login
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.
3
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, 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, 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
OFFSET
1,5
COMMENTS
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.
FORMULA
a(n) = A056239(n) - A356159(n) = A056239(n) - A056239(A046523(n)).
For all n, a(n) >= A358218(n). - Antti Karttunen, Nov 05 2022
EXAMPLE
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.
MATHEMATICA
{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 *)
PROG
(PARI)
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); }
A355930(n) = (A056239(n) - A056239(A046523(n)));
CROSSREFS
Cf. A025487 (positions of zeros), A046523, A056239.
Cf. also A319627, A328478, A358218.
Differs from A325799 for the first time at n=18, where a(18) = 1, while A325799(18) = 0.
Sequence in context: A349448 A194666 A325799 * A229946 A127460 A274021
KEYWORD
nonn
AUTHOR
Antti Karttunen as suggested by Don Reble, Oct 25 2022
STATUS
approved