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A243055
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Difference between the indices of the smallest and the largest prime dividing n: If n = p_i * ... * p_k, where p_i <= ... <= p_k, where p_h = A000040(h), then a(n) = (k-i), a(1) = 0 by convention.
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80
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0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 3, 1, 0, 0, 1, 0, 2, 2, 4, 0, 1, 0, 5, 0, 3, 0, 2, 0, 0, 3, 6, 1, 1, 0, 7, 4, 2, 0, 3, 0, 4, 1, 8, 0, 1, 0, 2, 5, 5, 0, 1, 2, 3, 6, 9, 0, 2, 0, 10, 2, 0, 3, 4, 0, 6, 7, 3, 0, 1, 0, 11, 1, 7, 1, 5, 0, 2, 0, 12, 0, 3, 4, 13, 8, 4, 0, 2
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OFFSET
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1,10
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COMMENTS
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For n>=1, A100484(n+1) gives the position where n occurs for the first time (setting also the records for the sequence).
a(n) = the difference between the largest and the smallest parts of the partition having Heinz number n. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436. Example: a(57) = 6; indeed, the partition having Heinz number 57 = 3*19 is [2, 8]. - Emeric Deutsch, Jun 04 2015
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LINKS
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FORMULA
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If n = p_i * ... * p_k, where p_i <= ... <= p_k are not necessarily distinct primes (sorted into nondescending order) in the prime factorization of n, where p_i = A000040(i), then a(n) = (k-i).
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MAPLE
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with(numtheory):
a:= n-> `if`(n=1, 0, (f-> pi(max(f[]))-pi(min(f[])))(factorset(n))):
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MATHEMATICA
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a[1]=0; a[n_] := Function[{f}, PrimePi[Max[f]] - PrimePi[Min[f]]][FactorInteger[n][[All, 1]]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jul 29 2015, after Alois P. Heinz *)
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PROG
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(Python)
from sympy import primepi, primefactors
def A243055(n): return primepi(max(p:=primefactors(n), default=0))-primepi(min(p, default=0)) # Chai Wah Wu, Oct 10 2023
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CROSSREFS
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Differs from A242411 for the first time at n=30.
A000961 gives the positions of zeros.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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