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A355930 Sum of the prime indices of n minus the sum of the prime indices of the smallest number with same prime signature as n, when the sum is taken with multiplicity, as in A056239.

%I A355930 #26 Nov 05 2022 11:09:44
%S A355930 0,0,1,0,2,0,3,0,2,1,4,0,5,2,2,0,6,1,7,1,3,3,8,0,4,4,3,2,9,0,10,0,4,5,
%T A355930 4,0,11,6,5,1,12,1,13,3,3,7,14,0,6,3,6,4,15,2,5,2,7,8,16,0,17,9,4,0,6,
%U A355930 2,18,5,8,2,19,0,20,10,4,6,6,3,21,1,4,11,22,1,7,12,9,3,23,1,7,7,10,13,8,0,24,5,5,2,25,4,26,4,3
%N A355930 Sum of the prime indices of n minus the sum of the prime indices of the smallest number with same prime signature as n, when the sum is taken with multiplicity, as in A056239.
%C A355930 a(n) gives the signature excitation of n (a concept proposed by _Allan C. Wechsler_, indicating the distance of n from the terms of A025487), when the primes in the "excited state", i.e., those present in A328478(n), are de-excited one by one, and the prime signature of n is preserved. See the example.
%H A355930 Antti Karttunen, <a href="/A355930/b355930.txt">Table of n, a(n) for n = 1..65537</a>
%H A355930 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A355930 a(n) = A056239(n) - A356159(n) = A056239(n) - A056239(A046523(n)).
%F A355930 For all n, a(n) >= A358218(n). - _Antti Karttunen_, Nov 05 2022
%e A355930 For n = 98 = 2*7*7, the other 7 is de-excited as 7 -> 5 -> 3 -> 2, and the other 7 is de-excited as 7 -> 5 -> 3, to get 2*2*3 = 12 = A046523(98). There are 3+2 de-excitations in total, therefore a(98) = 5.
%t A355930 {0}~Join~Array[Total@ Flatten[ConstantArray[PrimePi[#1], #2] & @@@ #] - Total@ Flatten@ MapIndexed[ConstantArray[First[#2], #1] &, ReverseSort[#[[All, -1]]]] &@ FactorInteger[#] &, 104, 2] (* _Michael De Vlieger_, Nov 02 2022 *)
%o A355930 (PARI)
%o A355930 A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
%o A355930 A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); }
%o A355930 A355930(n) = (A056239(n) - A056239(A046523(n)));
%Y A355930 Cf. A025487 (positions of zeros), A046523, A056239.
%Y A355930 Cf. also A319627, A328478, A358218.
%Y A355930 Differs from A325799 for the first time at n=18, where a(18) = 1, while A325799(18) = 0.
%K A355930 nonn
%O A355930 1,5
%A A355930 _Antti Karttunen_ as suggested by _Don Reble_, Oct 25 2022