A063919 -id:A063919 - cp's OEIS frontend

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

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 *)

A290480 Product of proper unitary divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 10, 1, 12, 1, 14, 15, 1, 1, 18, 1, 20, 21, 22, 1, 24, 1, 26, 1, 28, 1, 27000, 1, 1, 33, 34, 35, 36, 1, 38, 39, 40, 1, 74088, 1, 44, 45, 46, 1, 48, 1, 50, 51, 52, 1, 54, 55, 56, 57, 58, 1, 216000, 1, 62, 63, 1, 65, 287496, 1, 68, 69, 343000, 1, 72, 1, 74, 75, 76, 77, 474552, 1, 80

Offset: 1

Ilya Gutkovskiy, Aug 03 2017

nonn

			a(12) = 12 because 12 has 6 divisors {1, 2, 3, 4, 6, 12} among which 3 are proper unitary {1, 3, 4} and 1*3*4 = 12.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A001221, A007774 (fixed points), A007955, A007956, A034460, A061537, A062758, A063919, A078599, A087652, A126192, A136655, A183091.

with(numtheory):
a:= n-> mul(d, d=select(x-> igcd(x, n/x)=1, divisors(n) minus {n})):
seq(a(n), n=1..80);  # Alois P. Heinz, Aug 03 2017

Table[Product[d, {d, Select[Divisors[n], GCD[#, n/#] == 1 &]}]/n, {n, 80}]
Table[n^(2^(PrimeNu[n] - 1) - 1), {n, 80}]

A290480(n) = if(1==n,n,n^(2^(omega(n)-1)-1)); \\ Antti Karttunen, Aug 06 2018

from sympy import divisors, gcd, prod
def a(n): return prod(d for d in divisors(n) if gcd(d, n//d) == 1)//n
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Aug 04 2017

a(n) = A061537(n)/n.
a(n) = n^(2^(omega(n)-1)-1), where omega() is the number of distinct primes dividing n (A001221).
a(n) = 1 if n is a prime power.

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) ).

A290490 Numbers k such that (sum of proper unitary divisors of k) > (sum of proper unitary divisors of m) for all m < k.

Original entry on oeis.org

1, 2, 6, 10, 14, 18, 22, 26, 30, 42, 60, 66, 78, 102, 114, 138, 150, 174, 186, 210, 330, 390, 462, 510, 546, 570, 690, 798, 858, 870, 930, 1050, 1110, 1218, 1230, 1290, 1410, 1470, 1590, 1722, 1770, 1830, 2010, 2130, 2190, 2310, 2730, 3570, 3990, 4290, 4830, 5610, 6006, 6090, 6510, 7410, 7590, 7770

Offset: 1

Ilya Gutkovskiy, Aug 03 2017

nonn

Numbers k such that A034460(k) > A034460(m) for all m < k.

Amiram Eldar, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Unitary Divisor
Index entries for sequences related to sums of divisors

Cf. A002182, A002093, A034090, A034448, A034460, A063919, A280013, A281782, A285614.

mx = -1; t = {}; Do[u = DivisorSum[n, # &, GCD[#, n/#] == 1 &] - n; If[u > mx, mx = u; AppendTo[t, n]], {n, 8000}]; t

sumud(n) = sumdiv(n, d, if (gcd(d, n/d)==1, d)) - n;
lista(nn) = {lasts = -1; for (n=1, nn, if ((news = sumud(n)) > lasts, print1(n, ", "); lasts = news););} \\ Michel Marcus, Aug 04 2017

A291320 Numbers k such that uphi(k) is equal to the sum of the proper unitary divisors of k.

Original entry on oeis.org

2, 600, 25584, 97464, 826560, 1249920, 50725248, 1372734720, 702637447680

Offset: 1

Altug Alkan, Aug 22 2017

Or numbers k such that usigma(k) - k = uphi(k) where usigma(k) = A034448(k) and uphi(k) = A047994(k).

a(10) > 10^13. - Giovanni Resta, May 12 2020

			600 = 2^3*3*5^2 is a term because usigma(600) - uphi(600) = (2^3+1)*(3+1)*(5^2+1) - (2^3-1)*(3-1)*(5^2-1) = 600.

Cf. A034448, A047994, A063919.

ok[n_] := Block[{p = Power @@@ FactorInteger[n]}, Times @@ (p + 1) == n + Times @@ (p - 1)]; Select[Range[2, 10^6], ok] (* Giovanni Resta, Aug 22 2017 *)

usigma(n) = sumdivmult(n, d, if(gcd(d, n/d)==1, d));
uphi(n) = my(f=factor(n)~); prod(i=1, #f, f[1, i]^f[2, i]-1);
isok(n) = usigma(n)-uphi(n)==n;

list(lim)=my(v=List()); forfactored(n=2,lim\1, if(sumdivmult(n, d, if(gcd(d, n[1]/d)==1, d))-prod(i=1, #n[2]~, n[2][i,1]^n[2][i,2]-1)==n[1], listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Aug 22 2017

a(8) from Giovanni Resta, Aug 22 2017

a(9) from Giovanni Resta, May 12 2020

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A290480 Product of proper unitary divisors of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A290490 Numbers k such that (sum of proper unitary divisors of k) > (sum of proper unitary divisors of m) for all m < k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A291320 Numbers k such that uphi(k) is equal to the sum of the proper unitary divisors of k.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

PARI

Extensions