A098007 -id:A098007 - cp's OEIS frontend

A008886 Aliquot sequence starting at 42.

42, 54, 66, 78, 90, 144, 259, 45, 33, 15, 9, 4, 3, 1, 0

Offset: 0

The sum-of-divisor function A000203 and aliquot parts A001065 are defined only for positive integers, so the trajectory ends when 0 is reached, here at index 14. - M. F. Hasler, Feb 24 2018

R. K. Guy, Unsolved Problems in Number Theory, B6.

Christophe CLAVIER, Aliquot Sequences
Index entries for sequences related to aliquot parts.

Cf. A098007 (length of aliquot sequences); some other examples: A008885 (starting at 30) .. A008892 (starting at 276), A014360 (starting at 552) .. A014365 (starting at 1134), see link to index for a more complete list.

f := proc(n) option remember; if n = 0 then 42; else sigma(f(n-1))-f(n-1); fi; end:

Join[NestList[DivisorSigma[1,#]-#&,42,14],PadRight[{},60,0]] (* Harvey P. Dale, Apr 28 2014 *)

a(n,a=42)={for(i=1,n,a=sigma(a)-a);a} \\ M. F. Hasler, Feb 24 2018

a(n+1) = A001065(a(n)). - R. J. Mathar, Oct 11 2017

a(n) = A008885(n+1). - R. J. Mathar, Jan 12 2024

Edited by M. F. Hasler, Feb 24 2018

A008890 Aliquot sequence starting at 168.

Original entry on oeis.org

168, 312, 528, 960, 2088, 3762, 5598, 6570, 10746, 13254, 13830, 19434, 20886, 21606, 25098, 26742, 26754, 40446, 63234, 77406, 110754, 171486, 253458, 295740, 647748, 1077612, 1467588, 1956812

Offset: 0

N. J. A. Sloane

The sum-of-divisor function A000203 and aliquot parts A001065 are defined only for positive integers, so the trajectory ends when 0 is reached, here at index 175. - M. F. Hasler, Feb 24 2018

R. K. Guy, Unsolved Problems in Number Theory, B6.

T. D. Noe, Table of n, a(n) for n = 0..175 (a(176) removed by _Georg Fischer_, Apr 28 2019)
Christophe Clavier, Aliquot Sequences
Index entries for sequences related to aliquot parts.

Cf. A008885 (starting at 30), ..., A008892 (starting at 276), A098007 (length of aliquot sequences).

f := proc(n) option remember; if n = 0 then 168; else sigma(f(n-1))-f(n-1); fi; end:

NestList[DivisorSigma[1, #] - # &, 168, 175] (* Alonso del Arte, Feb 24 2018 *)

a(n,a=168)={for(i=1,n,a=sigma(a)-a);a} \\ M. F. Hasler, Feb 24 2018

a(n+1) = A001065(a(n)). - R. J. Mathar, Oct 11 2017

Edited by M. F. Hasler, Feb 24 2018

A014363 Aliquot sequence starting at 966.

Original entry on oeis.org

966, 1338, 1350, 2370, 3390, 4818, 5838, 7602, 9870, 17778, 17790, 24978, 27438, 30882, 30894, 34386, 40782, 52530, 82254, 82266, 82278, 121770, 241110, 450090, 750870, 1295226, 1572678, 1919538, 2760984, 4964136

Offset: 0

Paul Zimmermann

nonn

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B6, pp. 92-95.

Tyler Busby, Table of n, a(n) for n = 0..1109 (from factordb.com; terms 0..277 from T. D. Noe, terms 278..1035 from Amiram Eldar)
Christophe Clavier, Aliquot Sequences.
factordb.com, Aliquot sequence of 966.
Paul Zimmermann, Latest information.
Index entries for sequences related to aliquot parts.

Cf. A008885 (starting at 30) .. A008892 (starting at 276), A014360 (starting at 552) .. A014365 (starting at 1134), ..., A171103 (starting at 46758), A098007 (length of aliquot sequences).

Cf. A001065.

FixedPointList[If[# > 0, DivisorSigma[1, #] - #, 0]&, 966, 100] (* Jean-François Alcover, Mar 28 2020 *)

a(n,a=966)={for(i=1,n,a=sigma(a)-a);a} \\ M. F. Hasler, Feb 24 2018

a(n+1) = A001065(a(n)). - R. J. Mathar, Oct 11 2017

A014364 Aliquot sequence starting at 1074.

