A009194 -id:A009194 - cp's OEIS frontend

A073802 Number of common divisors of n and sigma(n).

1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 6, 1, 2, 1, 6, 1, 4, 1, 1, 2, 2, 1, 1, 1, 2, 1, 4, 1, 4, 1, 3, 2, 2, 1, 3, 1, 1, 2, 2, 1, 4, 1, 4, 1, 2, 1, 6, 1, 2, 1, 1, 1, 4, 1, 2, 2, 2, 1, 2, 1, 2, 1, 3, 1, 4, 1, 2, 1, 2, 1, 6, 1, 2, 2, 3, 1, 6, 2, 3, 1, 2, 2, 6, 1, 1, 2, 1, 1, 4, 1, 2, 2

Offset: 1

Labos Elemer, Aug 13 2002

nonn

From Jaroslav Krizek, Feb 18 2010: (Start)
Number of divisors d of number n such that d divides sigma(n).
a(n) = A000005(n) - A173438(n).
a(n) = A000005(n) for multiply-perfect numbers (A007691). (End)

			For n = 12: a(12) = 3; sigma(12) = 28, divisors of 12: 1, 2, 3, 4, 6, 12; d divides sigma(n) for 3 divisors d: 1, 2, 4.
n=96: d(96) = {1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96}, d(sigma(96)) = {1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 28, 36, 42, 63, 84, 126, 252}, CD(n, sigma(n)) = {1, 2, 3, 4, 6, 12} so a(96) = 6.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A062068, A073803, A073804.

[NumberOfDivisors(GCD(SumOfDivisors(n),n)): n in [1..100]]; // Vincenzo Librandi, Oct 09 2017

g1[x_] := Divisors[x]; g2[x_] := Divisors[DivisorSigma[1, x]]; ncd[x_] := Length[Intersection[g1[x], g2[x]]]; Table[ncd[w], {w, 1, 128}]
Table[Length[Intersection[Divisors[n], Divisors[DivisorSigma[1, n]]]], {n, 100}] (* Vincenzo Librandi, Oct 09 2017 *)
a[n_] := DivisorSigma[0, GCD[n, DivisorSigma[1, n]]]; Array[a, 100] (* Amiram Eldar, Nov 21 2024 *)

a(n)=numdiv(gcd(sigma(n),n)) \\ Charles R Greathouse IV, Mar 09 2014

See program.

a(n) = A000005(A009194(n)) = tau(gcd(n,sigma(n))). [Reinhard Zumkeller, Mar 12 2010]

A326048 a(n) = gcd(n-A050449(n), A082052(n)-n), where A050449 and A082052 give the sum of divisors of the form 4k+1, and not of that form, respectively.

Original entry on oeis.org

1, 1, 2, 1, 1, 5, 6, 1, 1, 2, 10, 1, 1, 1, 3, 1, 1, 1, 18, 2, 1, 1, 22, 1, 1, 2, 1, 27, 1, 12, 30, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 4, 42, 1, 3, 5, 46, 1, 1, 1, 3, 2, 1, 4, 1, 1, 1, 2, 58, 6, 1, 1, 2, 1, 1, 4, 66, 10, 1, 4, 70, 1, 1, 2, 2, 3, 1, 4, 78, 2, 1, 2, 82, 2, 1, 5, 3, 1, 1, 6, 7, 1, 1, 1, 1, 5, 1, 1, 14, 1, 1, 12, 102, 2, 9

Offset: 1

Antti Karttunen, Jun 04 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A050449, A082052, A326047, A326049, A326050.

Cf. also A009194, A325385, A325813, A325975.

A050449(n) = sumdiv(n, d, d*((d % 4) == 1)); \\ From A050449
A326049(n) = (n-A050449(n));
A082052(n) = sumdiv(n, d, if(1!=(d%4), d));
A326050(n) = (A082052(n)-n);
A326048(n) = gcd(A326049(n), A326050(n));

a(n) = gcd(A326049(n), A326050(n)) = gcd(n-A050449(n), A082052(n)-n).

a(2n-1) = A326047(2n-1) for all n.

