A286479 -id:A286479 - cp's OEIS frontend

A286480 Compound filter (prime signature of n & prime signature of n+d(n)): a(n) = P(A046523(n), A286479(n)), where P(n,k) is sequence A000027 used as a pairing function and d(n) is number of divisors of n (A000005).

2, 12, 5, 14, 5, 61, 12, 179, 109, 61, 5, 265, 23, 142, 27, 226, 5, 607, 23, 148, 42, 61, 12, 1509, 109, 601, 44, 148, 5, 625, 23, 698, 27, 61, 61, 1117, 23, 601, 27, 2509, 5, 850, 80, 265, 148, 142, 12, 1426, 109, 607, 61, 148, 23, 430, 27, 3765, 27, 61, 5, 8575, 80, 601, 148, 2144, 61, 625, 23, 148, 27, 1741, 5, 8587, 80, 601, 363, 148, 216, 625, 138, 5719

Offset: 1

Antti Karttunen, May 21 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pairing Function

Cf. A000005, A046523, A282231, A286479, A286530.

f[n_] := If[n == 1, 1, Times @@ MapIndexed[Prime[First[#2]]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]]]; Table[(2 + (#1 + #2)^2 - #1 - 3 #2)/2 & @@ {f@ n, f[n + DivisorSigma[0, n]]}, {n, 80}] (* Michael De Vlieger, May 21 2017 *)

A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A286479(n) = A046523(n+numdiv(n));
A286480(n) = (1/2)*(2 + ((A046523(n)+A286479(n))^2) - A046523(n) - 3*A286479(n));

from sympy import factorint, divisor_count
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def a286479(n): return a046523(n + divisor_count(n))
def a(n): return T(a046523(n), a286479(n)) # Indranil Ghosh, May 21 2017

(define (A286480 n) (* (/ 1 2) (+ (expt (+ (A046523 n) (A286479 n)) 2) (- (A046523 n)) (- (* 3 (A286479 n))) 2)))

a(n) = (1/2)*(2 + ((A046523(n)+A286479(n))^2) - A046523(n) - 3*A286479(n)).

A286529 a(n) = d(n+d(n)), where d(n) is the number of divisors of n (A000005).

Original entry on oeis.org

2, 3, 2, 2, 2, 4, 3, 6, 6, 4, 2, 6, 4, 6, 2, 4, 2, 8, 4, 4, 3, 4, 3, 6, 6, 8, 2, 4, 2, 4, 4, 4, 2, 4, 4, 6, 4, 8, 2, 10, 2, 6, 6, 6, 4, 6, 3, 4, 6, 8, 4, 4, 4, 4, 2, 7, 2, 4, 2, 12, 6, 8, 4, 2, 4, 4, 4, 4, 2, 8, 2, 12, 6, 8, 5, 4, 5, 4, 5, 12, 4, 4, 4, 12, 2, 12, 4, 12, 4, 8, 4, 6, 2, 6, 6, 12, 6, 8, 8, 2, 2, 8, 8, 10, 2, 8, 2, 16, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 1

Antti Karttunen, May 21 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Aleksandar Ivić, An asymptotic formula involving the enumerating function of finite Abelian groups, Publikacije Elektrotehničkog fakulteta, Serija Matematika 3 (1992), pp. 61-66.
Imre Kátai, On a problem of A. Ivic, Mathematica Pannonica, Vol. 18, No. 1 (2007), pp. 11-18.

Cf. A000005, A062249, A175304, A286479, A286530.

Table[DivisorSigma[0, n + DivisorSigma[0, n]], {n, 117}] (* Michael De Vlieger, May 21 2017 *)

PARI
```
A286529(n) = numdiv(n+numdiv(n));
```

from sympy import divisor_count as d
def a(n): return d(n + d(n)) # Indranil Ghosh, May 21 2017

(define (A286529 n) (A000005 (+ n (A000005 n))))

a(n) = A000005(A062249(n)) = A000005(n+A000005(n)).

Sum_{k=1..n} a(k) ~ D*n*log(n) + O(n*log(n)/log(log(n))), where D > 0 is a constant (conjectured with an error O(n) by Ivić, 1992; proven by Kátai, 2007). - Amiram Eldar, Jul 08 2020

cp's OEIS Frontend

A286480 Compound filter (prime signature of n & prime signature of n+d(n)): a(n) = P(A046523(n), A286479(n)), where P(n,k) is sequence A000027 used as a pairing function and d(n) is number of divisors of n (A000005).

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