A064510 -id:A064510 - cp's OEIS frontend

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

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.

A109883 Start subtracting from n its divisors beginning from 1 until one reaches a number smaller than the last divisor subtracted or reaches the last nontrivial divisor < n. Define this to be the perfect deficiency of n. Then a(n) = perfect deficiency of n.

Original entry on oeis.org

0, 1, 2, 1, 4, 0, 6, 1, 5, 2, 10, 2, 12, 4, 6, 1, 16, 6, 18, 8, 10, 8, 22, 0, 19, 10, 14, 0, 28, 3, 30, 1, 18, 14, 22, 11, 36, 16, 22, 10, 40, 9, 42, 4, 12, 20, 46, 12, 41, 7, 30, 6, 52, 15, 38, 20, 34, 26, 58, 2, 60, 28, 22, 1, 46, 21, 66, 10, 42, 31, 70, 9, 72, 34, 26, 12, 58, 27, 78

Offset: 1

Amarnath Murthy, Jul 11 2005

If n is a perfect number then a(n) = 0. But if a(n) = 0, n needs not be perfect, e.g., a(24) = 0, but 24 is not a perfect number. See A064510.

			a(14) = 4: 14-1 = 13, 13-2 = 11, 11-7 = 4.
a(6) = 0: 6-1 = 5, 5-2 = 3, 3-3 = 0. 6 is a perfect number.
a(35) = 22: 35-1 = 34, 34-5 = 29, 29-7 = 22.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A064510, A109884, A109886, A117552.

A109883:=proc(n)local d,j,k,m:if(n=1)then return 0:fi:j:=1:m:=n:d:=divisors(n);k:=nops(d):for j from 1 to k do m:=m-d[j]:if(mNathaniel Johnston, Apr 15 2011

subtract = If[ #1 < #2, Throw[ #1], #1 - #2]&;
a[n_] := Catch @ Fold[subtract, n, Divisors @ n]
Table[ a[n], {n, 80}] (* Bobby R. Treat (DrBob(AT)bigfoot.com), Jul 14 2005 *)

a(n) = {my(r = n); fordiv(n, d, if (r < d, return (r)); r -= d;); 0;} \\ Michel Marcus, Dec 28 2018

from sympy import divisors
def A109883(n):
    if n == 1: return 0
    s = n
    for d in divisors(n)[:-1]:
        if s < d: break
        s -= d
    return s
print([A109883(n) for n in range(1, 80)]) # Michael S. Branicky, Mar 31 2024

a(1) = 0, a(2^n) = 1.
a(p) = p-1, a(p^n) = (p^(n+1) - 2*p^n + 1)/(p-1), if p is a prime.
a(n) = n - A117552(n). - Ridouane Oudra, Jan 25 2024

More terms from Jason Earls and Robert G. Wilson v, Jul 12 2005

A194472 Erdős-Nicolas numbers.

Original entry on oeis.org

24, 2016, 8190, 42336, 45864, 392448, 714240, 1571328, 61900800, 91963648, 211891200, 1931236608, 2013143040, 4428914688, 10200236032, 214204956672

Offset: 1

Alonso del Arte, Aug 24 2011

Abundant numbers m such that the sum of the first k divisors is equal to m for some k, thus this is a subsequence of A064510. k has to be less than tau(m) - 1 for this sequence, whereas in A064510 k = tau(m) - 1 is allowed (and thus perfect numbers are in that sequence).
a(17) > 5*10^11. 104828758917120, 916858574438400, 967609154764800, 93076753068441600, 215131015678525440 and 1371332329173024768 are also terms. - Donovan Johnson, Dec 26 2012
a(17) > 10^12. - Giovanni Resta, Apr 15 2017
Equivalently, numbers whose abundancy equals 1 + the sum of the reciprocals of its first k divisors for some k > 1. - Charlie Neder, Feb 08 2019
96892692739248881664, 41407449045801454927872, 101616496263816777695232, 1346571992706422996646631651147776, 3304572752464376776401640967110656 are also terms. - Michel Marcus, Feb 09 2019
All known terms of A141643 (abundancy 5/2) are terms. - Michel Marcus, Feb 11 2019
Named after the Hungarian mathematician Paul Erdős (1913-1996) and the French mathematician Jean-Louis Nicolas. - Amiram Eldar, Jun 23 2021
Are all terms in this sequence even? - Jenaro Tomaszewski, May 07 2023

			The divisors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24 and 1 + 2 + 3 + 4 + 6 + 8 = 24, hence 24 is in the list.
The divisors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. The first seven of these add up to 36, but the first eight add up to 52, therefore 48 is not on the list.

