A327157 -id:A327157 - cp's OEIS frontend

A002827 Unitary perfect numbers: numbers k such that usigma(k) - k = k.

6, 60, 90, 87360, 146361946186458562560000

Offset: 1

d is a unitary divisor of k if gcd(d,k/d)=1; usigma(k) is their sum (A034448).
The prime factors of a unitary perfect number (A002827) are the Higgs primes (A057447). - Paul Muljadi, Oct 10 2005
It is not known if a(6) exists. - N. J. A. Sloane, Jul 27 2015
Frei proved that if there is a unitary perfect number that is not divisible by 3, then it is divisible by 2^m with m >= 144, it has at least 144 distinct odd prime factors, and it is larger than 10^440. - Amiram Eldar, Mar 05 2019
Conjecture: Subsequence of A083207 (Zumkeller numbers). Verified for all present terms. - Ivan N. Ianakiev, Jan 20 2020

			Unitary divisors of 60 are 1,4,3,5,12,20,15,60, with sum 120 = 2*60.
146361946186458562560000 = 2^18 * 3 * 5^4 * 7 * 11 * 13 * 19 * 37 * 79 * 109 * 157 * 313.

R. K. Guy, Unsolved Problems in Number Theory, Sect. B3.
F. Le Lionnais, Les Nombres Remarquables. Paris: Hermann, p. 59, 1983.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section III.45.1.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 147-148.

H. A. M. Frei, Über unitar perfekte Zahlen, Elemente der Mathematik, Vol. 33, No. 4 (1978), pp. 95-96.
Takeshi Goto, Upper Bounds for Unitary Perfect Numbers and Unitary Harmonic Numbers, Rocky Mountain Journal of Mathematics, Vol. 37, No. 5 (2007), pp. 1557-1576.
A. V. Lelechenko, The Quest for the Generalized Perfect Numbers, in Theoretical and Applied Aspects of Cybernetics, TAAC 2014, Kiev.
M. V. Subbarao, Letter to N. J. A. Sloane, Feb 18 1974
M. V. Subbarao, T. J. Cook, R. S. Newberry and J. M. Weber, On unitary perfect numbers, Delta, 3 (No. 1, 1972), 22-26.
G. Villemin's Almanac of Numbers, Nombres Unitairement Parfaits
C. R. Wall, Letter to P. Hagis, Jr., Jan 13 1972
C. R. Wall, The fifth unitary perfect number, Canad. Math. Bull., 18 (1975), 115-122.
C. R. Wall, On the largest odd component of a unitary perfect number, Fib. Quart., 25 (1987), 312-316.
Eric Weisstein's World of Mathematics, Unitary Perfect Number.
Wikipedia, Unitary perfect number

Cf. A034460, A034448, A057447.
Subsequence of the following sequences: A003062, A290466 (seemingly), A293188, A327157, A327158.
Gives the positions of ones in A327159.

usnQ[n_]:=Total[Select[Divisors[n],GCD[#,n/#]==1&]]==2n; Select[Range[ 90000],usnQ] (* This will generate the first four terms of the sequence; it would take a very long time to attempt to generate the fifth term. *) (* Harvey P. Dale, Nov 14 2012 *)

is(n)=sumdivmult(n, d, if(gcd(d, n/d)==1, d))==2*n \\ Charles R Greathouse IV, Aug 01 2016

If m is a term and omega(m) = A001221(m) = k, then m < 2^(2^k) (Goto, 2007). - Amiram Eldar, Jun 06 2020

A327159 Size of the cycle containing n in the map x -> usigma(x)-x or 0 if n is not a member of any finite cycle. Here usigma is the sum of unitary divisors of n (A034448).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0

Offset: 1

Antti Karttunen, Aug 28 2019

nonn

			Because A034460(6) = 6, a(6) = 1.
Because A034460(30) = 42, A034460(42) = 54, A034460(54) = 30, a(30) = a(42) = a(54) = 3.
Because A034460(90) = 90, a(90) = 1. Because A034460(78) = 90, a(78) = 0, as even though 78 ends into a cycle of one, it itself is not a part of that cycle.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A002827 (positions of ones), A063991 (of 2's), A319902 (of 4's), A097024 (of 5's), A319917 (of 6's), A319937 (of 10's), A097030 (of 14's), A327157 (of all nonzero terms).

Cf. also A034448, A034460, A097031, A318880, A318882, A327162.

a034460[0] = 0; (* avoids dividing by 0 when an iteration reaches 0 *)
a034460[n_] := Total[Select[Divisors[n], GCD[#, n/#] == 1 &]] - n /; n > 0
cycleL[k_] := Module[{nL=NestWhileList[a034460, k, UnsameQ, All]}, If[k==Last[nL], Length[nL]-1, 0]]
a327159[n_] := Map[cycleL, Range[n]]
a327159[120] (* Hartmut F. W. Hoft, Feb 04 2024 *)

A034460(n) = (sumdivmult(n, d, if(gcd(d, n/d)==1, d))-n); \\ From A034460
A327159(n,orgn=n,xs=Set([])) = if(1==n,0,if(vecsearch(xs,n), if(n==orgn,length(xs),0), xs = setunion([n],xs); A327159(A034460(n),orgn,xs)));

A336216 Irregular triangle of cycles of purely periodic unitary sigma aliquot sequences with their smallest member as starting number, read by rows.

Original entry on oeis.org

6, 30, 42, 54, 60, 90, 114, 126, 1140, 1260, 1482, 1878, 1890, 2142, 2178, 2418, 2958, 3522, 3534, 4146, 4158, 3906, 3774, 4434, 4446, 3954, 3966, 3978, 3582, 18018, 22302, 24180, 29580, 35220, 35340, 41460, 41580, 39060, 37740, 44340, 44460, 39540, 39660, 39780, 35820, 32130, 40446

Offset: 1

Hartmut F. W. Hoft, Jul 12 2020

For the definition of unitary divisors see A034448. This sequence is a permutation of A327157; the starting numbers of successive cycles are in increasing order; the numbers in a cycle are kept in the order of the iteration with the smallest number in the cycle as the starting number. In order to be consistent with A327157 the terminal 1-cycle consisting of 1 is not included in the sequence.
Sequence A336218 gives the cycle lengths, therefore the start of the k-th cycle in this sequence is at index 1 + Sum_{i=1..k-1} A336218(i). Sequence A336219 is the first column of the triangle.
From the formula of Vladeta Jovovic in A034448, it follows that all unitary aliquot sequences, and hence cycles, contain only odd numbers or only even numbers (except for the possible terminal 1). The table of Antti Karttunen in the link of A327157 includes just 2 odd cycles, the 2-cycles: 8619765, 9627915 and 17257695, 17578785.

			The first cycle of size 14 starting at position 16 is: 2418, 2958, 3522, 3534, 4146, 4158, 3906, 3774, 4434, 4446, 3954, 3966, 3978, 3582. Its 7th element is the first number in this sequence smaller than its predecessor.
Irregular triangle of cycles:
6
30    42   54
60
90
114   126
1140  1260
1482  1878 1890 2142 2178
2418  2958 3522 3534 4146 4158 3906 3774 4434 4446 3954 3966 3978 3582
18018 22302
...

Cf. A034448, A063919, A327157, A336218, A336219.

a063919[1] = 1; a063919[n_] := Total[Select[Divisors[n], GCD[#, n/#]==1&]] - n/;n>1 (* Jean-François Alcover *)
aliquotSequence[n_] := NestWhileList[a063919, n, UnsameQ, All]
a336216[n_] := Module[{list={}, listS={}, i, seq, seqS}, For[i=2, i<=n, i++, seq=aliquotSequence[i]; If[First[seq]==Last[seq], seqS=Sort[Most[seq]]; If[!MemberQ[listS, seqS], AppendTo[listS, seqS]; AppendTo[list, Most[seq]]]]]; list] (* list of cycles *)
Flatten[a336216[35000]] (* data - first 11 rows of triangle *)

A336218 a(n) is the length of the cycle of the purely periodic unitary sigma aliquot cycles listed in A336216.

