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Compound filter: a(n) = T(A046523(n), A109395(n)), where T(n,k) is sequence A000027 used as a pairing function.
3

%I #15 Feb 16 2025 08:33:44

%S 1,5,8,14,17,34,30,44,19,51,68,103,93,72,196,152,155,103,192,132,72,

%T 126,278,349,32,159,53,165,437,976,498,560,709,237,786,739,705,282,

%U 159,402,863,660,948,243,337,384,1130,1273,49,132,1546,288,1433,349,126,459,282,567,1772,2761,1893,636,165,2144,2421,1921,2280,390,2707,2046,2558,2773,2703

%N Compound filter: a(n) = T(A046523(n), A109395(n)), where T(n,k) is sequence A000027 used as a pairing function.

%H Antti Karttunen, <a href="/A286149/b286149.txt">Table of n, a(n) for n = 1..10000</a>

%H MathWorld, <a href="https://mathworld.wolfram.com/PairingFunction.html">Pairing Function</a>

%F a(n) = (1/2)*(2 + ((A046523(n)+A109395(n))^2) - A046523(n) - 3*A109395(n)).

%t Table[(2 + (#1 + #2)^2 - #1 - 3 #2)/2 & @@ {Times @@ MapIndexed[ Prime[First@ #2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]] - Boole[n == 1], Denominator[EulerPhi[n]/n]}, {n, 73}] (* _Michael De Vlieger_, May 04 2017 *)

%o (PARI)

%o A109395(n) = n/gcd(n, eulerphi(n));

%o 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 A286149(n) = (1/2)*(2 + ((A046523(n)+A109395(n))^2) - A046523(n) - 3*A109395(n));

%o for(n=1, 10000, write("b286149.txt", n, " ", A286149(n)));

%o (Scheme) (define (A286149 n) (* (/ 1 2) (+ (expt (+ (A046523 n) (A109395 n)) 2) (- (A046523 n)) (- (* 3 (A109395 n))) 2)))

%o (Python)

%o from sympy import factorint, totient, gcd

%o def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2

%o def P(n):

%o f = factorint(n)

%o return sorted([f[i] for i in f])

%o def a046523(n):

%o x=1

%o while True:

%o if P(n) == P(x): return x

%o else: x+=1

%o def a(n): return T(a046523(n), n/gcd(n, totient(n))) # _Indranil Ghosh_, May 05 2017

%Y Cf. A000027, A046523, A109395, A285729, A286142, A286143, A286144, A286152, A286154, A286160, A286161, A286162, A286163, A286164.

%K nonn,changed

%O 1,2

%A _Antti Karttunen_, May 04 2017