Jean-Marie De Koninck, Those Fascinating Numbers, Amer. Math. Soc., 2009, p. 141.

P. Erdős and J.-L. Nicolas, Répartition des nombres superabondants, Bull. Soc. Math. France, Vol. 103, No. 1 (1975), pp. 65-90.
Wikipedia, Erdős-Nicolas number.

Cf. A005835, A000396, A064510, A141643.

subtr = If[#1 < #2, Throw[#1], #1 - #2] &; selDivs[n_] := Catch@Fold[subtr, n, Drop[Divisors[n], -2]]; erdNickNums = {}; Do[If[selDivs[n] == 0, AppendTo[erdNickNums, n]], {n, 2, 10^5}]; erdNickNums (* Based on the program by Bobby R. Treat and Robert G. Wilson v for A064510 *)

isok(n) = {if (sigma(n) <= 2*n, return (0)); my(d = divisors(n), s = 0); for (k=1, #d-2, s += d[k]; if (s == n, return (1)); if (s > n, break);); return (0);} \\ Michel Marcus, Feb 09 2019

from itertools import accumulate, count, islice
from sympy import divisors
def A194472_gen(startvalue=1): # generator of terms >= startvalue
    return (n for n in count(max(startvalue,1)) if any(s == n for s in accumulate(divisors(n)[:-2])))
A194472_list = list(islice(A194472_gen(),5)) # Chai Wah Wu, Feb 18 2022

More terms from M. F. Hasler, Aug 24 2011

A185584 Positive numbers equal to the sum of the squares of their first k divisors for some k.

Original entry on oeis.org

1, 130, 1860, 148480, 3039520, 4172437680, 5788838100, 38341734200, 41251514850, 54116036100, 78936002964, 1059758860356

Offset: 1

Claudio Meller, Feb 04 2011

676358763167556, 3238006053537840, 343869932366333100, 408390181577292600, 466477390306128600 and 5447947283895097800 are also terms. - Donovan Johnson, Jan 20 2014

			130 --> (1, 2, 5, 10): 1^2+2^2+5^2+10^2 = 1+4+25+100 = 130.
1860 --> (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30).
148480 --> (1, 2, 4, 5, 8, 10, 16, 20, 29, 32, 40, 58, 64, 80, 116, 128, 145, 160, 232).
3039520 --> (1, 2, 4, 5, 8, 10, 11, 16, 20, 22, 32, 40, 44, 55, 80, 88, 110, 121, 157, 160, 176, 220, 242, 314, 352, 440, 484, 605, 628, 785, 880).

Cf. A064510.

See A185873 for the values of k.

Select[Range[5000000],MemberQ[Accumulate[Divisors[#]^2],#]&] (* Harvey P. Dale *)

isA185584(n) = fordiv(n,d,(n-=d^2)<0&return;n|return(1))
for(n=1,1e7,isA185584(n)&print1(n", ")) \\ M. F. Hasler

a(6) from Charles R Greathouse IV, Feb 05 2011
a(7) from Donovan Johnson, Feb 05 2011
Edited by N. J. A. Sloane, Feb 05 2011
a(8)-a(11) from Donovan Johnson, Feb 07 2011
a(12) from Donovan Johnson, Jan 20 2014

A378314 a(n) = number of subsets of {1, 2, ..., n} that represent the first k divisors of m for some positive integers m and 1 <= k <= A000005(m).

Original entry on oeis.org

0, 1, 2, 4, 6, 12, 16, 28, 36, 52, 70, 118, 150, 246, 318, 382, 430, 670, 798, 1278, 1566, 1886, 2270, 3230, 3742, 4702, 5854, 6814, 7966, 11806, 14878, 22558, 25630, 32542, 40222, 46366, 52510, 70942, 86302, 100126, 112414, 149278, 172318, 246046, 295198, 344350, 405790, 553246, 626974, 774430, 897310

Offset: 0

Max Alekseyev, Nov 22 2024

nonn

			a(4) = 6 enumerates subsets {1}, {1,2}, {1,2,3}, {1,2,4}, {1,2,3,4}, {1,3}.

Partial sums of A190940.

