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A375929
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Numbers k such that A002808(k+1) = A002808(k) + 1. In other words, the k-th composite number is 1 less than the next.
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0
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3, 4, 7, 8, 11, 12, 14, 15, 16, 17, 20, 21, 22, 23, 25, 26, 29, 30, 32, 33, 34, 35, 37, 38, 39, 40, 43, 44, 45, 46, 48, 49, 52, 53, 54, 55, 57, 58, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 72, 73, 76, 77, 80, 81, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94
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OFFSET
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1,1
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COMMENTS
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LINKS
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FORMULA
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EXAMPLE
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The composite numbers are 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, ... which increase by 1 after positions 3, 4, 7, 8, ...
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MATHEMATICA
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Join@@Position[Differences[Select[Range[100], CompositeQ]], 1]
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PROG
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(Python)
from sympy import primepi
def bisection(f, kmin=0, kmax=1):
while f(kmax) > kmax: kmax <<= 1
while kmax-kmin > 1:
kmid = kmax+kmin>>1
if f(kmid) <= kmid:
kmax = kmid
else:
kmin = kmid
return kmax
def f(x): return n+bisection(lambda y:primepi(x+2+y))-2
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CROSSREFS
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First differences are A373403 (except first).
The version for non-perfect-powers is A375740.
The version for nonprime numbers is A375926.
A046933 counts composite numbers between primes.
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KEYWORD
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nonn,new
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AUTHOR
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STATUS
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approved
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