Original entry on oeis.org

1074, 1086, 1098, 1320, 3000, 6360, 13080, 26520, 64200, 136680, 303960, 668040, 1448760, 2897880, 6778920, 14760600, 31761720, 75003840, 189623520, 475142400, 1262108388, 1723154620, 2250655556, 1742856988

Offset: 0

Paul Zimmermann

nonn

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B6, pp. 92-95.

Amiram Eldar, Table of n, a(n) for n = 0..2194 (terms 0..881 from R. J. Mathar)
Christophe Clavier, Aliquot Sequences.
factordb.com, Aliquot sequence of 1074.
Paul Zimmermann, Latest information.
Index entries for sequences related to aliquot parts.

Cf. A098007 (length of aliquot sequences). Some other examples: A008885 (starting at 30) .. A008892 (starting at 276), A014360 (starting at 552) .. A014365 (starting at 1134), ..., A171103 (starting at 46758). See link to index for a more complete list.

Cf. A000203.

FixedPointList[If[# > 0, DivisorSigma[1, #] - #, 0] &, 1074, 100] (* Jean-François Alcover, Mar 28 2020 *)

a(n, a=1074)={for(i=1, n, a=sigma(a)-a); a} \\ M. F. Hasler, Feb 24 2018

a(n+1) = A000203(a(n))-a(n). - R. J. Mathar, Oct 08 2017

A127161 Integers whose aliquot sequences terminate by encountering a prime number.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75

Offset: 1

Ant King, Jan 06 2007

nonn

This sequence is the same as A080907 from A080907's second term onwards.

			a(10)=12 because the tenth integer whose aliquot sequence terminates by encountering a prime as a member of its trajectory is 12. The complete aliquot sequence generated by iterating the proper divisors of 12 is 12->16->15->9->4->3->1->0

Benito, Manuel; Creyaufmueller, Wolfgang; Varona, Juan Luis; and Zimmermann, Paul; Aliquot Sequence 3630 Ends After Reaching 100 Digits; Experimental Mathematics, Vol. 11, No. 2, Natick, MA, 2002, pp. 201-206.

Manuel Benito and Juan L. Varona, Advances In Aliquot Sequences, Mathematics of Computation, Vol. 68, No. 225, (1999), pp. 389-393.
Wolfgang Creyaufmueller, Aliquot sequences.

Cf. A080907, A127162, A127163, A127164, A098007, A121507, A098008, A007906, A063769, A115060, A115350.

s[n_] := DivisorSigma[1, n] - n; g[n_] := If[n > 0, s[n], 0]; Trajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]]; Select[Range[2, 275], Last[Trajectory[ # ]] == 0 &]

Define s(i)=sigma(i)-i=A000203(i)-i. Then if the aliquot sequence obtained by repeatedly iterating s contains a prime as a member of its trajectory, i is included in this sequence

A127162 Composite numbers whose aliquot sequences terminate by encountering a prime number.

Original entry on oeis.org

4, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 96, 98, 99

Offset: 1

Ant King, Jan 06 2007

nonn

			a(5)=12 because the fifth composite number whose aliquot sequence terminates by encountering a prime as a member of its trajectory is 12. The complete aliquot sequence generated by iterating the proper divisors of 12 is 12->16->15->9->4->3->1->0

Benito, Manuel; Creyaufmueller, Wolfgang; Varona, Juan Luis; and Zimmermann, Paul; Aliquot Sequence 3630 Ends After Reaching 100 Digits; Experimental Mathematics, Vol. 11, No. 2, Natick, MA, 2002, pp. 201-206.

Manuel Benito and Juan L. Varona, Advances In Aliquot Sequences, Mathematics of Computation, Vol. 68, No. 225, (1999), pp. 389-393.
Wolfgang Creyaufmueller, Aliquot sequences.

Cf. A080907, A127161, A127163, A127164, A098007, A121507, A098008, A007906, A063769, A115060, A115350.

s[n_] := DivisorSigma[1, n] - n; g[n_] := If[n > 0, s[n], 0]; Trajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]]; Select[Range[2, 275], ! PrimeQ[ # ] && Last[Trajectory[ # ]] == 0 &]

Define s(i)=sigma(i)-i=A000203(i)-i. Then if i is composite and the aliquot sequence obtained by repeatedly applying the mapping i->s(i) contains a prime as a member of its trajectory, i is included in this sequence.

