A072892 -id:A072892 - cp's OEIS frontend

A090615 Smallest member of sociable quadruples.

1264460, 2115324, 2784580, 4938136, 7169104, 18048976, 18656380, 28158165, 46722700, 81128632, 174277820, 209524210, 330003580, 498215416, 1236402232, 1799281330, 2387776550, 2717495235, 2879697304, 3705771825, 4424606020, 4823923384, 5373457070, 8653956136

Offset: 1

Eric W. Weisstein, Dec 06 2003

nonn

120 sociable numbers of order 4 are known as of Feb. 2003.
142 were known in 2007 (http://amicable.homepage.dk/knwnc4.htm).
201 are known in 2012.
210 were known in April 2013. - Michel Marcus, Nov 10 2013
From Amiram Eldar, Mar 24 2024: (Start)
The terms were found by:
  a(1)-a(2) - Kenneth Dudley Fryer in 1965 (Honsberger, 1970; see also A072892)
  a(3)-a(7), a(9) - Cohen (1970)
  a(8) - Borho (1969)
  a(10)-a(13) - independently by Richard David (1972; Devitt et al., 1976, Guy 1977) and Steve C. Root (Beeler et al. 1972)
  a(14) - Steve C. Root in 1972
  a(15)-a(22) - Flammenkamp (1991)
  a(23)-a(24) - Moews and Moews (1991)
  a(25)-a(27) - Moews and Moews (1993)
(End)

Walter Borho, Über die Fixpunkte der k-fach iterierten Teilersummenfunktion, Mitt. Math. Gesellsch. Hamburg, Vol. 9, No. 5 (1969), pp. 34-48.
Richard David, Letter to D. H. Lehmer, February 25 , 1972.
John Stanley Devitt, Richard K. Guy, and John L. Selfridge, Third report on aliquot sequences, Proceedings of the Sixth Manitoba Conference on Numerical Mathematics, September 29 - October 2, 1976, Congressus Numerantium XVIII, University of Manitoba, Winnipeg, Manitoba, Utilitas Mathematics Publications, 1976, pp. 177-204.
Richard K. Guy, "Aliquot Sequences", in: Hans Zassenhaus (ed.), Number Theory and Algebra: Collected Papers Dedicated to Henry B. Mann, Arnold E. Ross, and Olga Taussky-Toddm, Academic Press Inc., 1977.
Ross Honsberger, Ingenuity in Mathematics, Mathematical Association of America, 1970.

Amiram Eldar, Table of n, a(n) for n = 1..48
Michael Beeler, R. William Gosper and Richard C. Schroeppel, HAKMEM, MIT Artificial Intelligence Laboratory report AIM-239, Feb. 29, 1972, Item 61; HTML transcription.
Karsten Blankenagel, Walter Borho, and Axel vom Stein, New amicable four-cycles, Math. Comp., Vol. 72, No. 244 (2003), pp. 2071-2076.
Karsten Blankenagel and Walter Borho, New amicable four-cycles II, International Journal of Mathematics and Scientific Computing, Vol. 5, No. 1 (2015), pp. 49-51.
Henri Cohen, On amicable and sociable numbers, Math. Comp., Vol. 24, No. 110 (1970), pp. 423-429.
Achim Flammenkamp, New sociable numbers, Mathematics of Computation, Vol. 56, No. 194 (1991), pp. 871-873.
Richard K. Guy and John L. Selfridge, Interim report on aliquot series, pp. 557-580 of Proceedings Manitoba Conference on Numerical Mathematics. University of Manitoba, Winnipeg, Oct 1971.
David Moews, A list of currently known aliquot cycles of length greater than 2.
David Moews and Paul C. Moews, Aliquot cycles below 10^10, Mathematics of Computation, Volume 57, No. 196 (1991), pp. 849-855.
David Moews and Paul C. Moews, A search for aliquot cycles and amicable pairs, Mathematics of computation, Vol. 61, No. 204 (1993), pp. 935-938.
Jan Munch Pedersen, Known Sociable Numbers of order four, Tables of Aliquot Cycles.
Eric Weisstein's World of Mathematics, Sociable Numbers.
Wikipedia, Sociable number.

Cf. A003023, A003416, A052470, A072892.

a(22)-a(24) from Flammenkamp (1991) and Moews and Moews (1991) added by Amiram Eldar, Mar 24 2024

A072890 The 28-cycle of the n => sigma(n)-n process, where sigma(n) is the sum of divisors of n (A000203).

Original entry on oeis.org

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, 14316

Offset: 1

Miklos Kristof, Jul 29 2002

Called a "sociable" chain.

One of the two aliquot cycles of length greater than 2 that were discovered by Belgian mathematician Paul Poulet (1887-1946) in 1918 (the second is A072891). They were the only known such cycles until 1965 (see A072892). - Amiram Eldar, Mar 24 2024

Albert H. Beiler, Recreations in the Theory of Numbers: The Queen of Mathematics Entertains, New York: Dover Publications, 1964, Chapter IV, pp. 28-29.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B7, p. 95.
C. Stanley Ogilvy, Tomorrow's math, unsolved problems for the amateur,Oxford University Press, 2nd ed., 1972, p. 113.
Paul Poulet, La chasse aux nombres I: Parfaits, amiables et extensions, Bruxelles: Stevens, 1929.

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.
Leonard Eugene Dickson, History of the Theory of Numbers, Vol. I: Divisibility and Primality, Washington, Carnegie Institution of Washington, 1919, p. 50.
Paul Poulet, Query 4865, L'Intermédiaire des Mathématiciens, Vol. 25 (1918), pp. 100-101.
Eric Weisstein's World of Mathematics, Sociable Numbers.
Wikipedia, Sociable number.

Cf. A000203, A001065, A072891, A072892.

NestList[DivisorSigma[1,#]-#&,14316,28] (* Harvey P. Dale, Oct 27 2013 *)

a(28+n) = a(n).

A072891 The 5-cycle of the n => sigma(n)-n process, where sigma(n) is the sum of divisors of n (A000203).

Original entry on oeis.org

12496, 14288, 15472, 14536, 14264, 12496

Offset: 1

Miklos Kristof, Jul 29 2002

Called a "sociable" chain.

One of the two aliquot cycles of length greater than 2 that were discovered by Belgian mathematician Paul Poulet (1887-1946) in 1918 (the second is A072890). They were the only known such cycles until 1965 (see A072892). - Amiram Eldar, Mar 24 2024

Albert H. Beiler, Recreations in the Theory of Numbers: The Queen of Mathematics Entertains, New York: Dover Publications, 1964, Chapter IV, p. 28.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B7, p. 95.
Paul Poulet, La chasse aux nombres I: Parfaits, amiables et extensions, Bruxelles: Stevens, 1929.

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.
Leonard Eugene Dickson, History of the Theory of Numbers, Vol. I: Divisibility and Primality, Washington, Carnegie Institution of Washington, 1919, p. 50.
Paul Poulet, Query 4865, L'Intermédiaire des Mathématiciens, Vol. 25 (1918), pp. 100-101.
Eric Weisstein's World of Mathematics, Sociable Numbers.
Wikipedia, Sociable number.

Cf. A000203, A001065, A003416, A072890, A072892.

NestWhileList[DivisorSigma[1, #] - # &, 12496, UnsameQ, All] (* Amiram Eldar, Mar 24 2024 *)

a(5+n) = a(n).

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

Eric Chen, Sep 13 2021

nonn

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).

			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;
...

Eric Chen, Table of n, a(n) for n = 0..2703 (all currently known terms, after the b-file of A008892)
Eric Weisstein's World of Mathematics, Aliquot sequence
Eric Weisstein's World of Mathematics, Catalan's Aliquot Sequence Conjecture
Wikipedia, Aliquot sequence

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).

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))

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A090615 Smallest member of sociable quadruples.

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A072890 The 28-cycle of the n => sigma(n)-n process, where sigma(n) is the sum of divisors of n (A000203).

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A072891 The 5-cycle of the n => sigma(n)-n process, where sigma(n) is the sum of divisors of n (A000203).

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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.

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