A380090 -id:A380090 - cp's OEIS frontend

A380087 The sum of the unitary divisors of n that are terms in A276078.

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

Offset: 1

Amiram Eldar, Jan 11 2025

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

Cf. A000720, A005117, A034448, A077610, A276078, A377517, A380085, A380086, A380090.

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

a(n) = A034448(A380085(n)).
Multiplicative with a(p^e) = p^e + 1 if e <= pi(p) = A000720(p), and 1 otherwise.
a(n) = 1 if and only if n is in A325127.
a(n) < A034448(n) if and only if n is in A276079.
a(n) = A034448(n) if and only if n is in A276078.
a(n) = A377517(n) if and only if n is squarefree (A005117).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{k>=1} (p(k)^(k+2) + p(k)^(k+1) + p(k)^k - p(k) - 1)/(p(k)^(k+1) * (p(k)+1)) = 1.03676..., where p(k) = prime(k).

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... .

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...

cp's OEIS Frontend

A380087 The sum of the 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.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula