A028235 -id:A028235 - cp's OEIS frontend

A332880 If n = Product (p_j^k_j) then a(n) = numerator of Product (1 + 1/p_j).

1, 3, 4, 3, 6, 2, 8, 3, 4, 9, 12, 2, 14, 12, 8, 3, 18, 2, 20, 9, 32, 18, 24, 2, 6, 21, 4, 12, 30, 12, 32, 3, 16, 27, 48, 2, 38, 30, 56, 9, 42, 16, 44, 18, 8, 36, 48, 2, 8, 9, 24, 21, 54, 2, 72, 12, 80, 45, 60, 12, 62, 48, 32, 3, 84, 24, 68, 27, 32, 72

Offset: 1

Ilya Gutkovskiy, Feb 28 2020

Numerator of sum of reciprocals of squarefree divisors of n.

(6/Pi^2) * A332881(n)/a(n) is the asymptotic density of numbers that are coprime to their digital sum in base n+1 (see A094387 and A339076 for bases 2 and 10). - Amiram Eldar, Nov 24 2022

			1, 3/2, 4/3, 3/2, 6/5, 2, 8/7, 3/2, 4/3, 9/5, 12/11, 2, 14/13, 12/7, 8/5, 3/2, 18/17, ...

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

Cf. A001615, A008683, A017665, A028235, A028236, A048250, A076512, A306695, A308443, A308462, A332881 (denominators), A332882.
Cf. also A348734, A348735.
Cf. A059956, A065463, A082020, A094387, A339076.

a:= n-> numer(mul(1+1/i[1], i=ifactors(n)[2])):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 28 2020

Table[If[n == 1, 1, Times @@ (1 + 1/#[[1]] & /@ FactorInteger[n])], {n, 1, 70}] // Numerator
Table[Sum[MoebiusMu[d]^2/d, {d, Divisors[n]}], {n, 1, 70}] // Numerator

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A332880(n) = numerator(A001615(n)/n);

Numerators of coefficients in expansion of Sum_{k>=1} mu(k)^2*x^k/(k*(1 - x^k)).
a(n) = numerator of Sum_{d|n} mu(d)^2/d.
a(n) = numerator of psi(n)/n.
a(p) = p + 1, where p is prime.
a(n) = A001615(n) / A306695(n) = A001615(n) / gcd(n, A001615(n)). - Antti Karttunen, Nov 15 2021
From Amiram Eldar, Nov 24 2022: (Start)
Asymptotic means:
Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A332881(k) = 15/Pi^2 = 1.519817... (A082020).
Limit_{m->oo} (1/m) * Sum_{k=1..m} A332881(k)/a(k) = Product_{p prime} (1 - 1/(p*(p+1))) = 0.704442... (A065463). (End)

A326689 Numerator of the fraction (Sum_{prime p | n} 1/p - 1/n).

Original entry on oeis.org

-1, 0, 0, 1, 0, 2, 0, 3, 2, 3, 0, 3, 0, 4, 7, 7, 0, 7, 0, 13, 3, 6, 0, 19, 4, 7, 8, 17, 0, 1, 0, 15, 13, 9, 11, 29, 0, 10, 5, 27, 0, 20, 0, 25, 23, 12, 0, 13, 6, 17, 19, 29, 0, 22, 3, 5, 7, 15, 0, 61, 0, 16, 29, 31, 17, 10, 0, 37, 25, 29, 0, 59, 0, 19, 13, 41

Offset: 1

Jonathan Sondow, Jul 18 2019

See Comments on denominators in A326690.

			-1/1, 0/1, 0/1, 1/4, 0/1, 2/3, 0/1, 3/8, 2/9, 3/5, 0/1, 3/4, 0/1, 4/7, 7/15, 7/16, 0/1, 7/9, 0/1, 13/20, 3/7, 6/11, 0/1, 19/24, 4/25, 7/13, 8/27, 17/28, 0/1, 1/1

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

Denominators are A326690. Cf. also A007850, A309132, A309235, A309378.

Cf. A028235.

PrimeFactors[n_] := Select[Divisors[n], PrimeQ];
g[n_] := Numerator[Sum[1/p, {p, PrimeFactors[n]}] - 1/n];
Table[ g[n], {n, 100}]

a(n) = numerator(sumdiv(n, d, isprime(d)/d) - 1/n); \\ Michel Marcus, Jul 19 2019

a(p) = 0 if p is a prime.

a(g) = 1 if g is a known Giuga number (see my 2nd comment in A007850).

A332882 If n = Product (p_j^k_j) then a(n) = numerator of Product (1 + 1/p_j^k_j).

Original entry on oeis.org

1, 3, 4, 5, 6, 2, 8, 9, 10, 9, 12, 5, 14, 12, 8, 17, 18, 5, 20, 3, 32, 18, 24, 3, 26, 21, 28, 10, 30, 12, 32, 33, 16, 27, 48, 25, 38, 30, 56, 27, 42, 16, 44, 15, 4, 36, 48, 17, 50, 39, 24, 35, 54, 14, 72, 9, 80, 45, 60, 2, 62, 48, 80, 65, 84, 24, 68, 45, 32, 72

Offset: 1

Ilya Gutkovskiy, Feb 28 2020

Numerator of sum of reciprocals of unitary divisors of n.

			1, 3/2, 4/3, 5/4, 6/5, 2, 8/7, 9/8, 10/9, 9/5, 12/11, 5/3, 14/13, 12/7, 8/5, 17/16, ...

Antti Karttunen, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Unitary Divisor.

Cf. A017665, A028235, A028236, A034448, A077610, A306633, A319676, A323166, A332880, A332883 (denominators).

Cf. also A348734, A348735.

a:= n-> numer(mul(1+i[1]^i[2], i=ifactors(n)[2])/n):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 28 2020

Table[If[n == 1, 1, Times @@ (1 + 1/#[[1]]^#[[2]] & /@ FactorInteger[n])], {n, 1, 70}] // Numerator
Table[Sum[If[GCD[d, n/d] == 1,  1/d, 0], {d, Divisors[n]}], {n, 1, 70}] // Numerator

a(n) = numerator(sumdiv(n, d, if (gcd(d, n/d)==1, 1/d))); \\ Michel Marcus, Feb 28 2020

a(n) = numerator of Sum_{d|n, gcd(d, n/d) = 1} 1/d.
a(n) = numerator of usigma(n)/n.
a(p) = p + 1, where p is prime.
a(n) = A034448(n) / A323166(n). - Antti Karttunen, Nov 13 2021
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A332883(k) = zeta(2)/zeta(3) = 1.368432... (A306633). - Amiram Eldar, Nov 21 2022

A373439 Numerator of sum of reciprocals of square divisors of n.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 5, 1, 1, 1, 21, 1, 10, 1, 5, 1, 1, 1, 5, 26, 1, 10, 5, 1, 1, 1, 21, 1, 1, 1, 25, 1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 21, 50, 26, 1, 5, 1, 10, 1, 5, 1, 1, 1, 5, 1, 1, 10, 85, 1, 1, 1, 5, 1, 1, 1, 25, 1, 1, 26, 5, 1, 1, 1, 21, 91, 1, 1, 5, 1

Offset: 1

Ilya Gutkovskiy, Jun 05 2024

			1, 1, 1, 5/4, 1, 1, 1, 5/4, 10/9, 1, 1, 5/4, 1, 1, 1, 21/16, 1, 10/9, 1, 5/4, 1, 1, 1, 5/4, 26/25, ...

