A360325 -id:A360325 - cp's OEIS frontend

A360327 a(n) is the sum of divisors of n that have only prime-indexed prime factors.

1, 1, 4, 1, 6, 4, 1, 1, 13, 6, 12, 4, 1, 1, 24, 1, 18, 13, 1, 6, 4, 12, 1, 4, 31, 1, 40, 1, 1, 24, 32, 1, 48, 18, 6, 13, 1, 1, 4, 6, 42, 4, 1, 12, 78, 1, 1, 4, 1, 31, 72, 1, 1, 40, 72, 1, 4, 1, 60, 24, 1, 32, 13, 1, 6, 48, 68, 18, 4, 6, 1, 13, 1, 1, 124, 1, 12

Offset: 1

Amiram Eldar, Feb 03 2023

Equivalently, a(n) is the sum of divisors of the largest divisor of n that has only prime-indexed prime factors.

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

Cf. A000203, A006450, A076610, A320628, A360325, A360326.

f[p_, e_] := If[PrimeQ[PrimePi[p]], (p^(e+1)-1)/(p-1), 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), p = f[,1], e = f[,2]); prod(i = 1, #p, if(isprime(primepi(p[i])), (p[i]^(e[i]+1)-1)/(p[i]-1), 1));}

a(n) = 1 if and only if n is in A320628.
a(n) = A000203(n) if and only if n is in A076610.
a(n) = A000203(A360325(n)).
Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if p is a prime-indexed prime (A006450), and 1 otherwise.

A360329 a(n) is the largest divisor of n that has only prime factors that are not prime-indexed primes.

Original entry on oeis.org

1, 2, 1, 4, 1, 2, 7, 8, 1, 2, 1, 4, 13, 14, 1, 16, 1, 2, 19, 4, 7, 2, 23, 8, 1, 26, 1, 28, 29, 2, 1, 32, 1, 2, 7, 4, 37, 38, 13, 8, 1, 14, 43, 4, 1, 46, 47, 16, 49, 2, 1, 52, 53, 2, 1, 56, 19, 58, 1, 4, 61, 2, 7, 64, 13, 2, 1, 4, 23, 14, 71, 8, 73, 74, 1, 76, 7

Offset: 1

Amiram Eldar, Feb 03 2023

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

Cf. A006450, A007821, A076610, A302590, A320628, A360325, A360330, A360331.

f[p_, e_] := If[PrimeQ[PrimePi[p]], 1, p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); for(i = 1, #f~, if(isprime(primepi(f[i,1])), f[i,1]=1)); factorback(f);}

a(n) = 1 if and only if n is in A076610.
a(n) = n if and only if n is in A320628.
a(n) = n/A360325(n).
Multiplicative with a(p^e) = 1 if p is a prime-indexed prime (A006450), and p^e otherwise (A007821).
Sum_{k=1..n} a(k) ~ (1/2) * c * n^2, where c = Product_{p in A006450} p/(p+1) < 0.4 (see A302590 for an estimate of 1/c).

A360326 a(n) is the number of divisors of n that have only prime-indexed prime factors.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Feb 03 2023

First differs from A322976 at n = 21.
Equivalently, a(n) is the number of divisors of the largest divisor of n that has only prime-indexed prime factors.
The asymptotic mean of this sequence is Product_{p in A006450} p/(p-1) > 3. See A076610 for a numerical estimate of the value of this product.

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

Cf. A000005, A006450, A076610, A320628, A322976, A360325, A360327.

f[p_, e_] := If[PrimeQ[PrimePi[p]], e+1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), p = f[,1], e = f[,2]); prod(i = 1, #p, if(isprime(primepi(p[i])), e[i]+1, 1));}

a(n) = 1 if and only if n is in A320628.
a(n) = A000005(n) if and only if n is in A076610.
a(n) = A000005(A360325(n)).
Multiplicative with a(p^e) = e+1 if p is a prime-indexed prime (A006450), and 1 otherwise.

cp's OEIS Frontend

A360327 a(n) is the sum of divisors of n that have only prime-indexed prime factors.

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A360329 a(n) is the largest divisor of n that has only prime factors that are not prime-indexed primes.

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A360326 a(n) is the number of divisors of n that have only prime-indexed prime factors.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula