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Search: a131884 -id:a131884
Displaying 1-6 of 6 results found. page 1
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A284279 Primitive terms of A131884. +20
0
276, 306, 396, 564, 660, 696, 780, 828, 888, 966, 996, 1074, 1086, 1098 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
In other words, terms of A131884 that cannot be written as k*t where k is integer, k > 1, t is a term of A131884.
LINKS
EXAMPLE
Since 276 is the first term of A131884, the number is in the sequence. But 131884(4) = 552 is not, because A131884(4) = 2 * A131884(1) = 2 * 276.
CROSSREFS
Cf. A131884.
KEYWORD
nonn,more
AUTHOR
Sergey Pavlov, Mar 24 2017
STATUS
approved
A063769 Aspiring numbers: numbers whose aliquot sequence terminates in a perfect number. +10
16
25, 95, 119, 143, 417, 445, 565, 608, 650, 652, 675, 685, 783, 790, 909, 913 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
There are many numbers whose aliquot sequences have not yet been completely computed, so this sequence is not fully known. In particular, 276 may, perhaps, be an element of this sequence, although this is very unlikely.
Numbers less than 1000 whose aliquot sequence is not known that could possibly be in this sequence are: 276, 306, 396, 552, 564, 660, 696, 780, 828, 888, 966, 996. - Robert Price, Jun 03 2013
REFERENCES
No number terminates at 28, the second perfect number.
LINKS
Eric Weisstein's World of Mathematics, Aspiring Number
EXAMPLE
The divisors of 95 less than itself are 1, 5 and 19. They sum to 25. The divisors of 25 less than itself are 1 and 5. They sum to 6, which is perfect.
MATHEMATICA
perfectQ[n_] := DivisorSigma[1, n] == 2*n; maxAliquot = 10^45; A131884 = {}; s[1] = 1; s[n_] := DivisorSigma[1, n] - n; selQ[n_ /; n <= 5] = False; selQ[n_] := NestWhile[s, n, If[{##}[[-1]] > maxAliquot, Print["A131884: ", n]; AppendTo[A131884, n]; False, Length[{##}] < 4 || {##}[[-4 ;; -3]] != {##}[[-2 ;; -1]]] &, All] // perfectQ; Reap[For[k = 1, k < 1000, k++, If[! perfectQ[k] && selQ[k], Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Nov 15 2013 *)
CROSSREFS
KEYWORD
hard,nice,nonn
AUTHOR
Tanya Khovanova and Alexey Radul, Aug 14 2001
EXTENSIONS
a(13)-a(16) from Robert Price, Jun 03 2013
STATUS
approved
A080907 Numbers whose aliquot sequence terminates in a 1. +10
14
1, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
All primes are in this set because s(p) = 1 for p prime. Perfect numbers are clearly not in this set. Neither are aspiring numbers (A063769), or numbers whose aliquot sequence is a cycle (such as 220 and 284).
There are some numbers whose aliquot sequences haven't been fully determined (such as 276).
LINKS
Jean-François Alcover, Table of n, a(n) for n = 1..1000
M. Benito, W. Creyaufmüller, J. L. Varona, and P. Zimmermann, Aliquot Sequence 3630 Ends After Reaching 100 Digits, Experimental Mathematics (2002), Volume 11, issue 2.
Markus Tervooren, Aliquot sequence of 840
Eric Weisstein's World of Mathematics, Aliquot Sequence
FORMULA
n is a member if n = 1 or s(n) is a member, where s(n) is the sum of the proper factors of n.
EXAMPLE
4 is in this set because its aliquot chain is 4->3->1. 6 is not in this set because it is perfect. 25 is not in this set because its aliquot chain is 25->6.
MATHEMATICA
maxAliquot = 10^50; A131884 = {}; s[1] = 1; s[n_] := DivisorSigma[1, n] - n; selQ[n_ /; n <= 5] = True; selQ[n_] := NestWhile[s, n, If[{##}[[-1]] > maxAliquot, Print["A131884: ", n]; AppendTo[A131884, n]; False, Length[{##}] < 4 || {##}[[-4 ;; -3]] != {##}[[-2 ;; -1]]] & , All] == 1; Select[Range[1, 1100], selQ] (* Jean-François Alcover, Nov 14 2013, updated Sep 10 2015 *)
CROSSREFS
Complement of A126016.