Original entry on oeis.org

1, 3, 1, 1, 2, 2, 5, 14, 2, 14, 2, 14, 2, 2, 2, 1, 2, 2, 2, 2, 4, 2, 2, 4, 2, 65, 2, 2, 2, 6, 25, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 39, 26, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Hartmut F. W. Hoft, Jul 12 2020

nonn

For the definition of unitary divisors see A034448. This sequence has been calculated from the table of 440 lines in the link of A327157 of Antti Karttunen. That table contains the numbers in 122 complete cycles and in 5 incomplete 2-cycles with values larger than number 27287260 in line 440 which results in the cumulative sum of 445 for the data listed in this sequence.

			The first cycle of size 14 = a(8) starts at position: 1 + (1+3+1+1+2+2+5) = 16 in A336216.

Hartmut F. W. Hoft, Plot of 25th through 52nd pure unitary sigma aliquot cycles

Cf. A034448, A327157, A336216, A336219.

A336219 a(n) is the smallest member of the n-th purely periodic unitary sigma aliquot cycle listed in A336216.

Original entry on oeis.org

6, 30, 60, 90, 114, 1140, 1482, 2418, 18018, 24180, 32130, 35238, 44772, 56430, 67158, 87360, 142310, 180180, 197340, 241110, 263820, 296010, 308220, 395730, 462330, 473298, 591030, 669900, 671580, 698130, 763620, 785148, 815100, 1004850, 1077890, 1080150, 1156870, 1177722

Offset: 1

Hartmut F. W. Hoft, Jul 12 2020

nonn

This is the first column of the irregular triangle in A336216.
From the formula of Vladeta Jovovic in A034448 we get for an even number n not divisible by 4 and odd prime p: usigma(2^m * p * n) = (2^(m+1) + 1) * (p + 1) * usigma(n) / 3  so that usigma(2^m * p * n) = (2^m * p * n) * usigma(n) when 3* 2^m * p = (2^(m+1) + 1) * (p + 1), and consequently, p = (2^(m+1) + 1) / (2^m - 1), i.e. p = 5 for m = 1, and p = 3 for m = 2.
Therefore, if all members a_1, a_2, ... , a_k, a_1 of a cycle are even and not divisible by 4 and 5 then 10*a_1, 10*a_2, ... , 10*a_k, 10*a_1 form a cycle, and if all members a_1, a_2, ... , a_k, a_1 of a cycle are even and not divisible by 3 and 4 then 12*a_1, 12*a_2, ... , 12*a_k, 12*a_1 form a cycle.
If all members a_1, a_2, ... , a_k, a_1 of a cycle are odd, divisible by 3, but not divisible by 5 and 9 then 15*a_1, 15*a_2, ... , 15*a_k, 15*a_1 form a cycle. No such cycles exist in the current data up to 27287260.

			Start numbers of cycles related by a factor of 10 or 12, respectively:
10:  (6, 60), (114, 1140), (2418, 24180), (18018, 180180), (67158, 671580), (1177722, 1777220), ...
12:  (142310, 1707720), (1077890, 12934680), (1156870, 13882440), (1475810, 17709720), ...

Cf. A034448, A063919, A327157, A336216, A336218.

(* a336216 and support functions in A336216 *)
Map[First, a336216[100000]] (* a(1..16) *)

a(n) = A336216( 1 + Sum_{i=1..n-1} A336218(i) ).

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A002827 Unitary perfect numbers: numbers k such that usigma(k) - k = k.

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A327159 Size of the cycle containing n in the map x -> usigma(x)-x or 0 if n is not a member of any finite cycle. Here usigma is the sum of unitary divisors of n (A034448).

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A336216 Irregular triangle of cycles of purely periodic unitary sigma aliquot sequences with their smallest member as starting number, read by rows.

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A336218 a(n) is the length of the cycle of the purely periodic unitary sigma aliquot cycles listed in A336216.

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A336219 a(n) is the smallest member of the n-th purely periodic unitary sigma aliquot cycle listed in A336216.

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