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A286360 Compound filter (prime signature & sum of the divisors): a(n) = P(A046523(n), A000203(n)), where P(n,k) is sequence A000027 used as a pairing function.

%I A286360 #19 Feb 16 2025 08:33:45
%S A286360 1,8,12,49,23,142,38,239,124,259,80,753,107,412,412,1051,173,1237,212,
%T A286360 1390,672,826,302,3427,565,1087,1089,2223,467,5080,530,4403,1384,1717,
%U A286360 1384,7911,743,2086,1836,6352,905,7780,992,4477,3928,2932,1178,14583,1774,5368,2932,5898,1487,10177,2932,10177,3576,4471,1832,25711,1955,5056,6567,18019,3922
%N A286360 Compound filter (prime signature & sum of the divisors): a(n) = P(A046523(n), A000203(n)), where P(n,k) is sequence A000027 used as a pairing function.
%H A286360 Antti Karttunen, <a href="/A286360/b286360.txt">Table of n, a(n) for n = 1..10000</a>
%H A286360 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PairingFunction.html">Pairing Function</a>
%H A286360 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A286360 a(n) = (1/2)*(2 + ((A046523(n)+A000203(n))^2) - A046523(n) - 3*A000203(n)).
%o A286360 (PARI)
%o A286360 A000203(n) = sigma(n);
%o A286360 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 A286360 A286360(n) = (1/2)*(2 + ((A046523(n)+A000203(n))^2) - A046523(n) - 3*A000203(n));
%o A286360 for(n=1, 10000, write("b286360.txt", n, " ", A286360(n)));
%o A286360 (Scheme) (define (A286360 n) (* (/ 1 2) (+ (expt (+ (A046523 n) (A000203 n)) 2) (- (A046523 n)) (- (* 3 (A000203 n))) 2)))
%o A286360 (Python)
%o A286360 from sympy import factorint, divisor_sigma as D
%o A286360 def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
%o A286360 def P(n):
%o A286360     f = factorint(n)
%o A286360     return sorted([f[i] for i in f])
%o A286360 def a046523(n):
%o A286360     x=1
%o A286360     while True:
%o A286360         if P(n) == P(x): return x
%o A286360         else: x+=1
%o A286360 def a(n): return T(a046523(n), D(n)) # _Indranil Ghosh_, May 12 2017
%Y A286360 Cf. A000027, A286359, A286460.
%Y A286360 Cf. A007503, A065608 (sequences matching to this filter), also A000203, A046523, A161942, A286034, A286357.
%K A286360 nonn
%O A286360 1,2
%A A286360 _Antti Karttunen_, May 10 2017