KEYWORD
nonn,nice
AUTHOR
Gabriel Cunningham (gcasey(AT)mit.edu), Mar 31 2003
EXTENSIONS
Edited by N. J. A. Sloane, Aug 14 2006
More terms from Franklin T. Adams-Watters, Dec 14 2006
The fact that 840 was missing from the sequence b-file was pointed out by Philip Turecek, Sep 10 2015
STATUS
approved
A126016 Numbers whose aliquot sequence does not terminate in 1. +10
6
6, 25, 28, 95, 119, 143, 220 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Sequence continues 276?, 284, 306?, 396?, 417, 445, 496, .... Because 276, 306 and 396 are all in the same family, either all 3 are present or none are. It is not known whether any aliquot sequence grows without bound; 276 is the smallest number for which this is unknown.
Additional tentative terms: 552, 562, 564, 565, 608, 650, 652, 660, 675, 685, 696, 780, 783, 790, 828, 840, 888, 909, 913, 966, 996, 1064, 1074, 1086, 1098, ... - Jean-François Alcover, Nov 14 2013
For additional terms, if the Goldbach Conjecture is assumed, take any odd term, subtract 1, and find two distinct primes that sum to it. For some numbers there will not be any pair of distinct primes. Multiply the two primes and the product is an element of the sequence. Note that this process does not work if the term - 1 is power of a prime. - Nathaniel J. Strout, Nov 25 2018
LINKS
Eric Weisstein's World of Mathematics, Aliquot Sequence
P. Zimmermann, Latest information
MATHEMATICA
maxAliquot = 10^45; A131884 = {}; s[1] = 1; s[n_] := DivisorSigma[1, n] - n; selQ[n_ /; n <= 5] = True; selQ[n_] := NestWhile[s, n, If[{##}[[-1]] > maxAliquot, Print["A131884: ", n]; AppendTo[A131884, n]; False, Length[{##}] < 4 || {##}[[-4 ;; -3]] != {##}[[-2 ;; -1]]] & , All] == 1; Reap[For[k = 1, k < 1100, k++, If[!selQ[k], Print[k]; Sow[k]]]][[2, 1]]
CROSSREFS
Complement of A080907. Includes A000396, A063990 and other sociable numbers, A063769, numbers whose aliquot sequence reaches a sociable number and numbers whose aliquot sequence grows without bound.
KEYWORD
hard,nonn
AUTHOR
STATUS
approved
A216072 Aliquot open end sequences which belong to distinct families. +10
3
276, 552, 564, 660, 966, 1074, 1134, 1464, 1476, 1488, 1512, 1560, 1578, 1632, 1734, 1920, 1992, 2232, 2340, 2360, 2484, 2514, 2664, 2712, 2982, 3270, 3366, 3408, 3432, 3564, 3678, 3774, 3876, 3906, 4116, 4224, 4290, 4350, 4380, 4788, 4800, 4842 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
These aliquot sequences are believed to grow forever without terminating in a prime or entering a cycle.
Sequence A131884 lists all the starting values of an aliquot sequence that lead to open-ending. It includes all values obtained by iterating from the starting values of this sequence. But this sequence lists only the values that are the lowest starting elements of open end aliquot sequences that are the part of different open-ending families. - V. Raman, Dec 08 2012
LINKS
Wolfgang Creyaufmüller, Aliquot Sequences
Paul Zimmermann, Aliquot Sequences.
CROSSREFS
KEYWORD
nonn
AUTHOR
V. Raman, Sep 01 2012
STATUS
approved
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. +10
1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
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).
LINKS
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
EXAMPLE
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;
...
PROG
(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))
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).
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).
KEYWORD
nonn
AUTHOR
Eric Chen, Sep 13 2021
STATUS
approved
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Last modified August 1 13:16 EDT 2024. Contains 374817 sequences. (Running on oeis4.)