, 34369920, 96445440, 197064960, 459818240, 557107200

Offset: 1

Michel Marcus, Feb 11 2019

3-perfect numbers (A005820) are terms.
All known terms of A055153 (abundancy 7/2) are terms.
1907020800 (with abundancy 23/6) is a term too.
A055153 is a subsequence, because no term of that sequence may be odd and so for each k in A055153 we have 2*k = sigma(k) - k - k/2. - Charlie Neder, Feb 12 2019

Cf. A005820 (3-perfect numbers), A055153 (abundancy 7/2).

Cf. A064510, A194472 (both with equal to m rather than to 2*m).

isok(n) = {if (sigma(n) < 2*n, return (0)); my(d = divisors(n), s = 0); for (k=1, #d-1, s += d[k]; if (s == 2*n, return (1)); if (s > 2*n, break);); return (0);}

is(n) = my(d = divisors(n), s = vecsum(d) - d[#d]); forstep(i = #d-1, 1, -1, if(s <= 2*n, return(s == 2*n)); s-=d[i]); 0 \\ David A. Corneth, Feb 11 2019

a(11)-a(13) from Jinyuan Wang, Feb 11 2019

A309843 Numbers m that equal the sum of their first k consecutive aliquot infinitary divisors, but not all of them (i.e k < A037445(m) - 1).

Original entry on oeis.org

24, 360, 4320, 14688, 1468800, 9547200, 50585472, 54198720, 189695520, 1680459264

Offset: 1

Amiram Eldar, Sep 14 2019

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

If all the aliquot infinitary divisors are permitted (i.e. k <= A037445(n) - 1), then the infinitary perfect numbers (A007357) are included.

			24 is in the sequence since its aliquot infinitary divisors are 1, 2, 3, 4, 6, 8, 12 and 24 and 1 + 2 + 3 + 4 + 6 + 8 = 24.

Cf. A007357, A077609, A037445, A049417, A194472, A293618.

idivs[x_] := If[x == 1, 1, Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[ x ] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]]; subtr = If[#1 < #2, Throw[#1], #1 - #2] &; selDivs[n_] := Catch@Fold[subtr, n, Drop[idivs[n], -2]]; s= {}; Do[If[selDivs[n] == 0, AppendTo[s, n]], {n, 2, 10^6}]; s(* after Alonso del Arte at A194472 *)

A334409 Numbers m such that the sum of the first k divisors and the last k divisors of m is equal to 2*m for some k that is smaller than half of the number of divisors of m.

Original entry on oeis.org

36, 152812, 6112576, 72702928, 154286848, 397955025, 15356519488, 23003680492, 35755623784, 93789539668, 302122464256, 351155553970, 1081806148665, 1090488143872, 1663167899025, 2233955122576

Offset: 1

Amiram Eldar, Apr 27 2020

If k is allowed to be equal to half of the number of divisors of m, then the perfect numbers (A000396) will be terms.

a(17) > 10^13. 3021194449732665786499072 is also a term. - Giovanni Resta, May 09 2020

			36 is a term since its divisors are {1, 2, 3, 4, 6, 9, 12, 18, 36} and the sum of the first 3 and last 3 divisors is (1 + 2 + 3) + (12 + 18 + 36) = 72 = 2 * 36.

Subsequence of A005835 and A334405.
A variant of A194472 and A318168.
Cf. A000396, A334410.

seqQ[n_] := Module[{d = Divisors[n]}, nd = Length[d]; nd2 = Ceiling[nd/2] - 1; s = Accumulate[d[[1 ;; nd2]] + n/d[[1 ;; nd2]]]; MemberQ[s, 2*n]]; Select[Range[10^6], seqQ]

from itertools import count, islice, accumulate
from sympy import divisors
def A334409_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        ds = divisors(n)
        if any(s==2*n for s in accumulate(ds[i]+ds[-1-i] for i in range((len(ds)-1)//2))):
            yield n
A334409_list = list(islice(A334409_gen(),2)) # Chai Wah Wu, Feb 19 2022

a(8)-a(16) from Giovanni Resta, May 06 2020

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 *)

cp's OEIS Frontend

A064510 Numbers m such that the sum of the first k divisors of m is equal to m for some k.

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A240698 Partial sums of divisors of n, cf. A027750.

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A293618 Numbers n that equal the sum of their first k consecutive aliquot bi-unitary divisors, but not all of them (i.e k < A286324(n)-1).

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A318168 Reverse Erdős-Nicolas numbers: abundant numbers m such that the sum of the last k proper divisors of m is equal to m for some k.

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A306373 Integers m such that the sum of the first k divisors is equal to 2*m for some k less than the number of divisors of m.

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A309843 Numbers m that equal the sum of their first k consecutive aliquot infinitary divisors, but not all of them (i.e k < A037445(m) - 1).

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A334409 Numbers m such that the sum of the first k divisors and the last k divisors of m is equal to 2*m for some k that is smaller than half of the number of divisors of m.

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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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