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Arithmetic mean of {sum of unitary divisors} and {sum of squarefree divisors}: a(n) = (1/2) * (A034448(n) + A048250(n)).
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%I #44 Feb 22 2024 02:18:16

%S 1,3,4,4,6,12,8,6,7,18,12,16,14,24,24,10,18,21,20,24,32,36,24,24,16,

%T 42,16,32,30,72,32,18,48,54,48,31,38,60,56,36,42,96,44,48,42,72,48,40,

%U 29,48,72,56,54,48,72,48,80,90,60,96,62,96,56,34,84,144,68,72,96,144,72,51,74,114,64,80,96,168,80,60,43,126

%N Arithmetic mean of {sum of unitary divisors} and {sum of squarefree divisors}: a(n) = (1/2) * (A034448(n) + A048250(n)).

%C This is not multiplicative: a(4) = 4, a(9) = 7, but a(36) = 31, not 28. However, the function acts multiplicatively on certain subsequences of natural numbers, like for example when restricted to A048107, where this sequence coincides with A326043.

%H Antti Karttunen, <a href="/A325973/b325973.txt">Table of n, a(n) for n = 1..20000</a>

%F a(n) = (1/2) * (A034448(n) + A048250(n)).

%F a(n) = A000203(n) - A325974(n).

%F a(n) = n + A325977(n).

%F a(A048107(n)) = A326043(A048107(n)).

%F For n >= 1, a(2^n) = A052548(n-1) = 2^(n-1) + 2.

%F For n >= 1, a(3^n) = A289521(n) = (3^n + 5)/2.

%F Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2)/zeta(3) + 1)/4 = 0.5921081944... . - _Amiram Eldar_, Feb 22 2024

%e For n = 36, its divisors are 1, 2, 3, 4, 6, 9, 12, 18, 36. Of these, unitary divisors are 1, 4, 9 and 36, so A034448(36) = 1+4+9+36 = 50, while the squarefree divisors are 1, 2, 3 and 6, so A048250(36) = 1+2+3+6 = 12, thus a(36) = (50+12)/2 = 31.

%e For n = 495, its divisors are 1, 3, 5, 9, 11, 15, 33, 45, 55, 99, 165, 495. Of these, unitary are 1, 5, 9, 11, 45, 55, 99, 495, whose sum is A034448(495) = 720, while the squarefree divisors are 1, 3, 5, 11, 15, 33, 55, 165, and their sum is A048250(495) = 288. Thus a(495) = (720+288)/2 = 504. Also for 495, whose prime factorization is 3^2 * 5^1 * 11^1 this can be computed faster as the average of ((3^2)+1)*(5+1)*(11+1) and (3+1)*(5+1)*(11+1), thus (1/2)*(3+(3^2)+2)*(5+1)*(11+1) = 504.

%t Array[(1/2) If[# == 1, 2, Times @@ (1 + Power @@@ #) + Times @@ (1 + #[[;; , 1]]) &@ FactorInteger[#]] &, 90] (* _Michael De Vlieger_, Jun 06 2019, after _Giovanni Resta_ at A034448 and _Amiram Eldar_ at A048250. *)

%o (PARI)

%o A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448

%o A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));

%o A325973(n) = ((A034448(n)+A048250(n))/2);

%o (PARI) A325973(n) = (1/2)*sumdiv(n, d, d*(issquarefree(d) + (1==gcd(d, n/d))));

%Y Cf. A000203, A034448, A048107, A048250, A052548, A289521, A306633, A325974, A325975, A325977, A325978, A325981, A326043.

%K nonn

%O 1,2

%A _Antti Karttunen_, Jun 02 2019