A009205 a(n) = gcd(d(n), sigma(n)).

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 2, 1, 1, 2, 2, 2, 2, 4, 4, 1, 2, 3, 2, 6, 4, 4, 2, 4, 1, 2, 4, 2, 2, 8, 2, 3, 4, 2, 4, 1, 2, 4, 4, 2, 2, 8, 2, 6, 6, 4, 2, 2, 3, 3, 4, 2, 2, 8, 4, 8, 4, 2, 2, 12, 2, 4, 2, 1, 4, 8, 2, 6, 4, 8, 2, 3, 2, 2, 2, 2, 4, 8, 2, 2, 1, 2, 2, 4, 4, 4, 4, 4, 2, 6, 4, 6, 4, 4, 4, 12, 2, 3, 6, 1, 2, 8, 2, 2, 8, 2, 2, 4, 2, 8, 4, 2, 2, 8, 4, 6, 2, 4, 4, 8

Offset: 1

David W. Wilson

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A009191, A009194, A009278, A064840, A087801, A286360.

Table[GCD[DivisorSigma[0,n],DivisorSigma[1,n]],{n,120}] (* Harvey P. Dale, Dec 05 2017 *)

A009205(n) = gcd(numdiv(n),sigma(n)); \\ Antti Karttunen, May 22 2017

from math import prod, gcd
from sympy import factorint
def A009205(n):
    f = factorint(n).items()
    return gcd(prod(e+1 for p, e in f),prod((p**(e+1)-1)//(p-1) for p,e in f)) # Chai Wah Wu, Jul 27 2023

a(n) = A064840(n)/A009278(n). - Amiram Eldar, Jan 31 2025

Data section extended to 120 terms by Antti Karttunen, May 22 2017

A326046 a(n) = gcd(n-A326039(n), A326040(n)-n).

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 3, 1, 1, 1, 1, 4, 12, 2, 1, 1, 8, 1, 3, 1, 5, 2, 1, 4, 1, 5, 1, 24, 28, 6, 15, 1, 1, 1, 1, 1, 36, 2, 1, 1, 40, 2, 3, 4, 4, 10, 1, 4, 1, 7, 15, 3, 4, 2, 19, 4, 1, 1, 1, 8, 60, 2, 1, 1, 1, 6, 3, 1, 1, 2, 35, 1, 72, 1, 1, 12, 1, 2, 3, 1, 1, 1, 1, 4, 1, 2, 1, 4, 8, 27, 5, 8, 29, 2, 7, 60, 48, 1, 1, 1, 100, 6, 3, 1, 1

Offset: 1

Antti Karttunen, Jun 06 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000203, A008833, A326039, A326040, A326044, A326045.

Cf. also A009194, A325385, A325813, A325975, A326047, A326048, A326056, A326057, A326060, A326062, A326129, A326130, A326140, A326144.

A008833(n) = (n/core(n));
A326039(n) = A008833(sigma(n));
A326040(n) = (sigma(n)-A008833(sigma(n)));
A326046(n) = gcd(n-A326039(n), A326040(n)-n);

a(n) = gcd(A326044(n), A326045(n)) = gcd(n-A326039(n), A326040(n)-n).

A326060 a(n) = gcd(n-A035316(n), A285309(n)-n), where A035316 and A285309 give respectively the sums of square and nonsquare divisors of n.

