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.

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.

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences computed from indices in prime factorization

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 A378532 A378531

Adjacent sequences: A355927 A355928 A355929 * A355931 A355932 A355933

Keyword

nonn

Author

Antti Karttunen as suggested by Don Reble, Oct 25 2022

Status

approved