%I #21 Jan 11 2023 08:47:21

%S 0,0,0,0,0,1,0,0,0,2,0,2,0,3,1,0,0,2,0,4,2,4,0,3,0,5,0,6,0,4,0,0,3,6,

%T 1,4,0,7,4,6,0,6,0,8,2,8,0,4,0,4,5,10,0,3,2,9,6,9,0,7,0,10,4,0,3,8,0,

%U 12,7,6,0,6,0,11,2,14,1,10,0,8,0,12,0,10,4,13,8,12,0,6,2,16,9,14,5,5,0,6,6,8,0,12,0,15,4,15,0,6,0,8,10,12,0,14,6,18,8,16,3,10

%N Sum of index differences between prime factors of n, summed over all unordered pairs of primes present (with multiplicity) in the prime factorization of n.

%H Antti Karttunen, <a href="/A261079/b261079.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%F a(n) = A304818(n) - A318283(n). - _Gus Wiseman_, Jan 09 2023

%F a(n) = 2*A304818(n) - A359362(n). - _Gus Wiseman_, Jan 09 2023

%e For n = 1 the prime factorization is empty, thus there is nothing to sum, so a(1) = 0.

%e For n = 6 = 2*3 = prime(1) * prime(2), a(6) = 1 because the (absolute value of) difference between prime indices of 2 and 3 is 1.

%e For n = 10 = 2*5 = prime(1) * prime(3), a(10) = 2 because the difference between prime indices of 2 and 5 is 2.

%e For n = 12 = 2*2*3 = prime(1) * prime(1) * prime(2), a(12) = 2 because the difference between prime indices of 2 and 3 is 1, and the pair (2,3) occurs twice as one can pick either one of the two 2's present in the prime factorization to be a pair of a single 3. Note that the index difference between 2 and 2 is 0, thus the pair (2,2) of prime divisors does not contribute to the sum.

%e For n = 36 = 2*2*3*3, a(36) = 4 because the index difference between 2 and 3 is 1, and the prime factor pair (2,3) occurs 2^2 = four times in total. As the index difference is zero between 2 and 2 as well as between 3 and 3, the pairs (2,2) and (3,3) do not contribute to the sum.

%t Table[Function[p, Total@ Map[Function[b, Times @@ {First@ Differences@ PrimePi@ b, Count[Subsets[p, {2}], c_ /; SameQ[c, b]]}], Subsets[Union@ p, {2}]]][Flatten@ Replace[FactorInteger@ n, {p_, e_} :> ConstantArray[p, e], 2]], {n, 120}] (* _Michael De Vlieger_, Mar 08 2017 *)

%o (Scheme, with Aubrey Jaffer's SLIB Scheme library)

%o (require 'factor)

%o (define (A261079 n) (let loop ((s 0) (pfs (factor n))) (cond ((or (null? pfs) (null? (cdr pfs))) s) (else (loop (fold-left (lambda (a p) (+ a (abs (- (A000720 (car pfs)) (A000720 p))))) s (cdr pfs)) (cdr pfs))))))

%Y Cf. A000720.

%Y Cf. A000961 (positions of zeros), A006094 (positions of ones).

%Y Cf. A243055, A242411.

%Y Cf. also A260737.

%Y A055396 gives minimum prime index, maximum A061395.

%Y A112798 list prime indices, length A001222, sum A056239.

%Y A304818 adds up partial sums of reversed prime indices, row sums of A359361.

%Y A318283 adds up partial sums of prime indices, row sums of A358136.

%Y Cf. A029931, A316413, A325362, A326836, A326844, A355536, A358137, A359362.

%K nonn

%O 1,10

%A _Antti Karttunen_, Sep 23 2015