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

Cf. A007406, A017665, A017667, A028235, A035316, A332880, A373440 (denominators).

nmax = 85; CoefficientList[Series[Sum[x^(k^2)/(k^2 (1 - x^(k^2))), {k, 1, nmax}], {x, 0, nmax}], x] // Rest // Numerator
f[p_, e_] := (p^2 - p^(-2*Floor[e/2]))/(p^2-1); a[1] = 1; a[n_] := Numerator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Jun 26 2024 *)

a(n) = numerator(sumdiv(n, d, if (issquare(d), 1/d))); \\ Michel Marcus, Jun 05 2024

Numerators of coefficients in expansion of Sum_{k>=1} x^(k^2)/(k^2*(1 - x^(k^2))).
a(n) is the numerator of Sum_{d^2|n} 1/d^2.
From Amiram Eldar, Jun 26 2024: (Start)
Let f(n) = a(n)/A373440(n). Then:
f(n) is multiplicative with f(p^e) = (p^2 - p^(-2*floor(e/2)))/(p^2-1).
Dirichlet g.f. of f(n): zeta(s) * zeta(2*s+2).
Sum_{k=1..n} f(k) ~ zeta(4) * n. (End)

A354432 a(n) is the numerator of the sum of the reciprocals of the nonprime divisors of n.

Original entry on oeis.org

1, 1, 1, 5, 1, 7, 1, 11, 10, 11, 1, 3, 1, 15, 16, 23, 1, 4, 1, 7, 22, 23, 1, 5, 26, 27, 31, 19, 1, 41, 1, 47, 34, 35, 36, 61, 1, 39, 40, 31, 1, 55, 1, 29, 6, 47, 1, 7, 50, 29, 52, 17, 1, 25, 56, 3, 58, 59, 1, 53, 1, 63, 74, 95, 66, 83, 1, 22, 70, 17, 1, 15, 1, 75, 28

Offset: 1

Ilya Gutkovskiy, May 28 2022

			1, 1, 1, 5/4, 1, 7/6, 1, 11/8, 10/9, 11/10, 1, 3/2, 1, 15/14, 16/15, 23/16, ...

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

Cf. A017665, A018252, A023890, A028235, A354433 (denominators).

Table[DivisorSum[n, 1/# &, !PrimeQ[#] &], {n, 75}] // Numerator

a(n) = numerator(sumdiv(n, d, if(!isprime(d), 1/d))) \\ Michael S. Branicky, May 28 2022

from fractions import Fraction
from sympy import divisors, isprime
def a(n): return sum(Fraction(1, d) for d in divisors(n, generator=True) if not isprime(d)).numerator
print([a(n) for n in range(1, 76)]) # Michael S. Branicky, May 28 2022

from math import prod
from fractions import Fraction
from sympy import factorint
def A354432(n):
    f = factorint(n)
    return (Fraction(prod(p**(e+1)-1 for p, e in f.items()),prod(p-1 for p in f)*n) - sum(Fraction(1,p) for p in f)).numerator # Chai Wah Wu, May 28 2022

a(p) = 1 for prime p. - Michael S. Branicky, May 28 2022

A379141 If n = Product (p_j^k_j) then a(n) = numerator of Sum 1/k_j.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 3, 1, 1, 2, 2, 2, 1, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 5, 1, 2, 3, 1, 2, 3, 1, 3, 2, 3, 1, 5, 1, 2, 3, 3, 2, 3, 1, 5, 1, 2, 1, 5, 2, 2, 2, 4, 1, 5, 2, 3, 2, 2, 2, 6, 1, 3, 3, 1, 1, 3, 1, 4, 3, 2, 1, 5, 1, 3

Offset: 1

Ilya Gutkovskiy, Dec 16 2024

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

Cf. A001222, A005361, A028235, A028236.

a:= n-> numer(add(1/i[2], i=ifactors(n)[2])):
seq(a(n), n=1..110);  # Alois P. Heinz, Dec 16 2024

Join[{0}, Table[Plus @@ (1/#[[2]] & /@ FactorInteger[n]), {n, 2, 110}]] // Numerator

a(n) = my(f=factor(n)); numerator(sum(k=1, #f~, 1/f[k,2])); \\ Michel Marcus, Dec 16 2024

A379967 Arithmetic derivative of {n divided by its largest squarefree divisor}: a(n) = A003415(A003557(n)).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 4, 1, 0, 0, 1, 0, 0, 0, 12, 0, 1, 0, 1, 0, 0, 0, 4, 1, 0, 6, 1, 0, 0, 0, 32, 0, 0, 0, 5, 0, 0, 0, 4, 0, 0, 0, 1, 1, 0, 0, 12, 1, 1, 0, 1, 0, 6, 0, 4, 0, 0, 0, 1, 0, 0, 1, 80, 0, 0, 0, 1, 0, 0, 0, 16, 0, 0, 1, 1, 0, 0, 0, 12, 27, 0, 0, 1, 0, 0, 0, 4, 0, 1, 0, 1, 0, 0, 0, 32, 0, 1, 1, 7, 0, 0, 0, 4

Offset: 1

Antti Karttunen, Jan 22 2025

nonn

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

Cf. A003415, A003557.

