A109883 -id:A109883 - cp's OEIS frontend

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.

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

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

A110002 Palindromes whose perfect deficiency (A109883) is also palindromic.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 22, 44, 222, 292, 414, 646, 717, 848, 979, 27072, 28882, 45954, 74247, 90109, 96569, 118811, 2376732, 5136315, 5185815, 5266625, 5635365, 5684865, 6344436, 7424247, 7481847, 7484847, 7929297, 9858589, 12333321, 21922912, 32255223

Offset: 1

Jason Earls, Sep 02 2005

Michael S. Branicky, Table of n, a(n) for n = 1..101 (all terms < 10^14)

Cf. A002113, A109883.

subtract = If[ #1 < #2, Throw[ #1], #1 - #2]&;d[n_] := Catch @ Fold[subtract, n, Divisors @ n];Select[Range[10000000],PalindromeQ[#]&&PalindromeQ[d[#]]&] (* James C. McMahon, Mar 31 2024 *)

# uses imports, function in A109883
from itertools import count, islice, product
def ispal(n): return (s:=str(n)) == s[::-1]
def pals(): # generator of palindromes
    digits = "0123456789"
    yield from map(int, digits)
    for d in count(2):
        for f in "123456789":
            for p in product(digits, repeat=d//2-1):
                left = f + "".join(p); right = left[::-1]
                for mid in [[""], digits][d%2]:
                    yield int(left + mid + right)
def agen(): yield from (p for p in pals() if p>0 and ispal(A109883(p)))
print(list(islice(agen(), 40))) # Michael S. Branicky, Mar 31 2024

a(38) and beyond from Michael S. Branicky, Mar 31 2024

A110277 Values of n such that the perfect deficiency (A109883) of n is a square.

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 14, 16, 17, 24, 28, 32, 37, 38, 42, 44, 64, 72, 92, 98, 101, 110, 128, 134, 152, 170, 172, 180, 186, 192, 197, 206, 248, 252, 256, 257, 284, 374, 398, 401, 410, 428, 434, 474, 480, 496, 512, 577, 590, 604, 616, 632, 638, 677, 752, 760, 864, 884

Offset: 1

Jason Earls, Jul 18 2005

nonn

Cf. A110278.

subtract = If[ #1 < #2, Throw[ #1], #1 - #2]&; f[n_] := Catch @ Fold[subtract, n, Divisors @ n]; Select[ Range[905], IntegerQ[ Sqrt[ f[ # ]]] &] (* Robert G. Wilson v, Jul 21 2005 *)

A110278 Values of n such that the perfect deficiency (A109883) of n and n+1 are both squares.

Original entry on oeis.org

1, 4, 5, 16, 37, 256, 65536, 80656, 3459600, 166926400

Offset: 1

Jason Earls, Jul 18 2005

Conjecture: sequence is infinite.

No more terms below 10^9. - Amiram Eldar, Dec 27 2018

			A109883(37)=36 & A109883(38)=16, both of which are squares, so 37 is a term.

Cf. A110277.

subtract = If[ #1 < #2, Throw[ #1], #1 - #2]&; f[n_] := Catch @ Fold[subtract, n, Divisors @ n]; a = False; Do[b = IntegerQ[ Sqrt[ f[ n]]]; If[{a, b} == {True, True}, Print[n - 1]]; a = b, {n, 10^7}] (* Robert G. Wilson v, Jul 21 2005 *)

a109883(n) = {my(r = n); fordiv(n, d, if (r < d, return (r)); r -= d;); 0;}
isok(n) = issquare(a109883(n)) && issquare(a109883(n+1)); \\ Michel Marcus, Dec 28 2018

a(10) from Amiram Eldar, Dec 27 2018

A108864 Numbers n such that the perfect deficiency of n (A109883) is <= 10.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 28, 30, 32, 40, 42, 44, 50, 52, 60, 64, 68, 72, 110, 120, 126, 128, 130, 136, 144, 150, 152, 180, 184, 204, 228, 256, 315, 462, 496, 512, 528, 592, 656, 750, 884, 1012, 1024, 1155, 1188, 1248

Offset: 1

Jason Earls, Jul 12 2005

nonn

Is 1155 the last odd number in this sequence?

A108865 Numbers n such that the perfect deficiency of n (A109883) is prime.

Original entry on oeis.org

3, 9, 10, 12, 25, 30, 36, 49, 50, 60, 70, 81, 121, 126, 136, 144, 150, 210, 225, 289, 324, 330, 338, 350, 390, 462, 484, 490, 510, 660, 676, 690, 750, 770, 780, 784, 800, 841, 961, 1058, 1089, 1156, 1190, 1225, 1250, 1380, 1470, 1500, 1610, 1650, 1682, 1750

Offset: 1

Jason Earls, Jul 12 2005

Consecutive terms are 9,10 and 49,50. Are there any more?

A110018 Numbers n such that the perfect deficiency of n (A109883) equals the perfect deficiency of n + 1.

Original entry on oeis.org

10611, 145215

Offset: 1

Jason Earls, Sep 03 2005

Are these the only terms?

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

Original entry on oeis.org

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

A117552 Largest partial sum of the increasingly ordered divisors of n, not exceeding n.

Original entry on oeis.org

1, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 10, 1, 10, 9, 15, 1, 12, 1, 12, 11, 14, 1, 24, 6, 16, 13, 28, 1, 27, 1, 31, 15, 20, 13, 25, 1, 22, 17, 30, 1, 33, 1, 40, 33, 26, 1, 36, 8, 43, 21, 46, 1, 39, 17, 36, 23, 32, 1, 58, 1, 34, 41, 63, 19, 45, 1, 58, 27, 39, 1, 63, 1, 40, 49, 64, 19, 51, 1, 66

Offset: 1

Leroy Quet, Mar 28 2006

nonn

			a(12)=10 because the increasingly ordered divisors of 12 are 1,2,3,4,6 and 12, with partial sums 1,3,6,10,16 and 28; the largest partial sum not exceeding 12 is 10.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sums of divisors

Cf. A117553, A109883, A377247 (corresponding largest divisor index).

with(numtheory): a:=proc(n) local div,j: if n=1 then 1 else div:=divisors(n): for j from 1 by 1 while sum(div[i],i=1..j)<=n do sum(div[k],k=1..j) od: fi: end: seq(a(n),n=1..90); # Emeric Deutsch, Apr 01 2006

Table[Last@ TakeWhile[Accumulate@ Divisors@ n, # <= n &], {n, 80}] (* Michael De Vlieger, Oct 30 2017 *)

A117552(n) = { my(divs=divisors(n), s=0); for(i=1,#divs,if((s+divs[i])>n,return(s),s+=divs[i])); s; }; \\ Antti Karttunen, Oct 30 2017

a(n) = n - A109883(n). - Ridouane Oudra, Jan 25 2024

More terms from Emeric Deutsch, Apr 01 2006

cp's OEIS Frontend

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

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A110002 Palindromes whose perfect deficiency (A109883) is also palindromic.

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A110277 Values of n such that the perfect deficiency (A109883) of n is a square.

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A110278 Values of n such that the perfect deficiency (A109883) of n and n+1 are both squares.

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A108864 Numbers n such that the perfect deficiency of n (A109883) is <= 10.

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A108865 Numbers n such that the perfect deficiency of n (A109883) is prime.

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A110018 Numbers n such that the perfect deficiency of n (A109883) equals the perfect deficiency of n + 1.

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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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A117552 Largest partial sum of the increasingly ordered divisors of n, not exceeding n.

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