A377519 -id:A377519 - cp's OEIS frontend

A377516 The number of divisors of n that are terms in A276078.

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

Offset: 1

Amiram Eldar, Oct 30 2024

The sum of these divisors is A377517(n), and the largest of them is A377515(n).

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

Cf. A000005, A000720, A013661, A276078, A377515, A377517, A377519.

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

a(n) = {my(f = factor(n)); prod(i = 1, #f~, min(primepi(f[i,1]), f[i,2]) + 1);}

a(n) = A000005(A377515(n)).
Multiplicative with a(p^e) = min(pi(p), e) + 1, where pi(n) = A000720(n).
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^((pi(p)+1)*s)).

A377518 The largest divisor of n that is a term in A207481.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 4, 9, 10, 11, 12, 13, 14, 15, 4, 17, 18, 19, 20, 21, 22, 23, 12, 25, 26, 27, 28, 29, 30, 31, 4, 33, 34, 35, 36, 37, 38, 39, 20, 41, 42, 43, 44, 45, 46, 47, 12, 49, 50, 51, 52, 53, 54, 55, 28, 57, 58, 59, 60, 61, 62, 63, 4, 65, 66, 67, 68, 69

Offset: 1

Amiram Eldar, Oct 30 2024

The number of these divisors is A377519(n), and their sum is A377520(n).

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

Cf. A207481, A368329, A377515, A377519, A377520.

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

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^min(f[i,1], f[i,2]));}

Multiplicative with a(p^e) = p^min(p, e).
a(n) = n if and only if n is in A207481.
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (p^((p+1)*s) - p^(p+1) - p^(p*s) + p^p)/p^((p+1)*s).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/(p^p * (p+1))) = 0.908130438292447963703... .

A377520 The sum of the divisors of n that are terms in A207481.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 7, 13, 18, 12, 28, 14, 24, 24, 7, 18, 39, 20, 42, 32, 36, 24, 28, 31, 42, 40, 56, 30, 72, 32, 7, 48, 54, 48, 91, 38, 60, 56, 42, 42, 96, 44, 84, 78, 72, 48, 28, 57, 93, 72, 98, 54, 120, 72, 56, 80, 90, 60, 168, 62, 96, 104, 7, 84, 144, 68

Offset: 1

Amiram Eldar, Oct 30 2024

First differs from A284341 at n = 81 = 3^4: a(81) = 40, while A284341(81) = 121.

The number of these divisors is A377519(n), and the largest of them is A377518(n).

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

Cf. A000203, A013661, A207481, A284341, A377517, A377518, A377519.

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

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^(min(f[i,1], f[i,2]) + 1) - 1)/(f[i,1] - 1));}

a(n) = A000203(A377518(n)).
Multiplicative with a(p^e) = (p^(min(p, e)+1) - 1)/(p - 1).
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (p^((p+1)*s) - p^(p+1))/p^((p+1)*s).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(2) * Product_{p prime} (1 - 1/p^(p+1)) = 1.42145673335960701365... .

A380089 The number of unitary divisors of n that are terms in A207481.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 1, 2, 4, 2, 4, 2, 4, 4, 1, 2, 4, 2, 4, 4, 4, 2, 2, 2, 4, 2, 4, 2, 8, 2, 1, 4, 4, 4, 4, 2, 4, 4, 2, 2, 8, 2, 4, 4, 4, 2, 2, 2, 4, 4, 4, 2, 4, 4, 2, 4, 4, 2, 8, 2, 4, 4, 1, 4, 8, 2, 4, 4, 8, 2, 2, 2, 4, 4, 4, 4, 8, 2, 2, 1, 4, 2, 8, 4, 4, 4

Offset: 1

Amiram Eldar, Jan 12 2025

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

Cf. A005117, A054743, A077610, A185359, A207481, A377519, A380086, A380088, A380090.

f[p_, e_] := If[e <= p, 2, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~,if(f[i,2] <= f[i,1], 2, 1));}

a(n) = A034444(A380088(n)).
Multiplicative with a(p^e) = 2 if e <= p, and 1 otherwise.
a(n) = 1 if and only if n is in A054743.
a(n) < A034444(n) if and only if n is in A185359.
a(n) = A034444(n) if and only if n is in A207481.
a(n) = A377519(n) if and only if n is squarefree (A005117).
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^s - 1/p^((p+1)*s)).
Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A034444(k) = Product_{p prime} (1 - 1/(2*p^(p+1))) = 0.93168306734008028353...

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A377516 The number of divisors of n that are terms in A276078.

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A377518 The largest divisor of n that is a term in A207481.

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A377520 The sum of the divisors of n that are terms in A207481.

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A380089 The number of unitary divisors of n that are terms in A207481.

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