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A375929 Numbers k such that A002808(k+1) = A002808(k) + 1. In other words, the k-th composite number is 1 less than the next.

%I A375929 #15 Sep 18 2024 06:44:24
%S A375929 3,4,7,8,11,12,14,15,16,17,20,21,22,23,25,26,29,30,32,33,34,35,37,38,
%T A375929 39,40,43,44,45,46,48,49,52,53,54,55,57,58,60,61,62,63,65,66,67,68,69,
%U A375929 70,72,73,76,77,80,81,83,84,85,86,87,88,89,90,91,92,93,94
%N A375929 Numbers k such that A002808(k+1) = A002808(k) + 1. In other words, the k-th composite number is 1 less than the next.
%C A375929 Positions of 1's in A073783 (see also A054546, A065310).
%F A375929 a(n) = A375926(n) - 1.
%e A375929 The composite numbers are 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, ... which increase by 1 after positions 3, 4, 7, 8, ...
%t A375929 Join@@Position[Differences[Select[Range[100],CompositeQ]],1]
%o A375929 (Python)
%o A375929 from sympy import primepi
%o A375929 def A375929(n):
%o A375929     def bisection(f,kmin=0,kmax=1):
%o A375929         while f(kmax) > kmax: kmax <<= 1
%o A375929         while kmax-kmin > 1:
%o A375929             kmid = kmax+kmin>>1
%o A375929             if f(kmid) <= kmid:
%o A375929                 kmax = kmid
%o A375929             else:
%o A375929                 kmin = kmid
%o A375929         return kmax
%o A375929     def f(x): return n+bisection(lambda y:primepi(x+2+y))-2
%o A375929     return bisection(f,n,n) # _Chai Wah Wu_, Sep 15 2024
%o A375929 (Python) # faster for initial segment of sequence
%o A375929 from sympy import isprime
%o A375929 from itertools import count, islice
%o A375929 def agen(): # generator of terms
%o A375929     pic, prevc = 0, -1
%o A375929     for i in count(4):
%o A375929         if not isprime(i):
%o A375929             if i == prevc + 1:
%o A375929                 yield pic
%o A375929             pic, prevc = pic+1, i
%o A375929 print(list(islice(agen(), 10000))) # _Michael S. Branicky_, Sep 17 2024
%Y A375929 Positions in A002808 of each element of A068780.
%Y A375929 The complement is A065890 shifted.
%Y A375929 First differences are A373403 (except first).
%Y A375929 The version for non-prime-powers is A375713, differences A373672.
%Y A375929 The version for prime-powers is A375734, differences A373671.
%Y A375929 The version for non-perfect-powers is A375740.
%Y A375929 The version for nonprime numbers is A375926.
%Y A375929 A000040 lists the prime numbers, differences A001223.
%Y A375929 A000961 lists prime-powers (inclusive), differences A057820.
%Y A375929 A002808 lists the composite numbers, differences A073783.
%Y A375929 A018252 lists the nonprime numbers, differences A065310.
%Y A375929 A046933 counts composite numbers between primes.
%Y A375929 Cf. A014689, A054546, A176246, A246655, A251092.
%K A375929 nonn
%O A375929 1,1
%A A375929 _Gus Wiseman_, Sep 12 2024