A073802 -id:A073802 - cp's OEIS frontend

A073810 Number of common divisors of sigma(2,n) and sigma(3,n).

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

Offset: 1

Labos Elemer, Aug 13 2002

			n=10: sigma(2,10)=130, sigma(3,10)=1134; Intersection[{1,2,5,10,13,26,65,130}, {1,2,3,6,7,9,14,18,21,27,42,54,63,81, 126,162,189,378,567,1134}]={1,2}, so a(10)=2.

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

Cf. A001157, A001158, A073802, A073808, A073809.

g1[x_] := Divisors[DivisorSigma[2, x]] g2[x_] := Divisors[DivisorSigma[3, x]] ncd[x_] := Length[Intersection[g1[x], g2[x]]] Table[ncd[w], {w, 1, 128}]
a[n_] := DivisorSigma[0, GCD[DivisorSigma[2, n], DivisorSigma[3, n]]]; Array[a, 100] (* Amiram Eldar, Oct 18 2019 *)

a(n) = Card[Intersection[D[A001157(n)], D[A001158(n)]]].

A073811 Number of common divisors of n and phi(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 1, 4, 1, 4, 1, 3, 2, 2, 1, 4, 2, 2, 3, 3, 1, 2, 1, 5, 1, 2, 1, 6, 1, 2, 2, 4, 1, 4, 1, 3, 2, 2, 1, 5, 2, 4, 1, 3, 1, 6, 2, 4, 2, 2, 1, 3, 1, 2, 3, 6, 1, 2, 1, 3, 1, 2, 1, 8, 1, 2, 2, 3, 1, 4, 1, 5, 4, 2, 1, 6, 1, 2, 1, 4, 1, 4, 1, 3, 2, 2, 1, 6, 1, 4, 2, 6, 1, 2, 1, 4, 2

Offset: 1

Labos Elemer, Aug 13 2002

nonn

Where records occur: 1, 4, 8, 16, 32, 36, 72, 108, 144, 216, 432, 648, 864, ... - David A. Corneth, Oct 21 2017

			For n = 24: phi(n) = 8, Intersection[{1,2,3,4,6,8,12,24},{1,2,4,8}] = {1,2,4,8}, so a(24) = 4.

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

Cf. A000005, A000010, A009195, A073802, A073808, A073809, A073810.

g1[x_] := Divisors[x] g2[x_] := Divisors[EulerPhi[x]] ncd[x_] := Length[Intersection[g1[x], g2[x]]] Table[ncd[w], {w, 1, 128}]
Table[Length[Intersection[Divisors[n],Divisors[EulerPhi[n]]]],{n,110}] (* Harvey P. Dale, Oct 03 2013 *)
a[n_] := DivisorSigma[0, GCD[n, EulerPhi[n]]]; Array[a, 100] (* Amiram Eldar, Jul 01 2022 *)

A073811(n) = sumdiv(eulerphi(n),d,!(n%d)); \\ Antti Karttunen, Oct 21 2017

a(n) = numdiv(gcd(eulerphi(n), n)) \\ David A. Corneth, Oct 21 2017

;; Implemented literally (naively) after the description. Either:
(define (A073811 n) (length (filter (lambda (d) (zero? (modulo n d))) (divisors (A000010 n)))))
;; Or:
(define (A073811 n) (let ((phn (A000010 n))) (length (filter (lambda (d) (zero? (modulo phn d))) (divisors n)))))
(define (divisors n) (cons 1 (proper-divisors n))) ;; This can be also memoized with definec.
(define (proper-divisors n) (let loop ((k n) (divs (list))) (cond ((= 1 k) divs) ((zero? (modulo n k)) (loop (- k 1) (cons k divs))) (else (loop (- k 1) divs)))))
;; Antti Karttunen, Oct 21 2017

a(n) = Card[Intersection[D[n], D[A000010(n)]]].
a(n) = Sum_{d|n, d|A000010(n)} 1. - Antti Karttunen, Oct 21 2017
a(n) = A000005(A009195(n)). - Antti Karttunen, Oct 21 2017, after David A. Corneth's PARI-program.

A073806 Number of divisors of sum of square of divisors.

Original entry on oeis.org

1, 2, 4, 4, 4, 6, 6, 4, 4, 8, 4, 16, 8, 8, 12, 4, 8, 8, 4, 16, 12, 8, 8, 12, 8, 12, 12, 24, 4, 18, 8, 16, 12, 12, 18, 12, 8, 8, 18, 16, 6, 15, 12, 16, 12, 12, 16, 16, 8, 16, 18, 32, 8, 18, 12, 16, 12, 8, 4, 48, 4, 16, 24, 4, 24, 18, 8, 32, 18, 24, 4, 16, 16, 12, 32, 16, 18, 24, 4, 16, 6

Offset: 1

Labos Elemer, Aug 13 2002

nonn

			n = 10: D = {1,2,5,10}, sigma(2,10) = 1 + 4 + 25 + 100 = 130, D(130) = {1,2,5,10,13,26,65,130}, so a(10) = 8.

