A073802 -id:A073802 - cp's OEIS frontend

A284229 a(n) is the least k such that A073802(k) = n.

1, 10, 12, 6, 336, 24, 5952, 168, 792, 496, 666624, 270, 10924032, 6720, 7344, 120, 3757637632, 4284, 45091651584, 2160, 79488, 1820672, 11544784011264, 672, 298080, 29331456, 106200, 13440, 53620880789471232, 10080, 1501384662105194496, 6552, 7022592, 7515275264

Offset: 1

Paolo P. Lava, Mar 23 2017

nonn

Composite numbers that have just 1 as divisor that satisfies the condition for which sigma(k) / d_i is an integer are the Duffinian numbers (A003624).

Alternative definition: Least k such that tau(gcd(k,sigma(k))) = n. - Giovanni Resta, Mar 23 2017

			The divisors of 12 are 1, 2, 3, 4, 6, 12 and sigma(12) = 28. Then:
1) 28 / 1 = 28;
2) 28 / 2 = 14;
3) 28 / 4 = 7;
and 12 is the least number to have this property. Therefore a(3) = 12.

Cf. A000203, A003624, A073802.

with(numtheory): P:=proc(q) local k,n; for k from 1 to q do
for n from 1 to q do if tau(gcd(n,sigma(n)))=k then
print(n); break; fi; od; od; end: P(10^9);

TakeWhile[#, # > 0 &] &@ Table[If[KeyExistsQ[#, n], First@ Lookup[#, n], -1], {n, Max@ Keys@ #}] &@ KeySort@ PositionIndex@ Table[DivisorSum[k, 1 &, IntegerQ[DivisorSigma[1, k]/#] &], {k, 10^6}] (* per Name, Version 10, or *)
TakeWhile[#, # > 0 &] &@ Table[If[KeyExistsQ[#, n], First@ Lookup[#, n], -1], {n, Max@ Keys@ #}] &@ KeySort@ PositionIndex@ Table[DivisorSigma[0, GCD[k, DivisorSigma[1, k]]], {k, 10^7}] (* faster, Version 10, Michael De Vlieger, Mar 24 2017 *)

nb(n) = my(s = sigma(n)); sumdiv(n, d, (s % d) == 0);
a(n) = k=1; while(nb(k) != n, k++); k; \\ Michel Marcus, Mar 24 2017

a(13), a(17), a(19) and from a(22) to a(34) from Giovanni Resta, Mar 23 2017

Name proposed by Michel Marcus, Mar 24 2017

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

A037197 Numbers k such that tau(sigma(k)) = tau(k) where tau(k) is the number of divisors of k and sigma(k) their sum.

Original entry on oeis.org

1, 2, 8, 12, 32, 52, 75, 84, 90, 98, 128, 150, 156, 338, 360, 392, 525, 528, 560, 600, 722, 867, 912, 972, 1050, 1352, 1452, 1456, 1525, 1734, 1922, 2064, 2160, 2340, 2400, 2888, 2890, 3050, 3120, 3216, 3698, 3744, 3872, 4080, 4144, 4200, 4500, 4575

Offset: 1

Naohiro Nomoto

nonn

			k = 75: divisors(75) = {1, 3, 5, 15, 25, 75}, divisors(sigma(75)) = divisors(124) = {1, 2, 4, 31, 62, 124}, both 75 and sigma(75) have 6 divisors, so 75 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Reinhard Zumkeller)

Cf. A000005, A000203, A062068, A073802, A073803, A073804.

Do[s=DivisorSigma[0, DivisorSigma[1, n]]; s0=DivisorSigma[0, n]; If[Greater[s0, s], Print[n]], {n, 1, 1000}]
Select[Range[4600],DivisorSigma[0,#]==DivisorSigma[0,DivisorSigma[1,#]]&] (* Harvey P. Dale, Feb 08 2025 *)

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

Solutions to A000005(x) = A062068(x) = A000005(A000203(x)).