A127652 Integers whose unitary aliquot sequences are longer than their ordinary aliquot sequences.

Original entry on oeis.org

25, 28, 36, 40, 50, 68, 70, 74, 94, 95, 98, 116, 119, 134, 142, 143, 154, 162, 170, 175, 182, 189, 190, 200, 220, 226, 242, 245, 262, 273

Offset: 1

Ant King, Jan 24 2007

Here the length of an aliquot sequence is defined to be the length of the transient part of its trajectory + the length of its terminal cycle.

			a(5)=50 because the fifth integer whose unitary aliquot sequence is longer than its ordinary aliquot sequence is 50.

Riele, H. J. J. te; Unitary Aliquot Sequences. MR 139/72, Mathematisch Centrum, 1972, Amsterdam.
Riele, H. J. J. te; Further Results On Unitary Aliquot Sequences. NW 2/73, Mathematisch Centrum, 1973, Amsterdam.

Manuel Benito and Juan L. Varona, Advances In Aliquot Sequences, Mathematics of Computation, Vol. 68, No. 225, (1999), pp. 389-393.
Wolfgang Creyaufmueller, Aliquot Sequences.

Cf. A127161, A127162, A127163, A127164, A063769, A063990, A097032, A098007, A097010, A127653, A098185, A127654, A063991, A127655, A097037, A097036.

UnitaryDivisors[n_Integer?Positive]:=Select[Divisors[n],GCD[ #,n/# ]==1&];sstar[n_]:=Plus@@UnitaryDivisors[n]-n;g[n_] := If[n > 0, sstar[n], 0];UnitaryTrajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]];s[n_]:=DivisorSigma[1,n]-n;h[n_] := If[n > 0, s[n], 0];OrdinaryTrajectory[n_] := Most[NestWhileList[h, n, UnsameQ, All]];Select[Range[275],Length[UnitaryTrajectory[ # ]]>Length[OrdinaryTrajectory[ # ]] &]

Sequence gives those values of n for which A097032(n)>A098007(n).

A234534 Terms of the cycles reached after iterations of numerator(sigma(n)/n) = A017665(n).

Original entry on oeis.org

1, 8, 15, 127, 128, 144, 255, 403, 448, 512, 1023, 29127, 47360

Offset: 1

Michel Marcus, Dec 27 2013

If all integers were in A014567, then this sequence would not exist and we would be looking at A216200; but some are in A069059, allowing the trajectories of A017665 to go down.
The term of the sequence correspond to the 5 cycles: [1], [15, 8], [448, 127, 128, 255, 144, 403], [1023, 512], [47360, 29127].
Are there some starting x's whose fate will remain unknown, like 276 for A098007?
Are there other cycles to be found?
No other cycles found with largest member less than 10^9.
There are no other cycles with the smallest member < 10^11. All numbers < 10^11 reach one of the five known cycles. - Donovan Johnson, Jan 07 2014

			Obviously 1 is a fixed point for A017665, so 1 is in the sequence.
A017665(8) = 15 and A017665(15) = 8, so both 8 and 15 are in the sequence.

Cf. A000203, A017665.

iscycle(v, nextn) = {for (i=1, #v, if (v[i] == nextn, return (1););); return (0);}
fcycle(n, known) = {v = vector(1); v[1] = n; first = n; while ((nextn = numerator(sigma(n)/n)) <= first, if (vecsearch(known, nextn), return([])); if (iscycle(v, nextn), return (v)); v = concat(v, nextn); n = nextn;); return ([]);}
fcycles(na, nb) = {known = []; known = [1, 8, 127, 512, 29127]; for (n = na, nb, v = fcycle(n, known); if (#v, print(v, ", "); return();););} \\ use empty vector for known to search for cycles from start; when a new cycle is found, insert its smallest term to vector known.

Missing terms 512 and 1023 noticed by Donovan Johnson added by Michel Marcus, Jan 02 2014

A323327 Numbers that start an unbounded aliquot-like sequence based on Dedekind psi function (A001615).