Cf. A001248, A027750, A064510, A180851, A194269, A307137, A378313.

a378314(n) = my(L=1); sum(i=1,n, L=lcm(L,i); sigma(L/i));

a(n) = Sum_{i=1..n} sigma(LCM(1,2,...,i)/i).

A109884 Indices k of members of A109883 such that A109883(k) is a divisor of k. Also k is a term if A109883(k) = 0.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 16, 18, 24, 28, 30, 32, 40, 44, 48, 60, 64, 72, 84, 120, 126, 128, 136, 140, 150, 152, 180, 184, 204, 216, 224, 234, 256, 270, 360, 420, 440, 462, 468, 496, 512, 520, 528, 546, 672, 700, 750, 752, 864, 870, 884

Offset: 1

Amarnath Murthy, Jul 11 2005

2^n is a term for all n. All perfect numbers are terms.

			18 is a term as A109883(18) = 6, 6 is a divisor of 18.
6 is a term as A109883(6) = 0.

Nathaniel Johnston, Table of n, a(n) for n = 1..463

Cf. A064510, A109883.

for n from 1 to 900 do if(A109883(n)=0 or type(n/A109883(n),integer))then print(n);fi:od: # Nathaniel Johnston, Apr 15 2011

With[{s = Table[Catch@ Fold[If[#1 < #2, Throw[#1], #1 - #2] &, n, Divisors@ n], {n, 10^3}]}, Select[MapIndexed[{First@ #2, #1 /. 0 -> 1} &, s], Divisible[#1, #2] & @@ # &][[All, 1]]] (* Michael De Vlieger, Aug 17 2017, after Bobby R. Treat at A109883 *)

Offset, a(11), and a(12) corrected by, and a(13) - a(52) from Nathaniel Johnston, Apr 15 2011

A378313 Numbers m such that such that Sum_{i=1..k} A027750(m,i)^i = m for some k <= A000005(m).

Original entry on oeis.org

1, 130, 135, 288, 5083, 8064, 10130, 374057639685, 3138436947541900183, 5386775810449231243, 74220449392444960903, 153525475816743446263, 1388286039882808958923, 8020029492466254993943, 8756593744534084572523, 16468366959402667137403

Offset: 1

Max Alekseyev, Nov 22 2024

There are many terms of the form 1 + p^2 + q^3 with primes p < q. Next known term not of this form is 1254382690393861635950014154836028.

Cf. A001248, A027750, A064510, A180851, A190940, A194269, A307137, A378314.

A185729 Numbers that are the sum of their first k non-divisors for some k.

Original entry on oeis.org

5, 8, 12, 51, 56, 85, 87, 105, 132, 164, 224, 249, 280, 321, 324, 343, 357, 363, 366, 405, 427, 428, 553, 583, 591, 618, 638, 654, 689, 699, 820, 918, 978, 1028, 1045, 1077, 1144, 1207, 1261, 1267, 1268, 1342, 1377, 1417, 1477, 1505, 1517, 1691, 1692, 1707, 1758, 1794, 1805, 1883, 1927, 2197, 2322, 2334, 2384, 2461, 2481, 2523, 2525, 2568

Offset: 1

Andrew Weimholt, Feb 05 2011

nonn

			51 is in the sequence because 51 = 2 + 4 + 5 + 6 + 7 + 8 + 9 + 10
(the sum of its first 8 non-divisors)

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A185811 (k-values associated with this sequence)

Cf. A064510.

Select[Range[2600],MemberQ[Accumulate[Complement[Range[#],Divisors[#]]],#]&]  (* Harvey P. Dale, Feb 16 2011 *)

is(n)=my(s,t=1); while(sCharles R Greathouse IV, Nov 23 2015

A194269 Numbers j such that Sum_{i=1..k} d(i)^i = j+1 for some k where d(i) is the sorted list of divisors of j.

Original entry on oeis.org

4, 9, 25, 49, 68, 121, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 16129, 17161, 17500, 18769, 19321, 22201, 22801, 24649, 26569, 27889

Offset: 1

Michel Lagneau, Aug 27 2011

nonn

Equivalently, numbers j such that Sum_{i=2..k} A027750(j,i)^i = j for some k.
The majority of these numbers are squares.
The sequence of numbers j such that Sum_{i=1..k} A027750(j,i)^i = j for some k is given by A378313.
All prime squares p^2 (A001248) are terms because the partial sum 1^1 + p^2 satisfy the condition. The terms that are not squares are given by A307137. - Michel Marcus, Mar 25 2019

			The divisors of 68 are 1, 2, 4, 17, 34, 68; 1^1 + 2^2 + 4^3 = 69, so 68 is a term.

