A277791 -id:A277791 - cp's OEIS frontend

A355815 a(n) = gcd(A276086(n), A277791(n)), where A276086 is primorial base exp-function and A277791 is the denominator of sum of reciprocals of proper divisors of n.

1, 1, 1, 1, 1, 1, 1, 1, 3, 5, 1, 1, 1, 1, 15, 1, 1, 1, 1, 5, 3, 1, 1, 1, 5, 1, 3, 1, 1, 1, 1, 1, 3, 1, 7, 1, 1, 1, 3, 5, 1, 7, 1, 1, 15, 1, 1, 1, 7, 25, 3, 1, 1, 1, 5, 1, 3, 1, 1, 1, 1, 1, 21, 1, 1, 1, 1, 1, 3, 35, 1, 1, 1, 1, 25, 1, 7, 1, 1, 1, 3, 1, 1, 7, 5, 1, 3, 1, 1, 1, 7, 1, 3, 1, 1, 1, 1, 49, 3, 5, 1, 1, 1, 1, 105

Offset: 1

Antti Karttunen, Jul 18 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 1..11550
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to primorial base

Sequence contains only terms of A048103.
Cf. A276086, A277791.
Cf. also A327858, A355003.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A277791(n) = denominator((sigma(n)-1)/n); \\ From A277791
A355815(n) = gcd(A276086(n), A277791(n));

from math import gcd
from sympy import nextprime, divisor_sigma
def A355815(n):
    m, p, c = 1, 2, n
    while c:
        c, a = divmod(c,p)
        m *= p**a
        p = nextprime(p)
    return gcd(m,n//gcd(n, divisor_sigma(n)-1)) # Chai Wah Wu, Jul 18 2022

a(n) = gcd(A276086(n), A277791(n)).

A355003 a(n) = gcd(A003415(n), A277791(n)), where A003415 is the arithmetic derivative and A277791 is the denominator of sum of reciprocals of proper divisors of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 4, 1, 1, 1, 8, 1, 3, 1, 4, 1, 1, 1, 4, 5, 1, 9, 4, 1, 1, 1, 16, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 4, 3, 1, 1, 16, 7, 5, 1, 4, 1, 27, 1, 4, 1, 1, 1, 4, 1, 1, 3, 32, 1, 1, 1, 4, 1, 1, 1, 12, 1, 1, 5, 4, 1, 1, 1, 16, 27, 1, 1, 4, 1, 1, 1, 4, 1, 3, 1, 4, 1, 1, 1, 16, 1, 7, 3, 5, 1, 1, 1, 4, 1

Offset: 1

Antti Karttunen, Jul 18 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003415, A277791.

Cf. also A355815.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A277791(n) = denominator((sigma(n)-1)/n); \\ From A277791
A355003(n) = gcd(A003415(n), A277791(n));

a(n) = gcd(A003415(n), A277791(n)).

a(p^k) = p^(k-1) for all primes p and exponents k > 0.

A355694 Dirichlet inverse of A277791, denominator of sum of reciprocals of proper divisors of n.

Original entry on oeis.org

1, -1, -1, -1, -1, -4, -1, -1, -2, -8, -1, 9, -1, -12, -13, -1, -1, 6, -1, 1, -19, -20, -1, -10, -4, -24, -4, 1, -1, 26, -1, -1, -31, -32, -33, 27, -1, -36, -37, 10, -1, 34, -1, 1, -12, -44, -1, 35, -6, 2, -49, 1, -1, -14, -53, 62, -55, -56, -1, 87, -1, -60, -18, -1, -63, 110, -1, 1, -67, 42, -1, -57, -1, -72, 12

Offset: 1

Antti Karttunen, Jul 18 2022

sign

Antti Karttunen, Table of n, a(n) for n = 1..12000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A277791.

A277791(n) = denominator((sigma(n)-1)/n); \\ From A277791
memoA355694 = Map();
A355694(n) = if(1==n,1,my(v); if(mapisdefined(memoA355694,n,&v), v, v = -sumdiv(n,d,if(dA277791(n/d)*A355694(d),0)); mapput(memoA355694,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA277791(n/d) * a(d).

A277790 Numerator of sum of reciprocals of proper divisors of n.

Original entry on oeis.org

0, 1, 1, 3, 1, 11, 1, 7, 4, 17, 1, 9, 1, 23, 23, 15, 1, 19, 1, 41, 31, 35, 1, 59, 6, 41, 13, 55, 1, 71, 1, 31, 47, 53, 47, 5, 1, 59, 55, 89, 1, 95, 1, 83, 77, 71, 1, 41, 8, 46, 71, 97, 1, 119, 71, 17, 79, 89, 1, 167, 1, 95, 103, 63, 83, 13, 1, 125, 95, 143, 1, 97, 1, 113, 41, 139, 95, 167, 1, 37

Offset: 1

Ilya Gutkovskiy, Oct 31 2016

			a(4) = 3 because 4 has 3 divisors {1,2,4} therefore 2 proper divisors {1,2} and 1/1 + 1/2 = 3/2.
0, 1, 1, 3/2, 1, 11/6, 1, 7/4, 4/3, 17/10, 1, 9/4, 1, 23/14, 23/15, 15/8, 1, 19/9, 1, 41/20, 31/21, 35/22, 1, 59/24, 6/5, 41/26, 13/9, 55/28, ...

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Eric Weisstein's World of Mathematics, Restricted Divisor Function
Index entries for sequences related to sums of divisors

Cf. A000203, A001065, A017665, A017666, A277791 (denominators).

with(numtheory): P:=proc(n) local a,k; a:=divisors(n) minus {n};
numer(add(1/a[k],k=1..nops(a))); end: seq(P(i),i=1..80); # Paolo P. Lava, Oct 17 2018

Table[Numerator[DivisorSigma[-1, n] - 1/n], {n, 1, 80}]
Table[Numerator[(DivisorSigma[1, n] - 1)/n], {n, 1, 80}]

a(n) = numerator((sigma(n)-1)/n); \\ Michel Marcus, Nov 01 2016

from math import gcd
from sympy import divisor_sigma
def A277790(n): return (m:=divisor_sigma(n)-1)//gcd(n,m) # Chai Wah Wu, Jul 18 2022

a(n) = numerator(Sum_{d|n, d

a(n) = numerator((sigma_1(n)-1)/n).

a(p) = 1 when p is prime.

a(p^k) = (p^k - 1)/(p - 1) when p is prime.

Dirichlet g.f.: (zeta(s) - 1)*zeta(s+1) (for fraction Sum_{d|n, d

A281086 Denominator of sum of reciprocals of numbers less than n that do not divide n.

Original entry on oeis.org

1, 1, 2, 3, 12, 20, 20, 70, 840, 504, 2520, 27720, 27720, 51480, 360360, 180180, 720720, 1361360, 4084080, 77597520, 15519504, 470288, 5173168, 356948592, 1784742960, 686439600, 26771144400, 80313433200, 80313433200, 2329089562800, 2329089562800, 36100888223400

Offset: 1

Ilya Gutkovskiy, Jan 14 2017

			a(6) = 20 because 6 has 4 divisors {1,2,3,6} therefore 2 non-divisors {4,5} and 1/4 + 1/5 = 9/20.
0, 0, 1/2, 1/3, 13/12, 9/20, 29/20, 59/70, 1163/840, 569/504, 4861/2520, 21341/27720, 58301/27720, 79139/51480, 619181/360360, 260041/180180, ...

Cf. A000203, A001008, A002805, A017665, A017666, A024816, A277791, A281085 (numerators).

Table[Denominator[HarmonicNumber[n] - DivisorSigma[-1, n]], {n, 1, 32}]
Table[Denominator[HarmonicNumber[n] - DivisorSigma[1, n]/n], {n, 1, 32}]
Table[Denominator[Total[1/Complement[Range[n],Divisors[n]]]],{n,40}] (* Harvey P. Dale, Jan 04 2020 *)

a(n) = denominator(H_n - Sum_{d|n} 1/d), where H_n is the n-th harmonic number.
a(n) = denominator(A001008(n)/A002805(n) - A000203(n)/n).
Denominators of coefficients in expansion of -log(1 - x)/(1 - x) - Sum_{k>=1} log(1/(1 - x^k)).

A372836 a(n) is the numerator of Sum_{d|n, 1 < d < n} 1/d.

Original entry on oeis.org

0, 0, 0, 1, 0, 5, 0, 3, 1, 7, 0, 5, 0, 9, 8, 7, 0, 10, 0, 21, 10, 13, 0, 35, 1, 15, 4, 27, 0, 41, 0, 15, 14, 19, 12, 3, 0, 21, 16, 49, 0, 53, 0, 39, 32, 25, 0, 25, 1, 21, 20, 45, 0, 65, 16, 9, 22, 31, 0, 107, 0, 33, 40, 31, 18, 7, 0, 57, 26, 73, 0, 61, 0, 39, 16, 63, 18, 89, 0, 21

Offset: 1

Ilya Gutkovskiy, May 14 2024

			0, 0, 0, 1/2, 0, 5/6, 0, 3/4, 1/3, 7/10, 0, 5/4, 0, 9/14, 8/15, 7/8, 0, 10/9, ...

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

Cf. A017665, A048050, A277790, A277791.

nmax = 80; CoefficientList[Series[Sum[x^(2 k)/(k (1 - x^k)), {k, 2, nmax}], {x, 0, nmax}], x] // Numerator // Rest

a(n) = numerator(sumdiv(n, d, if ((d>1) && (dMichel Marcus, May 14 2024

Numerators of coefficients in expansion of Sum_{k>=2} x^(2*k) / (k * (1 - x^k)).

Showing 1-6 of 6 results.

cp's OEIS Frontend

A355815 a(n) = gcd(A276086(n), A277791(n)), where A276086 is primorial base exp-function and A277791 is the denominator of sum of reciprocals of proper divisors of n.

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A355003 a(n) = gcd(A003415(n), A277791(n)), where A003415 is the arithmetic derivative and A277791 is the denominator of sum of reciprocals of proper divisors of n.

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A355694 Dirichlet inverse of A277791, denominator of sum of reciprocals of proper divisors of n.

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A277790 Numerator of sum of reciprocals of proper divisors of n.

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A281086 Denominator of sum of reciprocals of numbers less than n that do not divide n.

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A372836 a(n) is the numerator of Sum_{d|n, 1 < d < n} 1/d.

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