Conjecture: for n > 10^6, a(n) < n^2. - Benoit Cloitre, Aug 24 2002

Offset corrected by Reinhard Zumkeller, Jun 18 2009

Name edited by Michel Marcus, Nov 12 2023

A073803 Numbers k such that the number of divisors of k is smaller than that of sigma(k).

Original entry on oeis.org

3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91

Offset: 1

Labos Elemer, Aug 13 2002

nonn

			k = 96: divisors(96) = {1,2,3,4,6,8,12,16,24,32,48,96}, 12 divisors; divisors(sigma(96)) = {1,2,3,4,6,7,9,12,14,18,21,28,36,42,63,84,126,252}, 18 divisors; 12 < 18, so 96 is a term.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A062068, A073802, A073804, A037197.

filter:= proc(n) uses numtheory; tau(n) < tau(sigma(n)) end proc:
select(filter, [$1..100]); # Robert Israel, Aug 03 2020

Do[s=DivisorSigma[0, DivisorSigma[1, n]]; s0=DivisorSigma[0, n]; If[Greater[s0, s], Print[n]], {n, 1, 1000}]
Select[Range[100],DivisorSigma[0,#]Harvey P. Dale, Sep 22 2019 *)

isok(k) = {my(f = factor(k)); numdiv(f) < numdiv(sigma(f));} \\ Amiram Eldar, Mar 07 2025

Solutions to A000005(x) < A062068(x) = A000005(A000203(x)).

A073804 Numbers k such that the number of divisors of k is greater than that of sigma(k).

Original entry on oeis.org

4, 9, 16, 18, 25, 36, 48, 50, 64, 72, 80, 81, 100, 112, 144, 162, 180, 192, 196, 200, 208, 225, 240, 252, 256, 288, 289, 300, 320, 324, 336, 400, 432, 441, 448, 450, 468, 484, 512, 576, 578, 592, 624, 625, 648, 676, 700, 704, 720, 729, 768, 784, 800, 810, 832

Offset: 1

Labos Elemer, Aug 13 2002

nonn

			k = 25: divisors(25) = {1,5,25}, 3 divisors; divisors(sigma(25)) = {1,31}, 2 divisors; 2 < 3, so 25 is a term.
k = 48: divisors(48) = {1,2,3,4,6,8,12,16,24,48}, 10 divisors; divisors(sigma(48)) = {1,2,4,31,62,124}, 6 divisors, 6 < 10 so 48 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A000005, A000203, A062068, A073802, A073803, A037197.

Do[s=DivisorSigma[0, DivisorSigma[1, n]]; s0=DivisorSigma[0, n]; If[Greater[s0, s], Print[n]], {n, 1, 1000}]
Select[Range[900],DivisorSigma[0,#]>DivisorSigma[0,DivisorSigma[1,#]]&] (* Harvey P. Dale, Jan 18 2017 *)

isok(k) = {my(f = factor(k)); numdiv(f) > numdiv(sigma(f));} \\ Amiram Eldar, Mar 07 2025

Solutions to A000005(x) > A062068(x) = A000005(A000203(x)).

A073808 Number of common divisors of sigma_1(n) and sigma_2(n).

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Aug 13 2002

nonn

a(n) = Card[Intersection[D[A000203(n)], D[A001157(n)]]]. - This is the formula given by the original author. D[x] here means the set of divisors of x. - Antti Karttunen, Nov 23 2017

			n=10: sigma[1,10]=18, sigma[1,10]=130 Intersection[{1,2,3,6,9,18},{1,2,5,10,13,26,65,130}]={1,2}, so a(10)=2.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000005, A000203, A001157, A073802, A073809.

g1[x_] := Divisors[DivisorSigma[1, x]] g2[x_] := Divisors[DivisorSigma[2, x]] ncd[x_] := Length[Intersection[g1[x], g2[x]]] Table[ncd[w], {w, 1, 128}]
(* Second program: *)
Table[Length@ Apply[Intersection, Divisors@ Array[DivisorSigma[#, n] &, 2]], {n, 105}] (* Michael De Vlieger, Nov 23 2017 *)

A073808(n) = numdiv(gcd(sigma(n),sigma(n,2))); \\ Antti Karttunen, Nov 23 2017

a(n) = A000005(gcd(A000203(n), A001157(n))). - Antti Karttunen, Nov 23 2017

A073809 Number of common divisors of sigma_1(n) and sigma_3(n).