Original entry on oeis.org

318, 330, 498, 510, 534, 546, 636, 660, 786, 798, 942, 954, 978, 990, 996, 1020, 1068, 1092, 1110, 1122, 1254, 1272, 1320, 1398, 1410, 1470, 1494, 1506, 1518, 1530, 1572, 1596, 1602, 1614, 1626, 1638, 1734, 1884, 1908, 1938, 1950, 1956, 1980, 1992, 2040, 2046

Offset: 1

Amiram Eldar, Jan 11 2019

nonn

Let t(k) = psi(k) - k = A001615(k) - k be the sum of aliquot divisors d of k, such that k/d is squarefree. Penney & Pomerance proposed a problem to show that the aliquot-like sequence related to this function, i.e., the trajectory of an integer k under the repeated application of the map k -> t(k), can be unbounded. Since t(m^j * k) = m^j * t(k) if m|k, then if in the sequence a_0 = k, a_1 = t(k), a_2 = t(t(k)), ... there is a term a_{i1} = m^j * a_0 such that m|k and j > 0 then a_{i+i1} = m^j * a_i for all i and thus the sequence is unbounded.

			318 is in the sequence since t(318) = psi(318) - 318 = 330, t(330) = 534, etc., and this repeated mapping yields an unbounded sequence.

Jean-Marie De Koninck, Those Fascinating Numbers, Amer. Math. Soc., 2009, page 71, entry 318.

Kevin Brown and Charles Vanden Eynden, Pseudo-aliquot Sequences, Solution to Problem 10323, The American Mathematical Monthly, Volume 103, No. 8 (1996), pp. 697-698.
David E. Penney and Carl Pomerance, Problem 10323, The American Mathematical Monthly, Volume 100, No. 7 (1993), p. 688.
Eric Weisstein's World of Mathematics, Aliquot Sequence.
Wikipedia, Aliquot sequence.

Cf. A001615, A098007, A187778.

t[1]=0; t[n_] := (Times@@(1+1/Transpose[FactorInteger[n]][[1]])-1)*n; rt[n_] := Module[{f=FactorInteger[n]}, e=GCD@@f[[;;,2]]; Surd[n,e]]; divrootQ[n_, m_] := Divisible[n, rt[m]]; divQ[s_, n_] := If[n==0, 0, If[MemberQ[s, n], 1, If[ Length[Select[s, Divisible[n,#] && divrootQ[#, n/#] &]] > 0, 2, 3]]]; seqQ[n_] := Module[{n1=n}, s={}; While[divQ[s, n1] ==3, AppendTo[s, n1]; n1=t[n1]]; divQ[s, n1]] == 2; Select[Range[10000], seqQ]

A008887 Aliquot sequence starting at 60.

Original entry on oeis.org

60, 108, 172, 136, 134, 70, 74, 40, 50, 43, 1, 0

Offset: 0

N. J. A. Sloane

The sum-of-divisor function A000203 and aliquot parts A001065 are defined only for positive integers, so the trajectory ends when 0 is reached, here at index 11. - M. F. Hasler, Feb 24 2018

R. K. Guy, Unsolved Problems in Number Theory, B6.

Christophe CLAVIER, Aliquot Sequences
Index entries for sequences related to aliquot parts.

Cf. A008885 (starting at 30), ..., A008892 (starting at 276), A098007 (length of aliquot sequences).

f := proc(n) option remember; if n = 0 then 60; else sigma(f(n-1))-f(n-1); fi; end:

NestList[If[#==0,0,DivisorSigma[1,#]-#]&,60,80] (* Harvey P. Dale, Nov 29 2013 *)

a(n,a=60)=for(i=1,n,a=sigma(a)-a);a \\ Will raise an error for n > 11, in agreement with the definition. - M. F. Hasler, Feb 24 2018

a(n+1) = A001065(a(n)). - R. J. Mathar, Oct 11 2017

Edited by M. F. Hasler, Feb 24 2018

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A008886 Aliquot sequence starting at 42.

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A008890 Aliquot sequence starting at 168.

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A014363 Aliquot sequence starting at 966.

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A014364 Aliquot sequence starting at 1074.

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A127161 Integers whose aliquot sequences terminate by encountering a prime number.

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A127162 Composite numbers whose aliquot sequences terminate by encountering a prime number.

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A127652 Integers whose unitary aliquot sequences are longer than their ordinary aliquot sequences.

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