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

Cf. A000005, A000203, A001157, A062068, A073802-A073813.

a[n_] := DivisorSigma[0, DivisorSigma[2, n]]; Array[a, 81] (* Amiram Eldar, Aug 01 2019 *)

a(n) = A000005(A001157(n)) = d(sigma(2, n)).

A073807 Number of divisors of sum of cube of divisors.

Original entry on oeis.org

1, 3, 6, 2, 12, 18, 8, 12, 2, 20, 18, 12, 8, 24, 36, 4, 32, 6, 24, 24, 24, 30, 36, 72, 4, 24, 32, 16, 24, 60, 36, 16, 60, 48, 60, 4, 16, 72, 24, 80, 24, 72, 24, 36, 24, 60, 60, 24, 8, 12, 96, 16, 20, 96, 80, 96, 50, 40, 72, 72, 16, 108, 16, 8, 54, 100, 12, 64, 108, 100, 32, 24

Offset: 1

Labos Elemer, Aug 13 2002

nonn

a(n) = 2 for n in A063783. - Robert Israel, Jul 12 2023

			For n=10: D={1,2,5,10}, 1+8+125+1000=1134, divisors(1134)={1,2,3,6,7,9,14,18,21,27,42,54,63,81,126,162,189,378,567,1134} so a(10)=20.

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

Cf. A000005, A001158, A062068, A063783, A073802-A073813.

f:= n -> numtheory:-tau(numtheory:-sigma[3](n)):
map(f, [$1..100]); # Robert Israel, Jul 12 2023

Table[DivisorSigma[0,DivisorSigma[3,n]],{n,80}] (* Harvey P. Dale, Dec 13 2011 *)

a(n) = numdiv(sigma(n, 3)); \\ Michel Marcus, Jul 13 2023

a(n) = A000005(A001158(n)).

A073812 Number of common divisors of sigma(n) and phi(n).

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Aug 13 2002

			n=36: sigma(36)=91; phi(36)=12; Intersection[{1,7,13,91},{1,2,3,4,6,12}]={1}, so a(36)=1.

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

Cf. A000010, A000203, A073802, A073808, A073809, A073810, A073811.

g1[x_] := Divisors[DivisorSigma[1, x]] g2[x_] := Divisors[EulerPhi[x]] ncd[x_] := Length[Intersection[g1[x], g2[x]]] Table[ncd[w], {w, 1, 128}]
a[n_] := DivisorSigma[0, GCD[DivisorSigma[1, n], EulerPhi[n]]]; Array[a, 100] (* Amiram Eldar, Oct 18 2019 *)

a(n) = Card[Intersection[D[A000203(n)], D[A000010(n)]]].

A378267 Numbers k that have a record number of common divisors with sigma(k).

Original entry on oeis.org

1, 6, 24, 120, 672, 4320, 26208, 30240, 524160, 2178540, 8714160, 8910720, 17428320, 45532800, 132723360, 208565280, 240589440, 470564640, 668304000, 1307124000, 5228496000, 10805558400, 14182439040, 31998395520, 159991977600

Offset: 1

Amiram Eldar, Nov 21 2024

Indices of records in A073802.
This sequence is infinite since A073802 is unbounded. For example, for any odd number m we have A073802(2^(m-1)*(2^m-1)) >= A000005(m) and the number of divisors of odd numbers is unbounded.
The corresponding record values are 1, 4, 6, 16, 24, 40, 60, 96, 144, 216, 240, 336, ... .
a(26) <= 799959888000.

Cf. A000005, A000203 (sigma), A014567, A073802, A216793.

seq[kmax_] := Module[{d, dmax = 0, s = {}}, Do[d = DivisorSigma[0, GCD[k, DivisorSigma[1, k]]]; If[d > dmax, dmax = d; AppendTo[s, k]], {k, 1, kmax}]; s]; seq[10^6]

lista(kmax) = {my(d, dmax = 0); for(k = 1, kmax, d = numdiv(gcd(k, sigma(k))); if(d > dmax, dmax = d; print1(k, ", ")));}

cp's OEIS Frontend

A073810 Number of common divisors of sigma(2,n) and sigma(3,n).

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A073811 Number of common divisors of n and phi(n).

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PARI

PARI

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A073806 Number of divisors of sum of square of divisors.

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A073807 Number of divisors of sum of cube of divisors.

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Maple

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A073812 Number of common divisors of sigma(n) and phi(n).

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A378267 Numbers k that have a record number of common divisors with sigma(k).

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PARI