A285705 -id:A285705 - cp's OEIS frontend

A286385 a(n) = A003961(n) - A000203(n).

0, 0, 1, 2, 1, 3, 3, 12, 12, 3, 1, 17, 3, 9, 11, 50, 1, 36, 3, 21, 23, 3, 5, 75, 18, 9, 85, 43, 1, 33, 5, 180, 17, 3, 29, 134, 3, 9, 29, 99, 1, 69, 3, 33, 97, 15, 5, 281, 64, 54, 23, 55, 5, 255, 19, 177, 35, 3, 1, 147, 5, 15, 171, 602, 35, 51, 3, 45, 49, 87, 1, 480, 5, 9, 121, 67, 47, 87, 3, 381, 504, 3, 5, 271, 25, 9, 35, 171, 7, 291, 75, 93, 57, 15, 41, 963

Offset: 1

Antti Karttunen, May 09 2017

nonn

Are all terms nonnegative? This question is equivalent to the question posed in A285705.
From Antti Karttunen, Aug 05 2020: (Start)
The answer to the above question is yes. Because both A000203 and A003961 are multiplicative sequences, it suffices to prove that for any prime p, and e >= 1, q^e >= sigma(p^e) = ((p^(1+e))-1) / (p-1), where q = A151800(p), i.e., the next larger prime after p. If p is a lesser twin prime, then q = p+2 (and this difference can't be less than 2, apart from case p=2), and it is easy to see that (n+2)^e > ((n^(e+1)) - 1) / (n-1), for all n >= 2, e >= 1.
See comments in A326042.
(End)
This is the inverse Möbius transform of A337549, from which it is even easier to see that all terms are nonnegative. - Antti Karttunen, Sep 22 2020

Antti Karttunen, Table of n, a(n) for n = 1..16383

Cf. A000203, A001359, A003961, A001065, A031924, A033879, A048673, A151800, A285705, A326042, A336702, A336851, A336852, A337549 (Möbius transform).

Cf. A326057 [= gcd(a(n), A252748(n))].

Array[Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1] - DivisorSigma[1, #] &, 96] (* Michael De Vlieger, Oct 05 2020 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A286385(n) = (A003961(n) - sigma(n));
for(n=1, 16384, write("b286385.txt", n, " ", A286385(n)));

from sympy import factorint, nextprime, divisor_sigma as D
from operator import mul
def a048673(n):
    f = factorint(n)
    return 1 if n==1 else (1 + reduce(mul, [nextprime(i)**f[i] for i in f]))/2
def a(n): return 2*a048673(n) - D(n) - 1 # Indranil Ghosh, May 12 2017

(define (A286385 n) (- (A003961 n) (A000203 n)))

a(n) = A285705(A048673(n)) - 1 = 2*A048673(n) - A000203(n) - 1.
a(n) = A336852(n) - A336851(n). - Antti Karttunen, Aug 05 2020
a(n) = Sum_{d|n} A337549(d). - Antti Karttunen, Sep 22 2020
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((p^2-p)/(p^2-q(p))) - Pi^2/12 = 1.24152934..., where q(p) = nextprime(p) (A151800). - Amiram Eldar, Dec 21 2023

A285703 a(n) = A000203(A064216(n)).

Original entry on oeis.org

1, 3, 4, 6, 7, 8, 12, 12, 14, 18, 18, 20, 13, 15, 24, 30, 24, 24, 32, 36, 38, 42, 28, 44, 31, 42, 48, 32, 54, 54, 60, 42, 48, 62, 60, 68, 72, 39, 48, 74, 31, 80, 56, 72, 84, 72, 90, 72, 90, 56, 98, 102, 72, 104, 108, 96, 110, 80, 84, 84, 57, 114, 40, 114, 126, 128, 108, 60, 132, 138, 132, 96, 96, 93, 140, 150, 98, 120, 152, 144, 120, 158, 96, 164, 133, 126

Offset: 1

Antti Karttunen, Apr 26 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000203, A064216, A099774, A151799, A285702, A285704, A285705.

Table[DivisorSigma[1, #] &@ If[n == 1, 1, Apply[Times, FactorInteger[2 n - 1] /. {p_, e_} /; p > 2 :> NextPrime[p, -1]^e]], {n, 86}] (* Michael De Vlieger, Apr 26 2017 *)

(define (A285703 n) (A000203 (A064216 n)))

a(n) = A000203(A064216(n)).

Sum_{k=1..n} a(k) ~ c * n^2, where c = Product_{p prime} (p^3/((p+1)*(p^2-q(p)))) = 0.8168476756..., where q(p) = prevprime(p) (A151799) if p > 2 and q(2) = 1. - Amiram Eldar, Dec 21 2023

A285704 a(n) = A285703(n) - n = A000203(A064216(n)) - n.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 5, 4, 5, 8, 7, 8, 0, 1, 9, 14, 7, 6, 13, 16, 17, 20, 5, 20, 6, 16, 21, 4, 25, 24, 29, 10, 15, 28, 25, 32, 35, 1, 9, 34, -10, 38, 13, 28, 39, 26, 43, 24, 41, 6, 47, 50, 19, 50, 53, 40, 53, 22, 25, 24, -4, 52, -23, 50, 61, 62, 41, -8, 63, 68, 61, 24, 23, 19, 65, 74, 21, 42, 73, 64, 39, 76, 13, 80, 48, 40, 81, -10, 73, 84, 89, 88, 35, 18, -5

Offset: 1

Antti Karttunen, Apr 26 2017

sign

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000203, A001065, A064216, A151799, A285703, A285705.

Table[-n + DivisorSigma[1, #] &@ If[n == 1, 1, Apply[Times, FactorInteger[2 n - 1] /. {p_, e_} /; p > 2 :> NextPrime[p, -1]^e]], {n, 95}] (* Michael De Vlieger, Apr 26 2017 *)

Scheme
```
(define (A285704 n) (- (A285703 n) n))
```

a(n) = A285703(n) - n = A000203(A064216(n)) - n.

Sum_{k=1..n} a(k) ~ c * n^2, where c = -1/2 + Product_{p prime} (p^3/((p+1)*(p^2-q(p)))) = 0.3168476756..., where q(p) = prevprime(p) (A151799) if p > 2 and q(2) = 1. - Amiram Eldar, Dec 21 2023

A286459 Permuted compound filter: a(n) = A286458(A064216(n)).

Original entry on oeis.org

1, 1, 2, 13, 3, 24, 85, 25, 112, 201, 5, 242, 61, 15, 451, 723, 64, 87, 842, 393, 1107, 1355, 113, 1407, 137, 22, 1744, 204, 844, 2181, 2891, 313, 67, 3203, 657, 3703, 4056, 243, 196, 4424, 55, 4908, 196, 795, 5521, 2384, 2741, 1062, 6507, 250, 7574, 8460, 651, 8590, 9122, 1935, 9393, 597, 559, 1249, 833, 1507, 613, 9391, 4155, 12492, 5949, 513

Offset: 1

Antti Karttunen, May 14 2017

nonn

For all i, j: a(i) = a(j) => A285705(i) = A285705(j).

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A064216, A285705, A286458.

(define (A286459 n) (A286458 (A064216 n)))

a(n) = A286458(A064216(n)).

cp's OEIS Frontend

A286385 a(n) = A003961(n) - A000203(n).

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A285703 a(n) = A000203(A064216(n)).

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Mathematica

Scheme

Formula

A285704 a(n) = A285703(n) - n = A000203(A064216(n)) - n.

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Author

Keywords

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Programs

Mathematica

Scheme

Formula

A286459 Permuted compound filter: a(n) = A286458(A064216(n)).

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Author

Keywords

Comments

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Scheme

Formula