A008892 -id:A008892 - cp's OEIS frontend

A323328 Lexicographically earliest unbounded aliquot-like sequence based on the Dedekind psi function: a(1) = 318, a(n) = t(a(n-1)) where t(k) = A001615(k) - k.

318, 330, 534, 546, 798, 1122, 1470, 2562, 3390, 4818, 5838, 7602, 9870, 17778, 17790, 24978, 27438, 30882, 30894, 34386, 40782, 52530, 82254, 82266, 82278, 106074, 111654, 111690, 176022, 266346, 266382, 266490, 480006, 480330, 674406, 740826, 833814, 834138

Offset: 1

Amiram Eldar, Jan 11 2019

nonn

318 is the least number k whose repeated iteration of the mapping k -> A001615(k) - k yields an unbounded sequence. Since t(m^j * n) = m^j * t(n) if m|n, 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. Since a(13)=9870, after 19 iterations a(32) = 27 * 9870, 27 = 3^3 and 3|9870 then a(n+19) = 27 * a(n) for n >= 13.

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

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

Cf. A001615, A008892, A323327.

t[1] = 0; t[n_] := (Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]) - 1)*n; NestList[t, 318, 40]

A008891 Aliquot sequence starting at 180.

Original entry on oeis.org

180, 366, 378, 582, 594, 846, 1026, 1374, 1386, 2358, 2790, 4698, 6192, 11540, 12736, 12664, 11096, 11104, 10820, 11944, 10466, 5236, 6860, 9940, 14252, 14308, 15218, 10894, 6746, 3376, 3196, 2852, 2524, 1900, 2440, 3140, 3496, 3704, 3256, 3584, 4600, 6560, 9316, 8072, 7078, 3542, 3370, 2714, 1606, 1058, 601, 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 52. - 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..52
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 180; else sigma(f(n-1))-f(n-1); fi; end:

FixedPointList[If[# > 0, DivisorSigma[1, #] - #, 0]&, 180] // Most (* Jean-François Alcover, Mar 28 2020 *)

a(n,a=180)={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

A014361 Aliquot sequence starting at 564.

Original entry on oeis.org

564, 780, 1572, 2124, 3336, 5064, 7656, 13944, 26376, 49464, 88536, 187944, 295896, 443904, 812340, 1652304, 2767056, 4803888, 7914048, 13495104, 30725280, 79741440, 196505388, 300216656, 285162916, 237325596, 325831908, 434442572, 325831936, 347001764, 260735800, 434766560

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..3493 (from factordb.com; terms 0..600 from T. D. Noe, terms 601..3486 from Amiram Eldar)
Christophe Clavier, Aliquot Sequences.
factordb.com, Aliquot sequence of 564.
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. A001065.

FixedPointList[If[# > 0, DivisorSigma[1, #] - #, 0]&, 564, 100] (* Jean-François Alcover, Mar 28 2020 *)

a(n, a=564)={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

A008886 Aliquot sequence starting at 42.

Original entry on oeis.org

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

A146556 Natural growth of an aliquot sequence driven by a perfect number 2^(p-1)*((2^p) - 1).

Original entry on oeis.org

3, 5, 7, 9, 17, 19, 21, 43, 45, 111, 193, 195, 477, 927, 1777, 1779, 2973, 4963, 6397, 6399, 12961, 14983, 14985, 40191, 66993, 114063, 193233, 334959, 558273, 951999, 1586673, 3724815, 8255985, 18271887, 31279473, 66853647, 171456753, 339654927

Offset: 1

Sergio Pimentel, Oct 31 2008

This is the natural growth of an aliquot sequence that has a driver of the form 2^(p-1) * ((2^p) - 1) (Perfect Number). It will continue growing this way until it loses the driver, which can only happen when the next term and the driver are not coprimes (which hardly ever happens).

The natural growth of the aliquot sequence starting with p=5 at 2^(p-1)*(2^p-1)*3 = 496*3 = 1488 has the factors 3, 5, 7, 9, 17, 19, 21, 43, 45, 111, 193, 195, 477, 927, 1777, 1779, 2973, 4963, 6397, 6399, 12961, 14983, 14985, 40191, 66993, 114063, 193233, 334959, 558273, 951999, 1586673, 3564018 and "loses the driver" at the next term because it is not a multiple of 496. I complemented the terms therefore from p=7 and initial factor 3 which does not lose the driver early. - R. J. Mathar, Jan 22 2009

			The aliquot sequence starting at 1488 (2^4*31*3) is: 1488, 2480, 3472, 4464,8432, 9424 or: 496*3, 496*5, 496*7, 496*9, 496*17, 496*19, always keeping the 496 driver until reaching a term that is not coprime with 496.

Mathworld, Aliquot Sequence
Stern, Aliquot Sequences from the trenches [broken link?]

Cf. A000396, A008892, A215778, A216224.

p := 7: dr := 2^(p-1)*(2^p-1) ; f := 3 ; aliq := proc(n) option remember ; global dr,f ; local an_1 ; if n = 1 then dr*f ; else an_1 := procname(n-1) ; numtheory[sigma](an_1)-an_1 ; fi; end: A := proc(n) option remember ; global dr ; aliq(n)/dr ; end: for n from 1 to 70 do printf("%a,",A(n)) ; od: # R. J. Mathar, Jan 22 2009

NestList[2*DivisorSigma[1,#]-#&,3,40] (* Harvey P. Dale, Jul 16 2013 *)

A146556()=a=[3];until(#a==79,a=concat(a,a[#a]+2*(sigma(a[#a])-a[#a])));a

a(n)=if(n==1,3,2*sigma(a(n-1))-a(n-1)) \\ R. K. Guy, Jul 16 2013

a(n) = a(n-1) + 2*(sigma(a(n-1)) - a(n-1)). - Roderick MacPhee, Aug 21 2012

More terms, as derived from p=7, driver 8128. - R. J. Mathar, Jan 22 2009

A371423 Aliquot-like sequence based on the largest aliquot divisor of the sum of divisors of n (A371418) that starts with 222.

Original entry on oeis.org

222, 228, 280, 360, 585, 546, 672, 1008, 1612, 1568, 1197, 1040, 1302, 1536, 2046, 2304, 949, 518, 456, 600, 930, 1152, 1105, 756, 1120, 1512, 2400, 3906, 4992, 7140, 12096, 20320, 24192, 40800, 70308, 108416, 135660, 241920, 490560, 902208, 1235456, 1309440, 2354688

Offset: 1

Amiram Eldar, Mar 23 2024

nonn

222 is the least number k for which the repeated iterations of the mapping k -> A371418(k) seem to generate an unbounded sequence.

			a(1) = 222 by definition.
a(2) = A371418(a(1)) = A371418(222) = 228.
a(3) = A371418(a(2)) = A371418(228) = 280.

Amiram Eldar, Table of n, a(n) for n = 1..1389
Robert D. Carmichael, Empirical Results in the Theory of Numbers, The Mathematics Teacher, Vol. 14, No. 6 (1921), pp. 305-310; alternative link. See p. 309.

Cf. A371418, A371421, A371422.

Similar sequences: A008892, A323328, A361421.

r[n_] := n/FactorInteger[n][[1, 1]]; f[n_] := r[DivisorSigma[1, n]]; NestList[f, 222, 60]

f(n) = {my(s = sigma(n)); if(s == 1, 1, s/factor(s)[1, 1]);}
lista(nmax) = {my(m = 222); for(n = 1, nmax, print1(m, ", "); m = f(m));}

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

cp's OEIS Frontend

A323328 Lexicographically earliest unbounded aliquot-like sequence based on the Dedekind psi function: a(1) = 318, a(n) = t(a(n-1)) where t(k) = A001615(k) - k.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

A008891 Aliquot sequence starting at 180.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A014361 Aliquot sequence starting at 564.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A008886 Aliquot sequence starting at 42.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A008890 Aliquot sequence starting at 168.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A014363 Aliquot sequence starting at 966.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A014364 Aliquot sequence starting at 1074.

Views

Author

Keywords

References