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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).

%I A286480 #12 Feb 16 2025 08:33:45
%S A286480 2,12,5,14,5,61,12,179,109,61,5,265,23,142,27,226,5,607,23,148,42,61,
%T A286480 12,1509,109,601,44,148,5,625,23,698,27,61,61,1117,23,601,27,2509,5,
%U A286480 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
%N 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).
%H A286480 Antti Karttunen, <a href="/A286480/b286480.txt">Table of n, a(n) for n = 1..10000</a>
%H A286480 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PairingFunction.html">Pairing Function</a>
%F A286480 a(n) = (1/2)*(2 + ((A046523(n)+A286479(n))^2) - A046523(n) - 3*A286479(n)).
%t A286480 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 *)
%o A286480 (PARI)
%o A286480 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
%o A286480 A286479(n) = A046523(n+numdiv(n));
%o A286480 A286480(n) = (1/2)*(2 + ((A046523(n)+A286479(n))^2) - A046523(n) - 3*A286479(n));
%o A286480 (Scheme) (define (A286480 n) (* (/ 1 2) (+ (expt (+ (A046523 n) (A286479 n)) 2) (- (A046523 n)) (- (* 3 (A286479 n))) 2)))
%o A286480 (Python)
%o A286480 from sympy import factorint, divisor_count
%o A286480 def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
%o A286480 def P(n):
%o A286480     f = factorint(n)
%o A286480     return sorted([f[i] for i in f])
%o A286480 def a046523(n):
%o A286480     x=1
%o A286480     while True:
%o A286480         if P(n) == P(x): return x
%o A286480         else: x+=1
%o A286480 def a286479(n): return a046523(n + divisor_count(n))
%o A286480 def a(n): return T(a046523(n), a286479(n)) # _Indranil Ghosh_, May 21 2017
%Y A286480 Cf. A000005, A046523, A282231, A286479, A286530.
%K A286480 nonn
%O A286480 1,1
%A A286480 _Antti Karttunen_, May 21 2017