Cf. A001248, A027750, A064510, A180851, A190940, A307137, A378313, A378314.

isA194269 := proc(n) local dgs ,i,k; dgs := sort(convert(numtheory[divisors](n),list)) ; for k from 1 to nops(dgs) do if add(op(i,dgs)^i,i=1..k) = n+1 then return true; end if; end do; false ; end proc:
for n from 1 to 30000 do if isA194269(n) then print(n); end if; end do: # R. J. Mathar, Aug 27 2011

isok(n) = {my(d=divisors(n), s=0); for(k=1, #d, s += d[k]^k; if(s == n+1, return(1)); if(s > n+1, break););0;} \\ Michel Marcus, Mar 25 2019

Edited by Max Alekseyev, Nov 22 2024

A109886 Index of first occurrence of n in A109883, or -1 if n does not occur in A109883.

Original entry on oeis.org

1, 2, 3, 30, 5, 9, 7, 50, 20, 42, 11, 36, 13, 6510, 27, 54, 17, 70620, 19, 25, 46, 66, 23, 168630, 124, 98, 58, 78, 29

Offset: 0

Amarnath Murthy, Jul 11 2005

Sequence continues: a(29)=??, a(30)=??, 31, 70, 112, 100, 57, 200, 37, 484, 55, 102, 41, 49, 43, a(44)=??, 94, 114, 47, 225, 1264, 252, 104, 294, 53, 780, 87, 71940, 118, 138, 59, 4290, 61, 1470, 85, 306, 134, a(66)=??, 67, 6300, 142, 288, 71, 324, 73, a(74)=??, 712, 174, 158, 2940, 79, a(80)=??, 166, 186, 83, 1344210, 405, 242, 115, 1590, 89, a(90)=??, 141, 196, 406, 540, 119, 2310, 97, 390, 202, 222, ..., . - Jason Earls and Robert G. Wilson v, Jul 12 2005
Smallest number N with perfect deficiency n, that is, the first number such that A109883(N)=n. - Walter Kehowski, Sep 13 2005
a(29) > 10^9 if it exists. - Michel Marcus, Mar 30 2024

			a(7) = 50: divisors of 50 are 1,2,5,10,25,50; 50 - (1 + 2 + 5 + 10 + 25) = 7 and 50 is the smallest such number.

Cf. A064510, A109883, A109884.

with(numtheory); pdef := proc(n) local k, d, divd; d:=n; divd:=sort([op(divisors(n))]); for k in divd while d>=k do d:=d-k; od; end: PDL:=[]; for z from 1 to 1 do for pd from 0 to 60 do for n from 1 to 200000 do missed:=true; if pdef(n)=pd then PDL:=[op(PDL),n]; missed:=false; break fi od; if missed then PDL:=[op(PDL),-1] fi od od; PDL; # Walter Kehowski, Sep 13 2005

subtract = If[ #1 < #2, Throw[ #1], #1 - #2]&; f[n_] := Catch @ Fold[subtract, n, Divisors @ n] (* Bobby R. Treat, (DrBob(AT)bigfoot.com), Jul 14 2005 *)
t = Table[0, {60}]; Do[ a = f[n]; If[a < 60 && t[[a + 1]] == 0, t[[a + 1]] = n], {n, 10^8}] (* Robert G. Wilson v, Jul 14 2005 *)

Corrected and extended by Jason Earls, Jul 12 2005

More terms from Walter Kehowski, Sep 13 2005

cp's OEIS Frontend

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

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A109883 Start subtracting from n its divisors beginning from 1 until one reaches a number smaller than the last divisor subtracted or reaches the last nontrivial divisor < n. Define this to be the perfect deficiency of n. Then a(n) = perfect deficiency of n.

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A194472 Erdős-Nicolas numbers.

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A185584 Positive numbers equal to the sum of the squares of their first k divisors for some k.

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A378314 a(n) = number of subsets of {1, 2, ..., n} that represent the first k divisors of m for some positive integers m and 1 <= k <= A000005(m).

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A109884 Indices k of members of A109883 such that A109883(k) is a divisor of k. Also k is a term if A109883(k) = 0.

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A378313 Numbers m such that such that Sum_{i=1..k} A027750(m,i)^i = m for some k <= A000005(m).

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