Cf. also A028235, A341998, A342001.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factor(n)[, 1]));
A379967(n) = A003415(A003557(n));

a(n) = A003415(A003557(n)).

A380314 Numerator of sum of reciprocals of all prime divisors of all positive integers <= n.

Original entry on oeis.org

0, 1, 5, 4, 23, 71, 527, 316, 117, 283, 3183, 5737, 75736, 170777, 186793, 100904, 1730383, 1295397, 24782713, 13522987, 42878411, 91488457, 2113934201, 1149922463, 234446350, 494634185, 169835681, 89698402, 2608690087, 84946052281, 2639797313941, 1370038779503, 1412581913773

Offset: 1

Ilya Gutkovskiy, Jan 20 2025

Prime divisors counted without multiplicity.

			0, 1/2, 5/6, 4/3, 23/15, 71/30, 527/210, 316/105, 117/35, 283/70, 3183/770, 5737/1155, 75736/15015, ...

Robert Israel, Table of n, a(n) for n = 1..2346

Cf. A000720, A007947, A013939, A024451, A024924, A028235, A284648, A379367, A380315 (denominators).

N:= 100: # for a(1) .. a(N)
P:= select(isprime,[$1..N]):
f:= proc(n) local k;
  numer(add(floor(n/P[k])/P[k],k=1..numtheory:-pi(n)))
end proc:
map(f, [$1..N]); # Robert Israel, Jan 26 2025

Table[DivisorSum[n, 1/# &, PrimeQ[#] &], {n, 1, 33}] // Accumulate // Numerator
Table[Sum[Floor[n/Prime[k]]/Prime[k], {k, 1, n}], {n, 1, 33}] // Numerator
nmax = 33; CoefficientList[Series[1/(1 - x) Sum[x^Prime[k]/(Prime[k] (1 - x^Prime[k])), {k, 1, nmax}], {x, 0, nmax}], x] // Numerator // Rest

a(n) = my(vp=primes(primepi(n))); numerator(sum(k=1, #vp, (n\vp[k])/vp[k])); \\ Michel Marcus, Jan 26 2025

G.f. for fractions: (1/(1 - x)) * Sum_{k>=1} x^prime(k) / (prime(k)*(1 - x^prime(k))).

a(n) is the numerator of Sum_{k=1..pi(n)} floor(n/prime(k)) / prime(k).

A367202 If n = Product(p_i^e_i), a(n) = Sum_{i = 1..k}(rad(n)/p_i)^e_i, where rad is A007947.

Original entry on oeis.org

0, 1, 1, 1, 1, 5, 1, 1, 1, 7, 1, 11, 1, 9, 8, 1, 1, 7, 1, 27, 10, 13, 1, 29, 1, 15, 1, 51, 1, 31, 1, 1, 14, 19, 12, 13, 1, 21, 16, 127, 1, 41, 1, 123, 28, 25, 1, 83, 1, 9, 20, 171, 1, 11, 16, 345, 22, 31, 1, 241, 1, 33, 52, 1, 18, 61, 1, 291, 26, 59, 1, 31, 1

Offset: 1

David James Sycamore, Nov 10 2023

nonn

Diverges from A028235 at a(12).

			a(1) = 0, the empty sum.
rad(6) = rad(2*3) = 6 -->a(6) = (6/2)^1 + (6/3)^1 = 3 + 2 = 5.
rad(12) = rad(2^2*3) = 6 -->a(12) = (6/2)^2 + (6/3)^1 = 9 + 2 = 11.
rad(36) = rad(2^2*3^2) = 6 --> a(36) = (6/2)^2 +(6/3)^2 = 9 + 4 = 13.
rad(40) = rad(2^3*5^1) = 10 -->a(40) = (10/2)^3 + (10/5)^1 = 125 + 2 = 127.
n = 30 = 2*3*5 a squarefree number; a(30) = (30/2) + (30/3) + (30/5) = 15 + 10 +  6 = 31

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14, showing primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue, highlighting squareful numbers that are not prime powers in large light blue.

Cf. A000040, A001414, A007947, A028235.

Array[Function[{r, w}, Total[Power @@@ Transpose@ {r/w[[All, 1]], w[[All, -1]]}]] @@ {Times @@ #[[All, 1]], #} &@ FactorInteger[#] &, 120] (* Michael De Vlieger, Nov 10 2023 *)

rad(f) = factorback(f[, 1]);
a(n) = my(f=factor(n)); sum(i=1, #f~,(rad(f)/f[i,1])^f[i,2]); \\ Michel Marcus, Nov 10 2023

For n a prime power p^k, a(n) = (p/p)^1 = 1.
For n a squarefree semiprime a(n) = A001414(n).
For p,q distinct primes a(p*q^2) = q + p^2.
For n a squarefree number with prime divisors p_1,p_2..p_k, a(n) = Sum_{i = 1..k}(n/p_i) see Example

More terms from Michael De Vlieger, Nov 10 2023

A380315 Denominator of sum of reciprocals of all prime divisors of all positive integers <= n.

Original entry on oeis.org

1, 2, 6, 3, 15, 30, 210, 105, 35, 70, 770, 1155, 15015, 30030, 30030, 15015, 255255, 170170, 3233230, 1616615, 4849845, 9699690, 223092870, 111546435, 22309287, 44618574, 14872858, 7436429, 215656441, 6469693230, 200560490130, 100280245065, 100280245065

Offset: 1

Ilya Gutkovskiy, Jan 20 2025

Prime divisors counted without multiplicity.

Differs from A379370 first at n=15.

			0, 1/2, 5/6, 4/3, 23/15, 71/30, 527/210, 316/105, 117/35, 283/70, 3183/770, 5737/1155, 75736/15015, ...

Cf. A000720, A007947, A013939, A024924, A028235, A284650, A379370, A379368, A380314 (numerators).

Table[DivisorSum[n, 1/# &, PrimeQ[#] &], {n, 1, 33}] // Accumulate // Denominator
Table[Sum[Floor[n/Prime[k]]/Prime[k], {k, 1, n}], {n, 1, 33}] // Denominator
nmax = 33; CoefficientList[Series[1/(1 - x) Sum[x^Prime[k]/(Prime[k] (1 - x^Prime[k])), {k, 1, nmax}], {x, 0, nmax}], x] // Denominator // Rest

a(n) = my(vp=primes(primepi(n))); denominator(sum(k=1, #vp, (n\vp[k])/vp[k])); \\ Michel Marcus, Jan 26 2025

G.f. for fractions: (1/(1 - x)) * Sum_{k>=1} x^prime(k) / (prime(k)*(1 - x^prime(k))).

a(n) is the denominator of Sum_{k=1..pi(n)} floor(n/prime(k)) / prime(k).

cp's OEIS Frontend

A332880 If n = Product (p_j^k_j) then a(n) = numerator of Product (1 + 1/p_j).

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A326689 Numerator of the fraction (Sum_{prime p | n} 1/p - 1/n).

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A332882 If n = Product (p_j^k_j) then a(n) = numerator of Product (1 + 1/p_j^k_j).

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A373439 Numerator of sum of reciprocals of square divisors of n.

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A354432 a(n) is the numerator of the sum of the reciprocals of the nonprime divisors of n.

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A379141 If n = Product (p_j^k_j) then a(n) = numerator of Sum 1/k_j.

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A379967 Arithmetic derivative of {n divided by its largest squarefree divisor}: a(n) = A003415(A003557(n)).

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