Original entry on oeis.org

1, 2, 3, 1, 4, 6, 4, 4, 1, 6, 6, 6, 4, 8, 8, 2, 6, 2, 6, 8, 6, 9, 8, 12, 1, 8, 8, 4, 8, 12, 6, 3, 10, 8, 10, 1, 4, 12, 8, 12, 8, 12, 6, 6, 4, 12, 10, 6, 2, 2, 12, 4, 8, 16, 12, 16, 10, 12, 12, 16, 4, 12, 4, 2, 12, 15, 6, 12, 12, 15, 12, 8, 4, 8, 3, 12, 12, 16, 10, 8, 3, 12, 12, 12, 12, 12, 16

Offset: 1

Labos Elemer, Aug 13 2002

nonn

			n=10: sigma[1,10]=18, sigma[3,10]=1134; Intersection[{1,2,3,6,9,18},{1,2,3,6,7,9,14,18,21,27,42,54,63, 81,126,162,189,378,567,1134}]={1,2,3,6,9,18}, so a(10)=6.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000203, A001158, A073802, A073808.

g1[x_] := Divisors[DivisorSigma[1, x]] g2[x_] := Divisors[DivisorSigma[3, x]] ncd[x_] := Length[Intersection[g1[x], g2[x]]] Table[ncd[w], {w, 1, 128}]
(* Second program: *)
Table[Length@ Apply[Intersection, Divisors@ Array[DivisorSigma[2 # - 1, n] &, 2]], {n, 87}] (* Michael De Vlieger, Nov 23 2017 *)

A073809(n) = numdiv(gcd(sigma(n),sigma(n,3))); \\ Antti Karttunen, Nov 23 2017

a(n) = Card[Intersection[D[A000203(n)], D[A001158(n)]]] where D[x] is the set of divisors of x.

A173438 Number of divisors d of number n such that d does not divide sigma(n).

Original entry on oeis.org

0, 1, 1, 2, 1, 0, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 4, 1, 4, 3, 2, 1, 2, 2, 2, 3, 0, 1, 4, 1, 5, 2, 2, 3, 8, 1, 2, 3, 4, 1, 4, 1, 3, 4, 2, 1, 7, 2, 5, 2, 4, 1, 4, 3, 4, 3, 2, 1, 6, 1, 2, 5, 6, 3, 4, 1, 4, 2, 6, 1, 10, 1, 2, 5, 3, 3, 4, 1, 8, 4, 2, 1, 6, 3, 2, 2, 5, 1, 6, 2, 3, 3, 2, 2, 6, 1, 5

Offset: 1

Jaroslav Krizek, Feb 18 2010

nonn

a(n) = 0 for multiply-perfect numbers (A007691).

			For n = 12, a(12) = 3; sigma(12) = 28, divisors of 12: 1, 2, 3, 4, 6, 12; d does not divide sigma(n) for 3 divisors d: 3, 6 and 12.

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

Cf. A000005, A000203, A007691, A009194, A073802, A286570.

