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

Author

Antti Karttunen, May 21 2017

Keywords

• nonn

Links

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

Crossrefs

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

Programs

Mathematica

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 *)

PARI

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));

Python

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

Scheme

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

Formula

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