cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A098008 Length of transient part of aliquot sequence for n, or -1 if transient part is infinite.

Original entry on oeis.org

1, 2, 2, 3, 2, 0, 2, 3, 4, 4, 2, 7, 2, 5, 5, 6, 2, 4, 2, 7, 3, 6, 2, 5, 1, 7, 3, 0, 2, 15, 2, 3, 6, 8, 3, 4, 2, 7, 3, 4, 2, 14, 2, 5, 7, 8, 2, 6, 4, 3, 4, 9, 2, 13, 3, 5, 3, 4, 2, 11, 2, 9, 3, 4, 3, 12, 2, 5, 4, 6, 2, 9, 2, 5, 5, 5, 3, 11, 2, 7, 5, 6, 2, 6, 3, 9, 7, 7, 2, 10, 4, 6, 4, 4, 2, 9, 2, 3, 4, 5, 2, 18
Offset: 1

Views

Author

N. J. A. Sloane, Sep 09 2004

Keywords

Comments

See A098007 for further information.
a(n) = 0 if and only if n is perfect (A000396) or part of a cycle of length greater than 1. - Comment corrected by Antti Karttunen, Nov 02 2017.
It is believed that the first time a(n) = -1 is at n = 276 (see A008892). - N. J. A. Sloane, Nov 02 2017

Examples

			From _Antti Karttunen_, Nov 02 2017: (Start)
For n = 3, a(n) = 2, because A001065(3) = 1 and A001065(1) = 0, so it took two steps to end in zero.
For n = 25, a(n) = 1, because A001065(25) = 6, and A001065(6) = 6, so it took one step to enter into a cycle.
For n = 12496, a(n) = 0, because 12496 is a member of 5-cycle of map n -> A001065(n) (see A072891).
(End)

References

  • R. K. Guy, Unsolved Problems in Number Theory, B6.
  • R. K. Guy and J. L. Selfridge, Interim report on aliquot series, pp. 557-580 of Proceedings Manitoba Conference on Numerical Mathematics. University of Manitoba, Winnipeg, Oct 1971.

Links

Crossrefs

Cf. A001065, A098007, A044050, A003023, A008892. See A007906 for another version.
Cf. A206708 (gives a proper subset of zeros).

Programs

  • Mathematica
    g[n_] := If[n > 0, DivisorSigma[1, n] - n, 0]; f[n_] := NestWhileList[g, n, UnsameQ, All]; Table[ Length[ f[n]] - 2, {n, 102}] (* good only for n<220 *) (* Robert G. Wilson v, Sep 10 2004 *)
  • Scheme
    (define (A098008 n) (let loop ((visited (list n)) (i 1)) (let ((next (A001065 (car visited)))) (cond ((zero? next) i) ((member next visited) => (lambda (transientplus1) (- (length transientplus1) 1))) (else (loop (cons next visited) (+ 1 i))))))) ;; Good for at least n = 1..275.
(define (A001065 n) (- (A000203 n) n)) ;; For an implementation of A000203, see under that entry.
;; Antti Karttunen, Nov 02 2017

Extensions

More terms from Robert G. Wilson v, Sep 10 2004

A178601 a(n) = s(s(n)), where s(n) = sigma(n)-n = A001065(n).

Original entry on oeis.org

0, 0, 1, 0, 6, 0, 1, 3, 7, 0, 15, 0, 8, 4, 9, 0, 11, 0, 14, 1, 10, 0, 55, 6, 15, 1, 28, 0, 54, 0, 1, 9, 22, 1, 17, 0, 14, 1, 43, 0, 66, 0, 50, 15, 16, 0, 64, 7, 1, 11, 26, 0, 78, 1, 63, 1, 31, 0, 172, 0, 20, 1, 41, 1, 90, 0, 32, 13, 40, 0, 45, 0, 50, 8, 63, 1, 144, 0, 56, 50, 40, 0, 196, 1, 26, 15, 76, 0, 259, 11, 64, 13, 43, 6, 236, 0
Offset: 2

Views

Author

Roger L. Bagula, May 30 2010

Keywords

Links

Crossrefs

Cf. A001065, A000203, A051027, A072869.
Cf. A206708 (fixed points, union of A000396 and A063990). - Antti Karttunen, Nov 01 2017

Programs

  • Mathematica
    Table[Apply[ Plus, Divisors[Apply[ Plus, Divisors[n]] - n]] - (Apply[Plus, Divisors[n]] - n), {n, 0, 100}]
  • PARI
    A001065(n) = (sigma(n) - n);
A178601(n) = A001065(A001065(n)); \\ Antti Karttunen, Nov 01 2017

Extensions

Edited by N. J. A. Sloane, May 30 2010
Terms a(0) and a(1) removed and more terms added by Antti Karttunen, Nov 01 2017

A347770 Conjectured list of numbers which are perfect, amicable, or sociable.

Original entry on oeis.org

6, 28, 220, 284, 496, 1184, 1210, 2620, 2924, 5020, 5564, 6232, 6368, 8128, 10744, 10856, 12285, 12496, 14264, 14288, 14316, 14536, 14595, 15472, 17296, 17716, 18416, 19116, 19916, 22744, 22976, 31704, 45946, 47616, 48976, 63020, 66928, 66992, 67095, 69615, 71145, 76084, 79750
Offset: 1

Views

Author

Eric Chen, Sep 13 2021

Keywords

Comments

By definition, this is the union of A000396, A259180, and A122726. However, at present A122726 is not known to be complete. There is no proof that 564 (for example) is missing from this sequence. - N. J. A. Sloane, Sep 17 2021
Numbers m for which there exists k>=1 such that s^k(m) = m, where s is A001065.
Conjecture: There are no aliquot cycles containing even numbers and odd numbers simultaneously, i.e., every aliquot cycle either has only even numbers or has only odd numbers.

Examples

			Known aliquot cycles (sorted by smallest member):
{6}
{28}
{220, 284}
{496}
{1184, 1210}
{2620, 2924}
{5020, 5564}
{6232, 6368}
{8128}
{10744, 10856}
{12285, 14595}
{12496, 14288, 15472, 14536, 14264}
{14316, 19116, 31704, 47616, 83328, 177792, 295488, 629072, 589786, 294896, 358336, 418904, 366556, 274924, 275444, 243760, 376736, 381028, 285778, 152990, 122410, 97946, 48976, 45946, 22976, 22744, 19916, 17716}
{17296, 18416}
...

Links

Crossrefs

Cf. A000396, A002025, A002046, A003416, A063990, A122726, A206708, A259180.

Extensions

Edited with new definition (pointing out that the list is only conjectured to be complete) by N. J. A. Sloane, Sep 17 2021

A321328 a(n) is the smallest number k such that k = (sigma(n*(sigma(k)-k)) - n*(sigma(k)-k))/n.

Original entry on oeis.org

6, 20, 14, 4, 10, 26, 1012, 8, 1442, 68, 376, 38, 1660, 14, 506, 574, 352, 117, 590, 22, 254, 1292, 460, 82, 26108, 416, 266, 10, 3496, 15, 124, 32, 470, 5176, 658, 362, 104696, 152, 19305, 51, 12782, 62, 618770, 232, 15561, 1136, 4136, 1006, 8588, 49166, 154, 13988
Offset: 1

Views

Author

Paolo P. Lava, Nov 05 2018

Keywords

Comments

A sort of generalization of amicable numbers where x = n*(sigma(k)-k), y = (sigma(x)-x)/n = k and x >= y.
All the numbers that satisfy the equation for n=1 are listed in A206708.
a(n) = n for n = 4, 8, 14, 32, 128, 2366, 8193, 131072, etc.
In particular a(n) = n if n = 2^p where p is a Mersenne exponent (A000043).

Examples

			a(7) = 1012 because (sigma(7*(sigma(1012)-1012)) - 7*(sigma(1012)-1012))/7 = (sigma(7*1004) - 7*1004)/7 = (14112-7028)/7 = 7084/7 = 1012 and this is the least number to have this property.

Links

Crossrefs

Cf. A000040, A000043, A000203, A206708.