A173438 := proc(n)
    local sd,a;
    sd := numtheory[sigma](n) ;
    a := 0 ;
    for d in numtheory[divisors](n) do
        if modp(sd,d) <> 0 then
            a := a+1 ;
        end if;
    end do:
    a;
end proc: # R. J. Mathar, Oct 26 2015

Table[DivisorSum[n, 1 &, ! Divisible[DivisorSigma[1, n], #] &], {n, 98}] (* Michael De Vlieger, Oct 08 2017 *)

A173438(n) = (numdiv(n) - numdiv(gcd(sigma(n), n))); \\ (See PARI-code in A073802) - Antti Karttunen, Oct 08 2017

a(n) = A000005(n) - A073802(n).

a(n) = tau(n) - tau(gcd(n,sigma(n))). - Antti Karttunen, Oct 08 2017

A327154 a(n) = Product_{d|n, d>1} A008578(1+A286561(sigma(n),d)), where A286561(n,d) gives the highest exponent of d dividing n.

Original entry on oeis.org

1, 1, 1, 1, 1, 12, 1, 1, 1, 2, 1, 6, 1, 5, 2, 1, 1, 2, 1, 2, 1, 3, 1, 48, 1, 2, 1, 80, 1, 45, 1, 1, 2, 2, 1, 1, 1, 3, 1, 8, 1, 44, 1, 6, 2, 5, 1, 6, 1, 1, 3, 2, 1, 20, 1, 20, 1, 2, 1, 80, 1, 11, 1, 1, 1, 63, 1, 2, 2, 7, 1, 2, 1, 2, 1, 6, 1, 20, 1, 2, 1, 2, 1, 264, 1, 3, 2, 6, 1, 48, 2, 10, 1, 7, 2, 108, 1, 1, 2, 1, 1, 125, 1, 2, 2

Offset: 1

Antti Karttunen, Sep 18 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000040, A008578, A286561, A073802.

Cf. also A293514, A322312, A323155, A327155, A327156.

A327154(n) = { my(m=1,s=sigma(n),v); fordiv(n,d,if((d>1) && ((v = valuation(s,d))>0), m *= prime(v))); (m); };

a(n) = Product_{d|n, d>1} A008578(1+A286561(sigma(n),d)), where sigma = A000203.
Other identities. For all n >= 1:
1+A001222(a(n)) = A073802(n).

A327155 a(n) = Product_{d|sigma(n), d>1} A008578(1+A286561(n,d)), where A286561(n,d) gives the highest exponent of d dividing n.

Original entry on oeis.org

1, 1, 1, 1, 1, 8, 1, 1, 1, 2, 1, 6, 1, 2, 2, 1, 1, 3, 1, 3, 1, 2, 1, 80, 1, 2, 1, 48, 1, 8, 1, 1, 2, 2, 1, 1, 1, 2, 1, 20, 1, 8, 1, 6, 3, 2, 1, 21, 1, 1, 2, 3, 1, 20, 1, 20, 1, 2, 1, 48, 1, 2, 1, 1, 1, 8, 1, 3, 2, 2, 1, 3, 1, 2, 1, 6, 1, 8, 1, 7, 1, 2, 1, 48, 1, 2, 2, 10, 1, 48, 2, 6, 1, 2, 2, 264, 1, 1, 3, 1, 1, 8, 1, 5, 2

Offset: 1

Antti Karttunen, Sep 18 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000040, A008578, A286561, A073802.

Cf. also A293514, A322312, A323155, A327154, A327156.

A327155(n) = { my(m=1,v); fordiv(sigma(n),d,if((d>1) && ((v = valuation(n,d))>0), m *= prime(v))); (m); };

a(n) = Product_{d|sigma(n), d>1} A008578(1+A286561(n,d)), where sigma = A000203.
Other identities. For all n >= 1:
1+A001222(a(n)) = A073802(n).

cp's OEIS Frontend

A284229 a(n) is the least k such that A073802(k) = n.

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

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A037197 Numbers k such that tau(sigma(k)) = tau(k) where tau(k) is the number of divisors of k and sigma(k) their sum.

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A073803 Numbers k such that the number of divisors of k is smaller than that of sigma(k).

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A073804 Numbers k such that the number of divisors of k is greater than that of sigma(k).

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A073808 Number of common divisors of sigma_1(n) and sigma_2(n).

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A073809 Number of common divisors of sigma_1(n) and sigma_3(n).

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