A003624 -id:A003624 - cp's OEIS frontend

A373968 a(n) is the number of divisors of n that are Duffinian numbers (A003624).

0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 2, 1, 0, 0, 0, 4, 0, 0, 1, 3, 0, 0, 1, 2, 0, 1, 0, 1, 1, 0, 0, 3, 1, 2, 0, 1, 0, 2, 1, 2, 1, 0, 0, 1, 0, 0, 3, 5, 1, 0, 0, 1, 0, 1, 0, 4, 0, 0, 2, 1, 1, 1, 0, 3, 3, 0, 0, 2, 1, 0, 0, 2

Offset: 1

Marius A. Burtea, Jul 12 2024

nonn

			Since A003624(1) = 4 then a(1) = a(2) = a(3) = 0 and a(4) = 1.
a(8) = 2 because 8 has the divisors 4 = A003624(1) and 8 = A003624(2).

Cf. A000203, A003624, A009194.

f:=func; [#[d:d in Divisors(k)|f(d)]:k in [1..100]];

a[n_] := DivisorSum[n, 1 &, CompositeQ[#] && CoprimeQ[#, DivisorSigma[1, #]] &]; Array[a, 100] (* Amiram Eldar, Jul 19 2024 *)

a(p^k) = k - 1, for p prime and k >= 1.

A373892 a(n) is the smallest number that can be partitioned in exactly n ways as the sum of two Duffinian numbers (A003624).

Original entry on oeis.org

1, 8, 25, 43, 84, 71, 102, 160, 150, 219, 226, 196, 244, 350, 328, 300, 330, 354, 400, 386, 448, 408, 434, 390, 510, 536, 462, 546, 570, 624, 608, 740, 722, 690, 714, 770, 726, 660, 750, 804, 842, 858, 876, 870, 932, 914, 924, 840, 986, 1038, 966, 1108, 1050, 1056

Offset: 0

Marius A. Burtea, Jul 12 2024

nonn

			1 cannot be written as the sum of two Duffinian numbers, so a(0) = 1.
The numbers from 2 to 7 cannot be written as the sum of two Duffinian numbers and 8 = 4 + 4 = A003624(1) + A003624(1), so a(1) = 8.
25 = 4 + 21 = 9 + 16 and 4 = A003624(1), 9 = A003624(3), 16 = A003624(4), 21 = A003624(5) and the numbers 9 to 24 cannot be written in two ways as a sum of two Duffinian numbers. Thus a(2) = 25.

Cf. A003624.

f:=func; b:=[n: n in [1..2000] |f(n)]; a:=[]; for n in [0..60] do k:=1; while #RestrictedPartitions(k,2,Set(b)) ne n do k:=k+1; end while; Append(~a,k); end for; a;

dufQ[n_] := CompositeQ[n] && CoprimeQ[n, DivisorSigma[1, n]]; f[n_] := Sum[If[dufQ[k] && dufQ[n - k], 1, 0], {k, 1, Floor[n/2]}]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n] + 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[54, 2000] (* Amiram Eldar, Jul 19 2024 *)

A373969 The smallest number k whose divisors include exactly n Duffinian numbers (A003624).

Original entry on oeis.org

1, 4, 8, 16, 32, 64, 128, 256, 512, 576, 1152, 1600, 2304, 4608, 3600, 6300, 7200, 18900, 20736, 32725, 14400, 28800, 50400, 56700, 108900, 57600, 100800, 111321, 176400, 129600, 226800, 229075, 360000, 630000, 435600, 333963, 518400, 1374450, 871200, 1001889

Offset: 0

Marius A. Burtea, Jul 12 2024

nonn

Numbers of the form m = 2^(k+1), k >= 0, have exactly k divisors that are Duffinian numbers.

			Since A003624(1) = 4, a(0) = 1.
The numbers 2 and 3 have no divisors that are Duffinian numbers and 4 = A003624(1), so a(1) = 4.

Cf. A003624, A373968.

f:=func; a:=[]; for n in [0..38] do k:=1; while #[d:d in Divisors(k)|f(d)] ne n do  k:=k+1; end while; Append(~a,k); end for; a;

f[n_] := DivisorSum[n, 1 &, CompositeQ[#] && CoprimeQ[#, DivisorSigma[1, #]] &]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n] + 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[30, 10^7] (* Amiram Eldar, Jul 19 2024 *)

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

Original entry on oeis.org

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

A248020 Numbers which are coprime to the sum of their divisors, but are neither primes nor perfect powers.

Original entry on oeis.org

21, 35, 39, 50, 55, 57, 63, 65, 75, 77, 85, 93, 98, 111, 115, 119, 129, 133, 143, 155, 161, 171, 175, 183, 185, 187, 189, 201, 203, 205, 209, 215, 217, 219, 221, 235, 237, 242, 245, 247, 253, 259, 265, 275, 279, 291, 299, 301, 305, 309, 319, 323, 325, 327, 329, 333, 335, 338, 341

Offset: 1

Robert G. Wilson v, Sep 29 2014

Intersection of A003624 and A106543. - Michel Marcus, Sep 30 2014

Duffinian numbers (A003624) which are not perfect powers (A001597). - Robert G. Wilson v, Oct 02 2014

			21 is in the sequence since it is neither a prime nor a powerful number and its divisors 1, 3, 7, and 21 sum to 32, which is coprime to 21.
50 is in the sequence since it is neither a prime nor a powerful number and its divisors 1, 2, 5, 10, 25, and 50 sum to 93, which is coprime to 50.

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

perfectPowerQ[n_] := n == 1 || GCD @@ FactorInteger[n][[All, 2]] > 1; Select[ Range@ 350, !PrimeQ[ #] && GCD[#, DivisorSigma[1, #]] == 1 && !perfectPowerQ[ #] &]
cpQ[n_]:=CoprimeQ[n,DivisorSigma[1,n]]&&!PrimeQ[n]&&GCD@@ FactorInteger[ n][[All,2]]<2; Select[Range[2,400],cpQ] (* Harvey P. Dale, Oct 05 2020 *)

forcomposite(n=1, 1e3, if(gcd(n, sigma(n))==1, if(!ispower(n), print1(n, ", ")))) \\ Felix Fröhlich, Oct 25 2014

A248022 Achilles numbers which are coprime to the sum of their divisors.

Original entry on oeis.org

392, 800, 968, 1352, 2312, 2888, 3087, 3267, 3872, 4232, 5408, 6075, 6125, 6272, 6728, 7688, 7803, 9248, 10952, 11552, 12800, 13448, 14283, 14792, 15125, 15488, 16928, 17672, 19208, 20000, 21632, 22472, 22707, 25088, 26912, 27783, 27848, 29403, 29768, 30752, 33275

Offset: 1

Robert G. Wilson v, Sep 29 2014

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

Intersection of A052486 and A003624. - Michel Marcus, Sep 30 2014

achillesQ[n_] := Block[{ls = Last /@ FactorInteger@ n}, Min@ ls > 1 == GCD @@ ls]; Select[ Range@ 35000, achillesQ[ #] && GCD[#, DivisorSigma[1, #]] == 1 &]

A248023 Even numbers which are neither primes nor perfect powers and are coprime to the sum of their divisors.

Original entry on oeis.org

50, 98, 242, 338, 392, 578, 722, 800, 968, 1058, 1250, 1352, 1682, 1922, 2312, 2450, 2738, 2888, 3362, 3698, 3872, 4232, 4418, 4802, 5408, 5618, 6050, 6272, 6728, 6962, 7442, 7688, 8450, 8978, 9248, 10082, 10658, 10952, 11552, 12482, 12800, 13448, 13778, 14450

Offset: 1

Robert G. Wilson v, Sep 29 2014

			50 is in the sequence since it is neither a prime nor a powerful number and its divisors 1, 2, 5, 10, 25, and 50 sum to 93, which is coprime to 50.

G. C. Greubel, Table of n, a(n) for n = 1..1500

Cf. A003624, A106543, A248020.

perfectPowerQ[n_] := n == 1 || GCD @@ FactorInteger[n][[All, 2]] > 1; Select[ 2 Range@ 7500, !PrimeQ[ #] && GCD[#, DivisorSigma[1, #]] == 1 && !perfectPowerQ[ #] &]

lista(nn) = {forstep(n=4, nn, 2, if (!ispower(n) && (gcd(n, sigma(n))==1), print1(n, ", ")););} \\ Michel Marcus, Oct 02 2014

A280946 Numbers n such that n and number of proper divisors (A032741) of n are relatively prime and n is a nonprime (A018252).

Original entry on oeis.org

1, 8, 9, 10, 12, 14, 18, 22, 24, 25, 26, 28, 30, 32, 34, 35, 38, 40, 44, 46, 49, 52, 54, 55, 58, 60, 62, 63, 65, 66, 68, 72, 74, 76, 77, 78, 80, 81, 82, 84, 85, 86, 88, 90, 91, 92, 94, 95, 96, 98, 99, 102, 104, 106, 108, 110, 112, 114, 115, 116, 117, 118, 119, 121, 122, 124, 125, 126, 128, 130, 133, 134, 135, 136, 138

Offset: 1

Ilya Gutkovskiy, Jan 11 2017

Numbers n such that A125168(n) = 1 and A010051(n) = 0.

Numbers n such that gcd(n,A032741(n)) = 1 and A000005(n) != 2.

			12 is in the sequence because 12 is a nonprime, 12 has 5 proper divisors {1, 2, 3, 4, 6} and gcd(12,5) = 1.

Vincenzo Librandi, Table of n, a(n) for n = 1..3000

Cf. A000005, A003624, A010051, A014567, A032741, A046642, A125168.

Select[Range[139], CoprimeQ[#1, DivisorSigma[0, #1] - 1] && !PrimeQ[#1] & ]
Select[Range[139], GCD[#1, DivisorSigma[0, #1] - 1] == 1 && DivisorSigma[0, #1] != 2 &]

isok(n) = gcd(n, numdiv(n)-1) == 1; \\ Michel Marcus, Jan 14 2017

A280387 Composite numbers n such that sum of proper divisors of n divides sum of proper divisors of n^n.

Original entry on oeis.org

4, 8, 9, 16, 21, 25, 27, 32, 36, 45, 49, 64, 81, 87, 91, 99, 121, 125, 128, 144, 169, 196, 217, 243, 256, 289, 325, 343, 361, 400, 417, 481, 512, 529, 559, 625, 685, 697, 703, 721, 729, 745, 749, 775, 801, 841, 925, 931, 961, 1024, 1156, 1157, 1261, 1331

Offset: 1

Altug Alkan, Jan 01 2017

Terms are 2^2, 2^3, 3^2, 2^4, 3*7, 5^2, 3^3, 2^5, 2^2*3^2, 3^2*5, 7^2, 2^6, 3^4, 3*29, 7*13, 3^2*11, 11^2, 5^3, ...

Terms that are not Duffinian numbers are 45, 87, 91, 99, 196, 703, 745, 775, 801, 931, ...

			Composite number 21 is a term because (sigma(21) - 21) = 11 divides (sigma(21^21) - 21^21) = 4381940263463668467705506011

Cf. A000203, A001065, A003624, A275123.

Select[Range[10^3], And[CompositeQ@ #, Divisible @@ Map[DivisorSigma[1, #] - # &, {#^#, #}]] &] (* Michael De Vlieger, Jan 02 2017 *)

is(n) = !isprime(n) && (sigma(n^n)-n^n)%(sigma(n)-n)==0;

More terms from Amiram Eldar, Feb 19 2019

cp's OEIS Frontend

A373968 a(n) is the number of divisors of n that are Duffinian numbers (A003624).

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A373892 a(n) is the smallest number that can be partitioned in exactly n ways as the sum of two Duffinian numbers (A003624).

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A373969 The smallest number k whose divisors include exactly n Duffinian numbers (A003624).

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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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A248020 Numbers which are coprime to the sum of their divisors, but are neither primes nor perfect powers.

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A248022 Achilles numbers which are coprime to the sum of their divisors.

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A248023 Even numbers which are neither primes nor perfect powers and are coprime to the sum of their divisors.

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