A069059 -id:A069059 - cp's OEIS frontend

A009194 a(n) = gcd(n, sigma(n)).

1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 4, 1, 2, 3, 1, 1, 3, 1, 2, 1, 2, 1, 12, 1, 2, 1, 28, 1, 6, 1, 1, 3, 2, 1, 1, 1, 2, 1, 10, 1, 6, 1, 4, 3, 2, 1, 4, 1, 1, 3, 2, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 1, 1, 1, 6, 1, 2, 3, 2, 1, 3, 1, 2, 1, 4, 1, 6, 1, 2, 1, 2, 1, 28, 1, 2, 3, 4, 1, 18, 7, 4, 1, 2, 5, 12, 1, 1, 3, 1, 1, 6, 1, 2

Offset: 1

David W. Wilson

nonn

LCM of common divisors of n and sigma(n). It equals n if n is multiply perfect (A007691). - Labos Elemer, Aug 14 2002

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
P. Pollack, On the greatest common divisor of a number and its sum of divisors, Michigan Math. J. Volume 60, Issue 1 (2011), 199-214.

Cf. A000203, A003624, A007691, A014567, A017666, A063906, A069059, A073802, A082062, A179931, A205523, A216793 (positions of records), A234367, A249917.

Cf. also A009191, A009205, A009242, A274382, A286591, A286594.

a009194 n = gcd (a000203 n) n  -- Reinhard Zumkeller, Mar 23 2013

Table[GCD[n,DivisorSigma[1,n]],{n,110}] (* Harvey P. Dale, Aug 23 2015 *)

a(n) = gcd(n, sigma(n)); \\ Michel Marcus, Oct 23 2013

A000005(a(n)) = A073802(n). - Reinhard Zumkeller, Mar 12 2010
A006530(a(n)) = A082062(n). - Reinhard Zumkeller, Jul 10 2011
a(A014567(n)) = 1; A069059(a(n)) > 1. - Reinhard Zumkeller, Mar 23 2013
a(n) = n/A017666(n). - Antti Karttunen, May 22 2017

A014567 Numbers k such that k and sigma(k) are relatively prime, where sigma(k) = sum of divisors of k (A000203).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 35, 36, 37, 39, 41, 43, 47, 49, 50, 53, 55, 57, 59, 61, 63, 64, 65, 67, 71, 73, 75, 77, 79, 81, 83, 85, 89, 93, 97, 98, 100, 101, 103, 107, 109, 111, 113, 115, 119, 121, 125, 127, 128, 129, 131, 133

Offset: 1

Eric W. Weisstein

Related to "solitary numbers": n is solitary if there is no other integer m such that sigma(m)/m = sigma(n)/n.
It is easy to show that if n and sigma(n) are relatively prime then n is solitary. But the converse is not true; for example, 18, 45, 48 and 52 are solitary. Probably also 10, 14, 15, 20, 22 and many others are solitary, but I do not think that will ever be proved. - Dean Hickerson
From Daniel Forgues, Jun 23 2009: (Start)
Union of unit, primes and Duffinian numbers.
Duffinian numbers (A003624) are the composite numbers (including, among others, the proper prime powers) for which (n, sigma(n)) = 1. (End)
A009194(a(n)) = 1. - Reinhard Zumkeller, Mar 23 2013
These numbers satisfy (denominator of sigma(n)/n) = n. - Michel Marcus, Oct 27 2013
The asymptotic density of this sequence is 0 (Dressler, 1974; Luca, 2007). - Amiram Eldar, Jul 23 2020
If m*n is in this sequence and gcd(m,n) = 1, then m and n are both in this sequence. - Jianing Song, Aug 07 2022

			sigma(21) = 1 + 3 + 7 + 21 = 32 is relatively prime to 21, so 21 is in the sequence.

T. D. Noe, Table of n, a(n) for n = 1..1000
C. W. Anderson and D. Hickerson, Problem 6020: Friendly Integers, Amer. Math. Monthly 84, 65-66, 1977.
Robert E. Dressler, On a theorem of Niven, Canadian Mathematical Bulletin, Vol. 17, No. 1 (1974), pp. 109-110.
Andrew Feist, Fun with the sigma(n) function, Missouri Journal of Mathematical Sciences 15:3 (2003), pp. 173-177.
P. A. Loomis, New families of solitary numbers, J. Algebra and Applications, 14 (No. 9, 2015), #1540004 (6 pages).
Florian Luca, On the densities of some subsets of integers, Missouri Journal of Mathematical Sciences 19:3 (2007), pp. 167-170.
Eric Weisstein's World of Mathematics, Solitary Number.

Cf. A003624.
Cf. A069059 (complement).
Includes A000961 as a subsequence.

a014567 n = a014567_list !! (n-1)
a014567_list = filter ((== 1) . a009194) [1..]
-- Reinhard Zumkeller, Mar 23 2013

lst={};Do[d=DivisorSigma[1, n];If[GCD[d, n]==1, AppendTo[lst, n]], {n, 6!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 01 2008 *)
Select[Range[150],CoprimeQ[#,DivisorSigma[1,#]]&] (* Harvey P. Dale, Jan 23 2015 *)

is(n)=gcd(n,sigma(n))==1 \\ Charles R Greathouse IV, Feb 13 2013

from math import gcd
from sympy import divisor_sigma
def ok(n): d = divisor_sigma(n, 1); return gcd(n, d) == 1
print([k for k in range(1, 134) if ok(k)]) # Michael S. Branicky, Mar 28 2022

a(n) << n log n. Can this be improved? - Charles R Greathouse IV, Feb 13 2013

a(n) >> n log log log n, see Luca. - Charles R Greathouse IV, Feb 17 2014

More terms from Labos Elemer

A372566 Numbers k such that k, sigma(k) and A003961(k) have a common divisor larger than 1, where A003961(n) is fully multiplicative function with a(prime(i)) = prime(i+1).

Original entry on oeis.org

6, 18, 24, 30, 42, 54, 60, 66, 72, 78, 90, 96, 102, 114, 120, 126, 132, 135, 138, 140, 150, 162, 168, 174, 180, 186, 198, 204, 210, 216, 222, 234, 240, 246, 258, 264, 270, 276, 282, 285, 288, 294, 306, 312, 318, 330, 342, 348, 354, 360, 366, 378, 384, 390, 396, 402, 408, 414, 420, 426, 435, 438, 450, 455, 456, 462

Offset: 1

Antti Karttunen, May 19 2024

nonn

			24 = 2^3 * 3, sigma(24) = 60 = 2^2 * 3 * 5, and A003961(24) = 135 = 3^3 * 5, have 3 as their common divisor, therefore 24 is present in this sequence.

Cf. A000203, A003961.
Positions of terms > 1 in A372565.
Subsequence of each of the following sequences: A069059, A104210, A349166, A379477.
Cf. A372567 (odd terms), A379475 (characteristic function).

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA372566(n) = (1A003961(n)]));

A234534 Terms of the cycles reached after iterations of numerator(sigma(n)/n) = A017665(n).

Original entry on oeis.org

1, 8, 15, 127, 128, 144, 255, 403, 448, 512, 1023, 29127, 47360

Offset: 1

Michel Marcus, Dec 27 2013

If all integers were in A014567, then this sequence would not exist and we would be looking at A216200; but some are in A069059, allowing the trajectories of A017665 to go down.
The term of the sequence correspond to the 5 cycles: [1], [15, 8], [448, 127, 128, 255, 144, 403], [1023, 512], [47360, 29127].
Are there some starting x's whose fate will remain unknown, like 276 for A098007?
Are there other cycles to be found?
No other cycles found with largest member less than 10^9.
There are no other cycles with the smallest member < 10^11. All numbers < 10^11 reach one of the five known cycles. - Donovan Johnson, Jan 07 2014

			Obviously 1 is a fixed point for A017665, so 1 is in the sequence.
A017665(8) = 15 and A017665(15) = 8, so both 8 and 15 are in the sequence.

Cf. A000203, A017665.

iscycle(v, nextn) = {for (i=1, #v, if (v[i] == nextn, return (1););); return (0);}
fcycle(n, known) = {v = vector(1); v[1] = n; first = n; while ((nextn = numerator(sigma(n)/n)) <= first, if (vecsearch(known, nextn), return([])); if (iscycle(v, nextn), return (v)); v = concat(v, nextn); n = nextn;); return ([]);}
fcycles(na, nb) = {known = []; known = [1, 8, 127, 512, 29127]; for (n = na, nb, v = fcycle(n, known); if (#v, print(v, ", "); return();););} \\ use empty vector for known to search for cycles from start; when a new cycle is found, insert its smallest term to vector known.

Missing terms 512 and 1023 noticed by Donovan Johnson added by Michel Marcus, Jan 02 2014

A262432 Regular triangle read by rows: T(n, k) gives the number of times that the denominator of sigma(x,-1) (A017666) is equal to k when x goes from 1 to n.

Original entry on oeis.org

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

Offset: 1

Michel Marcus, Sep 22 2015

The sum of terms of the n-th row is n.
T(n, n) = 1 when n is in A014567.
T(n, n) = 0 when n is in A069059.
T(n, 1) increases when n is a multiperfect number A007691.
For a given k, the first index n for which T(n,k) = 1 is A162657(k).

			The first 6 terms of A017666 are 1, 2, 3, 4, 5, 1 where 1 appears twice, 2 to 5 appear once and 6 is absent; giving the 6th row: 2, 1, 1, 1, 1, 0.
Triangle starts
1;
1, 1;
1, 1, 1;
1, 1, 1, 1;
1, 1, 1, 1, 1;
2, 1, 1, 1, 1, 0;
2, 1, 1, 1, 1, 0, 1;
2, 1, 1, 1, 1, 0, 1, 1;
2, 1, 1, 1, 1, 0, 1, 1, 1;
2, 1, 1, 1, 2, 0, 1, 1, 1, 0;
...

Michel Marcus, Table of n, a(n) for n = 1..5050

Cf. A007691, A014567, A017666, A069059, A162657.

Table[Length@ Select[Range@ n, Denominator[DivisorSigma[-1, #]] == k &], {n, 13}, {k, n}] // Flatten (* Michael De Vlieger, Sep 22 2015 *)

tabl(nn) = {vds = vector(nn, n, denominator(sigma(n,-1))); for (n=1, nn, vin = vector(n, k, vds[k]); rown = vector(n, k, #select(x->x==k, vin)); for(k=1, n, print1(rown[k], ", ")); print(););}

A234367 Numbers such that gcd(sigma(n), n) != 1 and numerator(sigma(n)/n) > n.

Original entry on oeis.org

20, 70, 80, 104, 160, 208, 272, 320, 350, 416, 464, 490, 544, 550, 650, 770, 832, 928, 1088, 1184, 1190, 1280, 1300, 1312, 1332, 1430, 1610, 1664, 1696, 1700, 1750, 1856, 1870, 1952, 2170, 2196, 2210, 2368, 2420, 2530, 2560, 2590, 2624, 2628, 2750, 2990, 3010

Offset: 1

Michel Marcus, Dec 28 2013

For numbers in A014567, we have A017665(n) = numerator(sigma(n)/n) = sigma(n) = A000203(n), so A017665(n) > n.
For numbers in A069059, since both terms of the fraction are divisible by their GCD, A009194(n), we will have A017665(n) < A000203(n).
Here we are interested in terms of A069059 for which we still have A017665(n) > n, despite the division by the GCD.
Numbers such that sigma(n)/n > gcd(sigma(n), n) > 1. - Charlie Neder, Sep 08 2018

			For n=20, we have A000203(20) = sigma(20) = 42, and since gcd(42, 20) != 1, then A017665(20) = numerator(42/20) = numerator(21/10) = 21 < sigma(20), but still A017665(20) > 20.

Robert G. Wilson v, Table of n, a(n) for n = 1..10519 (First 6892 terms from Charlie Neder).

Cf. A000203, A009194, A014567, A017665, A069059.

gnQ[n_]:=Module[{s=DivisorSigma[1,n]},GCD[s,n]!=1&&Numerator[s/n]>n]; Select[ Range[ 3100],gnQ] (* Harvey P. Dale, Jan 03 2018 *)

isok(n) = (gcd(sigma(n), n) != 1) && (numerator(sigma(n)/n) > n);

A259917 All friendly numbers, with smallest member of each club listed just before the second-smallest one.

Original entry on oeis.org

6, 28, 30, 140, 80, 200, 40, 224, 12, 234, 84, 270, 66, 308, 78, 364, 102, 476, 496, 114, 532, 240, 600, 138, 644, 120, 672, 150, 700, 174, 812, 135, 819, 186, 868, 864, 936, 222, 1036, 246, 1148, 60, 1170, 258, 1204, 282, 1316, 560, 1400, 318, 1484, 1488, 330

Offset: 1

Jeppe Stig Nielsen, Jul 08 2015

nonn

Run through all natural numbers i = 1, 2, 3, ... in order, and record for each the abundancy index sigma(i)/i. When we reach an abundancy that has been seen before, output first the "old" number which had that abundancy (unless that number has already been output earlier), and output secondly the current i.
By construction, no number can occur more than once in the sequence.
Friendly numbers that are not smallest in their club, appear in increasing order. Friendly numbers that are smallest in their club, appear just before the second-smallest member.
If we were to "forget" to output the smallest member in each club, we would get instead A095301.
Oppositely, if we output the smallest members only, we get instead A259918.
It is not known whether the number 10 belongs to this sequence.

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..6000
Doyon Kim, Friends of 12, arXiv:1608.06834 [math.HO], 2016.
Jeffrey Ward, Does Ten Have a Friend?, arXiv:0806.1001 [math.NT], 2008.
Eric Weisstein's World of Mathematics, Friendly Pair
Wikipedia, Friendly number.

Cf. A050973, A259918, A095301, A014567, A074902.

Terms form a subset of A069059.

known=List(); for(i=1,10^5,a=sigma(i)/i; match=0; for(j=1,#known,if(known[j][1]==a,match=j;break())); if(match,old=known[match][2]; if(old,print1(old,", "); known[match]=[a,0]); print(i,","),listput(known,[a,i])))

A248021 Perfect powers which are not coprime to the sum of their divisors.

Original entry on oeis.org

196, 216, 441, 1000, 1521, 1728, 1764, 2744, 3249, 3375, 3969, 4900, 5832, 5929, 6084, 7056, 7776, 8000, 8649, 9604, 9801, 10648, 11025, 12321, 12544, 12996, 13689, 13824, 15376, 15876, 16641, 17576, 17689, 21952, 23716, 24025, 24336, 25921, 27000, 28224, 29241, 33124, 33489, 34596

Offset: 1

Robert G. Wilson v, Sep 29 2014

Intersection of A001597 and A069059. - Michel Marcus, Oct 25 2014

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001597 (perfect powers), A069059 (gcd(n, sigma(n)) != 1).

perfectPowerQ[n_] := n == 1 || GCD @@ FactorInteger[n][[All, 2]] > 1; Select[ Range@ 35000, perfectPowerQ[ #] && GCD[#, DivisorSigma[1, #]] > 1 &]

for(n=1, 1e5, if(ispower(n), if(gcd(n, sigma(n)) > 1, print1(n, ", ")))) \\ Felix Fröhlich, Oct 25 2014

A273157 Numbers which have nonpositive entries in the difference table of their divisors (complement of A273130).

Original entry on oeis.org

6, 10, 12, 14, 15, 18, 20, 22, 24, 26, 28, 30, 34, 35, 36, 38, 40, 42, 44, 45, 46, 48, 50, 52, 54, 56, 58, 60, 62, 63, 66, 68, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 86, 88, 90, 91, 92, 94, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 112, 114, 116, 117

Offset: 1

Peter Luschny, May 16 2016

nonn

Primorial numbers (A002110) greater than 2 are in this sequence.

			30 is in this sequence because the difference table of the divisors of 30 is:
[1, 2, 3, 5, 6, 10, 15, 30]
[1, 1, 2, 1, 4, 5, 15]
[0, 1, -1, 3, 1, 10]
[1, -2, 4, -2, 9]
[-3, 6, -6, 11]
[9, -12, 17]
[-21, 29]
[50]

Cf. A069059, A187202, A273102, A273103, A273109, A273130 (complement).

def nsf(z):
    D = divisors(z)
    T = matrix(ZZ, len(D))
    for m, d in enumerate(D):
        T[0, m] = d
        for k in range(m-1, -1, -1) :
            T[m-k, k] = T[m-k-1, k+1] - T[m-k-1, k]
            if T[m-k, k] <= 0: return True
    return False
print([n for n in range(1, 100) if nsf(n)])

cp's OEIS Frontend

A009194 a(n) = gcd(n, sigma(n)).

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A014567 Numbers k such that k and sigma(k) are relatively prime, where sigma(k) = sum of divisors of k (A000203).

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A372566 Numbers k such that k, sigma(k) and A003961(k) have a common divisor larger than 1, where A003961(n) is fully multiplicative function with a(prime(i)) = prime(i+1).

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A234534 Terms of the cycles reached after iterations of numerator(sigma(n)/n) = A017665(n).

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A262432 Regular triangle read by rows: T(n, k) gives the number of times that the denominator of sigma(x,-1) (A017666) is equal to k when x goes from 1 to n.

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A234367 Numbers such that gcd(sigma(n), n) != 1 and numerator(sigma(n)/n) > n.

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A259917 All friendly numbers, with smallest member of each club listed just before the second-smallest one.

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A248021 Perfect powers which are not coprime to the sum of their divisors.

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