A375341 -id:A375341 - cp's OEIS frontend

A375339 If n has exactly one non-unitary prime factor then a(n) is the exponent of the highest power of this prime that divides n, otherwise a(n) = 0.

0, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 3, 2, 0, 3, 2, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 2, 2, 0, 0, 4, 2, 2, 0, 2, 0, 3, 0, 3, 0, 0, 0, 2, 0, 0, 2, 6, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 4, 4, 0, 0, 2, 0, 0, 0, 3, 0, 2, 0, 2, 0, 0, 0, 5, 0, 2, 2, 0, 0, 0, 0, 3, 0

Offset: 1

Amiram Eldar, Aug 12 2024

First differs from A212172, A275812 and A372603 at n = 36.
If n = m * p^e, such that m is squarefree, p is a prime that does not divide m and e >= 2, then a(n) = e, otherwise a(n) = 0.
By definition all the positive terms are larger than 1.
The asymptotic density of 0's in this sequence is 1 - Sum_{p prime} (1/(p^2-1)) / zeta(2) = 1 - A059956 * A154945 = 0.66461069244308962639... .
The asymptotic density of the occurrences of k >= 2 in this sequence is Sum_{p prime} (1/(p^(k-1)*(p+1))) / zeta(2). E.g., 0.200755... (A271971) for k = 2, 0.0741777... for k = 3, and 0.0320652... for k = 4.

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

Cf. A005361, A048109, A051903, A060687, A082293, A190641, A359466, A375341 (the positive terms).
Cf. A013661, A059956, A154945, A271971, A375340.
Cf. A212172, A275812, A372603.

a[n_] := Module[{e = Select[FactorInteger[n][[;; , 2]], # > 1 &]}, If[Length[e] == 1, e[[1]], 0]]; Array[a, 100]

a(n) = {my(e = select(x -> x > 1, factor(n)[,2])); if(#e == 1, e[1], 0);}

a(n) = A051903(n) * A359466(n).
a(n) = A005361(n) * A359466(n).
a(A190641(n)) >= 2.
a(n) = 2 if and only if n is in A060687.
a(n) = 3 if and only if n is in A048109.
a(n) <= 3 if and only if n is in A082293.
Asymptotic mean:  = Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} (2*p-1)/((p-1)^2*(p+1)) / zeta(2) = A375340 / A013661 = 0.92105359989459565838... .
Asymptotic second raw moment:  = Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)^2 = Sum_{p prime} (4*p^2-3*p+1)/((p-1)^3*(p+1)) / zeta(2) = 3.04027120804428071157... .
The asymptotic second central moment, or variance, is  - ^2 = 2.19193147416548680815... and the asymptotic standard deviation is sqrt( - ^2) = 1.48051729951577627898... .

A375340 Decimal expansion of Sum_{p prime} ((2p-1)/((p+1)(p-1)^2)).

Original entry on oeis.org

1, 5, 1, 5, 0, 7, 2, 4, 4, 3, 8, 5, 9, 8, 1, 6, 4, 2, 0, 6, 1, 8, 2, 2, 3, 1, 0, 1, 2, 1, 8, 2, 3, 5, 2, 1, 6, 7, 8, 7, 0, 5, 0, 5, 4, 6, 7, 5, 5, 6, 0, 1, 1, 0, 7, 7, 0, 0, 5, 2, 0, 6, 6, 9, 0, 3, 4, 9, 6, 6, 2, 5, 3, 0, 3, 2, 3, 8, 3, 6, 3, 4, 3, 3, 1, 5, 8, 0, 3, 0, 0, 9, 6, 0, 7, 1, 0, 9, 9, 8, 3, 5, 1, 3, 9

Offset: 1

Amiram Eldar, Aug 12 2024

			1.515072443859816420618223101218235216787050546755601...

Cf. A013661, A028242, A154945, A375339, A375341.

PARI
```
sumeulerrat((2*p-1)/((p+1)*(p-1)^2))
```

Equals Sum_{k>=2} A028242(k) * P(k), where P is the prime zeta function.
Equals zeta(2) * Limit_{m->oo} (1/m) * Sum_{k=1..m} A375339(k).
Equals A154945 * Limit_{m->oo} (1/m) * Sum_{k=1..m} A375341(k).

A375342 The maximum exponent in the prime factorization of the numbers whose powerful part is a power of a squarefree number that is larger than 1.

Original entry on oeis.org

2, 3, 2, 2, 4, 2, 2, 3, 2, 3, 2, 5, 2, 3, 2, 2, 4, 2, 2, 2, 3, 3, 2, 2, 6, 2, 2, 2, 4, 4, 2, 3, 2, 2, 5, 2, 2, 2, 3, 4, 2, 2, 3, 2, 2, 3, 2, 7, 2, 3, 3, 2, 2, 2, 2, 3, 2, 2, 5, 4, 2, 3, 2, 2, 2, 2, 4, 2, 3, 2, 3, 6, 2, 2, 2, 2, 4, 2, 3, 2, 5, 2, 2, 3, 2, 2, 4, 2, 5, 2, 2, 3, 3, 2, 8, 2, 2, 3, 2, 3, 4, 2, 2, 2, 3

Offset: 1

Amiram Eldar, Aug 12 2024

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

Cf. A051903, A057521, A067259, A375142, A375341.

s[n_] := Module[{e = Select[FactorInteger[n][[;; , 2]], # > 1 &]}, If[Length[e] > 0 && SameQ @@ e, e[[1]], Nothing]]; Array[s, 300]

lista(kmax) = {my(e); for(k = 1, kmax, e = select(x -> x > 1, factor(k)[,2]); if(#e > 0 && vecmin(e) == vecmax(e), print1(e[1], ", ")));}

a(n) = A051903(A375142(n)).
a(n) = 2 if and only if A375142(n) is in A067259.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=2} k*d(k) / Sum_{k>=2} d(k) = 2.70113273169250927084..., where d(k) = (f(k)-1)/zeta(2) is the asymptotic density of terms m of A375142 with A051903(m) = k, f(k) = zeta(k) * Product_{p prime} (1 + Sum_{i=k+1..2*k-1} (-1)^i/p^i), if k is even, and f(k) = (zeta(2*k)/zeta(k)) * Product_{p prime} (1 + 2/p^k + Sum_{i=k+1..2*k-1} (-1)^(i+1)/p^i) if k is odd > 1.

cp's OEIS Frontend

A375339 If n has exactly one non-unitary prime factor then a(n) is the exponent of the highest power of this prime that divides n, otherwise a(n) = 0.

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A375340 Decimal expansion of Sum_{p prime} ((2p-1)/((p+1)(p-1)^2)).

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PARI

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A375342 The maximum exponent in the prime factorization of the numbers whose powerful part is a power of a squarefree number that is larger than 1.

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cp's OEIS Frontend

A375339 If n has exactly one non-unitary prime factor then a(n) is the exponent of the highest power of this prime that divides n, otherwise a(n) = 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A375340 Decimal expansion of Sum_{p prime} ((2*p-1)/((p+1)*(p-1)^2)).

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

A375342 The maximum exponent in the prime factorization of the numbers whose powerful part is a power of a squarefree number that is larger than 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A375340 Decimal expansion of Sum_{p prime} ((2p-1)/((p+1)(p-1)^2)).