Original entry on oeis.org

1, 1, 2, 1, 4, 5, 6, 1, 1, 1, 10, 1, 12, 1, 2, 1, 16, 1, 18, 1, 10, 1, 22, 1, 1, 5, 1, 23, 28, 1, 30, 1, 2, 1, 2, 1, 36, 1, 2, 5, 40, 1, 42, 1, 1, 5, 46, 1, 1, 1, 10, 1, 52, 4, 2, 1, 2, 1, 58, 1, 60, 1, 1, 1, 2, 1, 66, 1, 2, 1, 70, 1, 72, 1, 1, 1, 2, 1, 78, 1, 1, 1, 82, 1, 2, 5, 2, 1, 88, 2, 10, 1, 2, 1, 2, 15, 96, 1, 1, 1, 100, 1, 102, 1, 2

Offset: 1

Antti Karttunen, Jun 06 2019

nonn

Below 2^27 there are following numbers k such that a(k) is equal to A326059(k), and quotient A326058(k)/A326059(k) is odd: 6, 28, 496, 1625, 2057, 8128, 33550336, 107452235. The odd terms are factored as: 1625 = 5^3 * 13, 2057 = 11^2 * 17, 107452235 = 5 * 11^2 * 97 * 1831.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A035316, A285309, A326058, A326059.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326056, A326057, A326062.

A035316(n) = sumdiv(n,d,issquare(d)*d);
A326058(n) = (n-A035316(n));
A285309(n) = sumdiv(n,d,(!issquare(d))*d);
A326059(n) = (A285309(n)-n);
A326060(n) = gcd(A326058(n), A326059(n));

a(n) = gcd(A326058(n), A326059(n)) = gcd(n-A035316(n), A285309(n)-n).

A326062 a(1) = gcd((sigma(n)-A032742(n))-n, n-A032742(n)), where A032742 gives the largest proper divisor of n.

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 6, 1, 1, 1, 10, 2, 12, 1, 2, 1, 16, 3, 18, 2, 2, 1, 22, 12, 1, 1, 2, 14, 28, 3, 30, 1, 2, 1, 2, 1, 36, 1, 2, 10, 40, 3, 42, 2, 6, 1, 46, 4, 1, 1, 2, 2, 52, 3, 2, 4, 2, 1, 58, 6, 60, 1, 2, 1, 2, 3, 66, 2, 2, 1, 70, 3, 72, 1, 2, 2, 2, 3, 78, 2, 1, 1, 82, 14, 2, 1, 2, 4, 88, 9, 2, 2, 2, 1, 2, 12, 96, 1, 6, 1, 100, 3, 102, 2, 2

Offset: 1

Antti Karttunen, Jun 06 2019

nonn

See comments in A326063 and A326064.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000203, A032742, A060681, A318505, A326061, A326063, A326064.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326056, A326057, A326060.

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A326062(n) = gcd((sigma(n)-A032742(n))-n, n-A032742(n));

a(1) = 1; for n > 1, a(n) = gcd(A060681(n), A318505(n)).

a(n) = gcd((A000203(n)-A032742(n))-n, n-A032742(n)).

A326056 a(n) = gcd(sigma(n)-A008833(n)-n, n-A008833(n)), where sigma is the sum of divisors of n, and A008833 is the largest square dividing n.

Original entry on oeis.org

1, 1, 2, 1, 4, 5, 6, 1, 5, 1, 10, 4, 12, 1, 2, 1, 16, 3, 18, 2, 10, 1, 22, 4, 19, 5, 2, 24, 28, 1, 30, 1, 2, 1, 2, 19, 36, 1, 2, 2, 40, 1, 42, 4, 12, 5, 46, 4, 41, 1, 10, 6, 52, 3, 2, 4, 2, 1, 58, 8, 60, 1, 2, 1, 2, 1, 66, 2, 2, 1, 70, 3, 72, 1, 2, 12, 2, 1, 78, 2, 41, 1, 82, 8, 2, 5, 2, 4, 88, 27, 10, 8, 2, 1, 2, 20, 96, 1, 6

Offset: 1

Antti Karttunen, Jun 05 2019

nonn

Composite numbers n such that a(n) = A326055(n) start as: 6, 28, 336, 496, 792, 8128, 31968, 3606912, ...

