A380089 -id:A380089 - cp's OEIS frontend

A380086 The number of unitary divisors of n that are terms in A276078.

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

Offset: 1

Amiram Eldar, Jan 11 2025

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

Cf. A000720, A005117, A034444, A077610, A276078, A325127, A377516, A380085, A380087, A380089.

f[p_, e_] := If[e <= PrimePi[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] <= primepi(f[i,1]), 2, 1));}

a(n) = A034444(A380085(n)).
Multiplicative with a(p^e) = 2 if e <= pi(p) = A000720(p), and 1 otherwise.
a(n) = 1 if and only if n is in A325127.
a(n) < A034444(n) if and only if n is in A276079.
a(n) = A034444(n) if and only if n is in A276078.
a(n) = A377516(n) if and only if n is squarefree (A005117).
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^s - 1/p^((pi(p)+1)*s)).
Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A034444(k) = Product_{p prime} (1 - 1/(2*p^(pi(p)+1))) = 0.85808348184674088116... .

A380088 The largest unitary divisor of n that is a term in A207481.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 1, 9, 10, 11, 12, 13, 14, 15, 1, 17, 18, 19, 20, 21, 22, 23, 3, 25, 26, 27, 28, 29, 30, 31, 1, 33, 34, 35, 36, 37, 38, 39, 5, 41, 42, 43, 44, 45, 46, 47, 3, 49, 50, 51, 52, 53, 54, 55, 7, 57, 58, 59, 60, 61, 62, 63, 1, 65, 66, 67, 68, 69, 70

Offset: 1

Amiram Eldar, Jan 12 2025

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

Cf. A005117, A054743, A077610, A185359, A207481, A377518, A380085, A380089, A380090.

f[p_, e_] := If[e <= p, 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~, if(f[i, 2] <= f[i, 1], f[i, 1]^f[i, 2], 1));}

Multiplicative with a(p^e) = p^e if e <= p, and 1 otherwise.
a(n) = 1 if and only if n is in A054743.
a(n) < n if and only if n is in A185359.
a(n) = n if and only if n is in A207481.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (p^(2*(p+1)) + p^(2*p+1) - p^(p+1) - p^p + 1)/(p^(2*p+1) * (p+1)) = 0.87453068804586281444... .

A380090 The sum of the unitary divisors of n that are terms in A207481.

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 1, 10, 18, 12, 20, 14, 24, 24, 1, 18, 30, 20, 30, 32, 36, 24, 4, 26, 42, 28, 40, 30, 72, 32, 1, 48, 54, 48, 50, 38, 60, 56, 6, 42, 96, 44, 60, 60, 72, 48, 4, 50, 78, 72, 70, 54, 84, 72, 8, 80, 90, 60, 120, 62, 96, 80, 1, 84, 144, 68, 90, 96

Offset: 1

Amiram Eldar, Jan 12 2025

First differs from A371242 at n = 27.

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

Cf. A005117, A054743, A077610, A185359, A207481, A371242, A377520, A380087, A380088, A380089.

f[p_, e_] := If[e <= p, p^e, 0] + 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], f[i,1]^f[i,2], 0) + 1);}

a(n) = A034448(A380088(n)).
Multiplicative with a(p^e) = p^e + 1 if e <= p, and 1 otherwise.
a(n) = 1 if and only if n is in A054743.
a(n) < A034448(n) if and only if n is in A185359.
a(n) = A034448(n) if and only if n is in A207481.
a(n) = A377520(n) if and only if n is squarefree (A005117).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (p^(p+2) + p^(p+1) + p^p - p - 1)/(p^(p+1) * (p+1)) = 1.2078161... .

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A380086 The number of unitary divisors of n that are terms in A276078.

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A380088 The largest unitary divisor of n that is a term in A207481.

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A380090 The sum of the unitary divisors of n that are terms in A207481.

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