A083346 -id:A083346 - cp's OEIS frontend

A359593 Multiplicative with a(p^e) = 1 if p divides e, p^e otherwise.

1, 2, 3, 1, 5, 6, 7, 8, 9, 10, 11, 3, 13, 14, 15, 1, 17, 18, 19, 5, 21, 22, 23, 24, 25, 26, 1, 7, 29, 30, 31, 32, 33, 34, 35, 9, 37, 38, 39, 40, 41, 42, 43, 11, 45, 46, 47, 3, 49, 50, 51, 13, 53, 2, 55, 56, 57, 58, 59, 15, 61, 62, 63, 1, 65, 66, 67, 17, 69, 70, 71, 72, 73, 74, 75, 19, 77, 78, 79, 5, 81, 82, 83, 21

Offset: 1

Antti Karttunen, Jan 09 2023

Each term a(n) is a multiple of both A083346(n) and A327938(n).

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

Cf. A072873 (positions of 1's), A359594.

Cf. also A083346, A327938, A342014, A359552.

f[p_, e_] := If[Divisible[e, p], 1, p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 09 2023 *)

A359593(n) = { my(f = factor(n)); prod(k=1, #f~, f[k, 1]^(f[k,2]*!!(f[k, 2]%f[k, 1]))); };

from math import prod
from sympy import factorint
def A359593(n): return prod(p**e for p, e in factorint(n).items() if e%p) # Chai Wah Wu, Jan 10 2023

a(n) = n / A359594(n).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1 - p^(p-1)*(p-1)/(p^(2*p)-1)) = 0.4225104173... . - Amiram Eldar, Jan 11 2023

A188901 Integers in the sequences (arithmetic derivative of k) divided by k.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 3, 2, 5, 4, 3, 1, 6, 5, 4, 2, 7, 3, 6, 5, 3, 8, 4, 2, 7, 6, 4, 9, 5, 3, 8, 4, 7, 5, 1, 10, 6, 4, 9, 5, 3, 8, 6, 2, 11, 7, 5, 10, 6, 4, 2, 9, 7, 3, 5, 12, 8, 6, 2, 11, 7, 5, 3, 10, 8, 4, 6, 4, 13, 9, 7, 3, 12, 8, 6, 4, 11

Offset: 1

Giorgio Balzarotti, Apr 16 2011

nonn

Integers in the sequence (k'/k) = A003415(k)/k. See A072873 for the values of k where k'/k is an integer.

			1' = 0, 0/1 = 0 -> a(1) = 0;
4' = 4 ,4/4 = 1 -> a(2) = 1;
16' = 32, 32/16 = 2 -> a(3) = 2.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 2500 terms from Nathaniel Johnston)

Cf. A003415, A068311, A072873, A083346, A083346, A085731, A086130, A189100, A189102.

A267143 Primes q such that Sum_(q-1; i=1..m) e(i)/p(i) is an integer k, where the prime factorization of n is Product_(n; i=1..m) p(i)^e(i).

Original entry on oeis.org

5, 17, 109, 257, 433, 2917, 65537, 746497, 1350001, 1769473, 3294173, 5038849, 5400001, 8503057, 21600001, 28311553, 57395629, 113246209, 145800001, 210827009, 984150001, 1811939329, 2500000001, 3936600001, 4218750001, 5692329217, 9331200001, 16875000001

Offset: 1

Jaroslav Krizek, Jan 11 2016

nonn

Primes from the set {A072873(n) + 1: n>1}.
Fermat primes > 3 from A019434 are in the sequence.
Corresponding values of k: 1, 2, 2, 4, 3, 3, 8, 7, 4, 9, 2, 7, 5, ...

			Prime 433 is a term because 432 = 2^4 * 3^3 and 4/2 + 3/3 = 3 (integer).

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 185 terms from Jaroslav Krizek)

Cf. A019434, A072873, A083345, A083346.

[n: n in [3..10^8] | IsPrime(n) and Denominator(&+[p[2]/p[1]: p in Factorization(n-1)]) eq 1];

isA072873(n)=my(f=factor(n)); for(i=1, #f~, if(f[i, 2]%f[i, 1], return(0))); 1
lista(nn) = {forprime(p=2, nn, if (isA072873(p-1), print1(p, ", ")););} \\ Michel Marcus, Jan 21 2016

A381959 Denominator of the sum of the reciprocals of the indices of distinct prime factors of n.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 1, 2, 3, 5, 2, 6, 4, 6, 1, 7, 2, 8, 3, 4, 5, 9, 2, 3, 6, 2, 4, 10, 6, 11, 1, 10, 7, 12, 2, 12, 8, 3, 3, 13, 4, 14, 5, 6, 9, 15, 2, 4, 3, 14, 6, 16, 2, 15, 4, 8, 10, 17, 6, 18, 11, 4, 1, 2, 10, 19, 7, 18, 12, 20, 2, 21, 12, 6, 8, 20, 3, 22, 3, 2, 13, 23, 4, 21, 14, 5, 5, 24, 6, 12, 9, 22, 15, 24

Offset: 1

Ilya Gutkovskiy, Mar 11 2025

			0, 1, 1/2, 1, 1/3, 3/2, 1/4, 1, 1/2, 4/3, 1/5, 3/2, 1/6, 5/4, 5/6, 1, 1/7, 3/2, 1/8, 4/3, ...

Index entries for sequences computed from indices in prime factorization

Cf. A000720, A007947, A066328, A083346, A318574, A381958 (numerators).

Join[{1}, Table[Plus @@ (1/PrimePi[#[[1]]] & /@ FactorInteger[n]), {n, 2, 95}] // Denominator]
nmax = 95; CoefficientList[Series[Sum[x^Prime[k]/(k (1 - x^Prime[k])), {k, 1, nmax}], {x, 0, nmax}], x] // Denominator // Rest

a(n) = my(f=factor(n)); denominator(sum(k=1, #f~, 1/primepi(f[k,1]))); \\ Michel Marcus, Mar 11 2025

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

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

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A359593 Multiplicative with a(p^e) = 1 if p divides e, p^e otherwise.

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A188901 Integers in the sequences (arithmetic derivative of k) divided by k.

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A267143 Primes q such that Sum_(q-1; i=1..m) e(i)/p(i) is an integer k, where the prime factorization of n is Product_(n; i=1..m) p(i)^e(i).

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A381959 Denominator of the sum of the reciprocals of the indices of distinct prime factors of n.

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