Nonsquare odd numbers n such that a(n) = abs(A326054(n)) start as: 21, 153, 301, 697, 1333, 1909, 1917, 2041, 3901, 4753, 24601, 24957, 26977, 29161, 29637, 56953, 67077, 96361, ...

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000203, A008833, A326053, A326054, A326055.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326057, A326060, A326062, A326129, A326130, A326140, A326144.

A008833(n) = (n/core(n));
A326053(n) = (sigma(n)-A008833(n));
A326054(n) = (A326053(n)-n);
A326055(n) = (n-A008833(n));
A326056(n) = gcd(A326054(n), A326055(n));

a(n) = gcd(A326054(n), A326055(n)) = gcd((A000203(n)-A008833(n))-n, n-A008833(n)).

A326129 a(n) = gcd(A326127(n), A326128(n)).

Original entry on oeis.org

1, 1, 2, 1, 4, 0, 6, 1, 1, 2, 10, 1, 12, 4, 6, 1, 16, 1, 18, 1, 10, 8, 22, 6, 1, 10, 2, 21, 28, 12, 30, 1, 18, 14, 22, 1, 36, 16, 22, 10, 40, 12, 42, 1, 4, 20, 46, 1, 1, 1, 30, 3, 52, 12, 38, 2, 34, 26, 58, 3, 60, 28, 2, 1, 46, 12, 66, 1, 42, 4, 70, 1, 72, 34, 2, 3, 58, 12, 78, 1, 1, 38, 82, 7, 62, 40, 54, 2, 88, 2, 70, 1, 58, 44, 70, 30

Offset: 1

Antti Karttunen, Jun 09 2019

nonn

Question: Are there any other numbers than those in A000396 that satisfy a(k) = A326128(k)?

See also comments in A336641, where all such k should reside. - Antti Karttunen, Jul 29 2020

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000396, A007913, A326126, A326127, A326128.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326056, A326057, A326060, A326062, A326130, A326140, A326144, A336641, A336645.

A326127(n) = (sigma(n)-core(n)-n);
A326128(n) = (n-core(n));
A326129(n) = gcd(A326127(n), A326128(n));

a(n) = n - A336645(n). - Antti Karttunen, Jul 29 2020

A326144 a(n) = gcd(A066503(n), A326143(n)) = gcd(n - A007947(n), sigma(n) - A007947(n) - n).

Original entry on oeis.org

1, 1, 2, 1, 4, 0, 6, 1, 1, 2, 10, 2, 12, 4, 6, 1, 16, 3, 18, 2, 10, 8, 22, 6, 1, 10, 2, 14, 28, 12, 30, 1, 18, 14, 22, 1, 36, 16, 22, 10, 40, 12, 42, 2, 6, 20, 46, 14, 1, 1, 30, 2, 52, 12, 38, 2, 34, 26, 58, 6, 60, 28, 2, 1, 46, 12, 66, 2, 42, 4, 70, 3, 72, 34, 2, 2, 58, 12, 78, 2, 1, 38, 82, 14, 62, 40, 54, 2, 88, 6, 70, 2

Offset: 1

Antti Karttunen, Jun 09 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000203, A007947, A066503, A326142, A326143, A326145.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326056, A326057, A326060, A326062, A326129, A326130, A326140.

rad[n_] := Times @@ (First /@ FactorInteger@n);
a[n_] := GCD[n - rad[n], DivisorSigma[1, n] - rad[n] - n];
Array[a, 100] (* Jean-François Alcover, Dec 01 2021 *)

A007947(n) = factorback(factorint(n)[, 1]);
A066503(n) = (n - A007947(n));
A326143(n) = (sigma(n)-A007947(n)-n);
A326144(n) = gcd(A066503(n), A326143(n));

a(n) = gcd(A066503(n), A326143(n)) = gcd(n-A007947(n), A000203(n)-A007947(n)-n).

A003624 Duffinian numbers: composite numbers k relatively prime to sigma(k).

Original entry on oeis.org

4, 8, 9, 16, 21, 25, 27, 32, 35, 36, 39, 49, 50, 55, 57, 63, 64, 65, 75, 77, 81, 85, 93, 98, 100, 111, 115, 119, 121, 125, 128, 129, 133, 143, 144, 155, 161, 169, 171, 175, 183, 185, 187, 189, 201, 203, 205, 209, 215, 217, 219, 221, 225, 235, 237, 242, 243, 245, 247

Offset: 1

N. J. A. Sloane, Mira Bernstein, Sep 19 1994

nonn

All prime powers greater than 1 are in the sequence. No factorial number can be a term. - Arkadiusz Wesolowski, Feb 16 2014
Even terms are in A088827. Any term also in A005153 is either an even square or twice an even square not divisible by 3. - Jaycob Coleman, Jun 08 2014
All primes satisfy the second condition since gcd(p, p+1) = 1, thus making this sequence a proper subset of A014567. - Robert G. Wilson v, Oct 02 2014

			4 is in the sequence since it is not a prime, its divisors 1, 2, and 4 sum to 7, and gcd(7, 4) = 1.
21 is in the sequences since it is not a prime, and its divisors 1, 3, 7, and 21 sum to 32, which is coprime to 21.

T. Koshy, Elementary number theory with applications, Academic Press, 2002, p. 141, exerc. 6,7,8 and 9.
L. Richard Duffy, The Duffinian numbers, Journal of Recreational Mathematics 12 (1979), pp. 112-115.
Peter Heichelheim, There exist five Duffinian consecutive integers but not six, Journal of Recreational Mathematics 14 (1981-1982), pp. 25-28.
J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 64.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Florian Luca, On the densities of some subsets of integers, Missouri Journal of Mathematical Sciences 19:3 (2007), pp. 167-170.
Rose Mary Zbiek, What can we say about the Duffinian numbers?, The Pentagon 42:2 (1983), pp. 99-109.

Cf. A000203, A002808, A014567, A025475.

a003624 n = a003624_list !! (n-1)
a003624_list = filter ((== 1) . a009194) a002808_list
-- Reinhard Zumkeller, Mar 23 2013

fQ[n_] := n != 1 && !PrimeQ[n] && GCD[n, DivisorSigma[1, n]] == 1; Select[ Range@ 280, fQ]

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

from math import gcd
from itertools import count, islice
from sympy import isprime, divisor_sigma
def A003624_gen(startvalue=2): # generator of terms
    return filter(lambda k:not isprime(k) and gcd(k,divisor_sigma(k))==1,count(max(startvalue,2)))
A003624_list = list(islice(A003624_gen(),30)) # Chai Wah Wu, Jul 06 2023

A009194(a(n)) * (1 - A010051(a(n))) = 1. - Reinhard Zumkeller, Mar 23 2013

a(n) >> n log log log n, see Luca. (Clearly excluding the primes only makes the n-th term larger.) - Charles R Greathouse IV, Feb 17 2014

cp's OEIS Frontend

A073802 Number of common divisors of n and sigma(n).

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A326048 a(n) = gcd(n-A050449(n), A082052(n)-n), where A050449 and A082052 give the sum of divisors of the form 4k+1, and not of that form, respectively.

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A009205 a(n) = gcd(d(n), sigma(n)).

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A326046 a(n) = gcd(n-A326039(n), A326040(n)-n).

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A326060 a(n) = gcd(n-A035316(n), A285309(n)-n), where A035316 and A285309 give respectively the sums of square and nonsquare divisors of n.

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A326062 a(1) = gcd((sigma(n)-A032742(n))-n, n-A032742(n)), where A032742 gives the largest proper divisor of n.

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A326056 a(n) = gcd(sigma(n)-A008833(n)-n, n-A008833(n)), where sigma is the sum of divisors of n, and A008833 is the largest square dividing n.

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A326129 a(n) = gcd(A326127(n), A326128(n)).

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