Programs

  • Maple
    with(numtheory): P:=proc(q) local k,n; for n from 1 to q do
for k from 1 to q do if (sigma(n*(sigma(k)-k))-n*(sigma(k)-k))/n=k
then print(k); break; fi; od; od; end: P(10^6);
  • Mathematica
    s[n_] := DivisorSigma[1,n]-n; a[n_] := Module[{k=2}, While[k != s[n*s[k]]/n, k++];k]; Array[a, 52] (* Amiram Eldar, Nov 06 2018 *)
  • PARI
    f(n,k) = {my(sk = sigma(k)-k); iferr((sigma(n*sk)-n*sk)/n, E, 0);}
a(n) = {my(k=1); while (k != f(n,k), k++); k;} \\ Michel Marcus, Nov 06 2018

A347769 a(0) = 0; a(1) = 1; for n > 1, a(n) = A001065(a(n-1)) = sigma(a(n-1)) - a(n-1) (the sum of aliquot parts of a(n-1)) if this is not yet in the sequence; otherwise a(n) is the smallest number missing from the sequence.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 16, 15, 13, 14, 17, 18, 21, 19, 20, 22, 23, 24, 36, 55, 25, 26, 27, 28, 29, 30, 42, 54, 66, 78, 90, 144, 259, 45, 33, 31, 32, 34, 35, 37, 38, 39, 40, 50, 43, 41, 44, 46, 47, 48, 76, 64, 63, 49, 51, 52, 53, 56, 57, 58, 59, 60, 108, 172
Offset: 0

Views

Author

Eric Chen, Sep 13 2021

Keywords

Comments

This sequence is a permutation of the nonnegative integers iff Catalan's aliquot sequence conjecture (also called Catalan-Dickson conjecture) is true.
a(563) = 276 is the smallest number whose aliquot sequence has not yet been fully determined.
As long as the aliquot sequence of 276 is not known to be finite or eventually periodic, a(563+k) = A008892(k).

Examples

			a(0) = 0, a(1) = 1;
since A001065(a(1)) = 0 has already appeared in this sequence, a(2) = 2;
since A001065(a(2)) = 1 has already appeared in this sequence, a(3) = 3;
...
a(11) = 11;
since A001065(a(11)) = 1 has already appeared in this sequence, a(12) = 12;
since A001065(a(12)) = 16 has not yet appeared in this sequence, a(13) = A001065(a(12)) = 16;
since A001065(a(13)) = 15 has not yet appeared in this sequence, a(14) = A001065(a(13)) = 15;
since A001065(a(14)) = 9 has already appeared in this sequence, a(15) = 13;
...

Links

Crossrefs

Cf. A032451.
Cf. A001065 (sum of aliquot parts).
Cf. A003023, A044050, A098007, A098008: ("length" of aliquot sequences, four versions).
Cf. A007906.
Cf. A115060 (maximum term of aliquot sequences).
Cf. A115350 (termination of the aliquot sequences).
Cf. A098009, A098010 (records of "length" of aliquot sequences).
Cf. A290141, A290142 (records of maximum term of aliquot sequences).
Cf. A000396, A063990, A122726, A206708, A347770.
Cf. A080907, A063769, A121507, A121508, A131884, A126016.
Aliquot sequences starting at various numbers: A000004 (0), A000007 (1), A033322 (2), A010722 (6), A143090 (12), A143645 (24), A010867 (28), A008885 (30), A143721 (38), A008886 (42), A143722 (48), A143723 (52), A008887 (60), A143733 (62), A143737 (68), A143741 (72), A143754 (75), A143755 (80), A143756 (81), A143757 (82), A143758 (84), A143759 (86), A143767 (87), A143846 (88), A143847 (96), A143919 (100), A008888 (138), A008889 (150), A008890 (168), A008891 (180), A203777 (220), A008892 (276), A014360 (552), A014361 (564), A074907 (570), A014362 (660), A269542 (702), A045477 (840), A014363 (966), A014364 (1074), A014365 (1134), A074906 (1521), A143930 (3630), A072891 (12496), A072890 (14316), A171103 (46758), A072892 (1264460).

Programs

  • PARI
    A347769_list(N)=print1(0, ", "); if(N>0, print1(1, ", ")); v=[0, 1]; b=1; for(n=2, N, if(setsearch(Set(v), sigma(b)-b), k=1; while(k<=n, if(!setsearch(Set(v), k), b=k; k=n+1, k++)), b=sigma(b)-b); print1(b, ", "); v=concat(v, b))
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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