A375929 - OEIS

%I #15 Sep 18 2024 06:44:24

%S 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 39,40,43,44,45,46,48,49,52,53,54,55,57,58,60,61,62,63,65,66,67,68,69,

%U 70,72,73,76,77,80,81,83,84,85,86,87,88,89,90,91,92,93,94

%N 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 Positions of 1's in A073783 (see also A054546, A065310).

%F a(n) = A375926(n) - 1.

%e 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 Join@@Position[Differences[Select[Range[100],CompositeQ]],1]

%o (Python)

%o from sympy import primepi

%o def A375929(n):

%o     def bisection(f,kmin=0,kmax=1):

%o         while f(kmax) > kmax: kmax <<= 1

%o         while kmax-kmin > 1:

%o             kmid = kmax+kmin>>1

%o             if f(kmid) <= kmid:

%o                 kmax = kmid

%o             else:

%o                 kmin = kmid

%o         return kmax

%o     def f(x): return n+bisection(lambda y:primepi(x+2+y))-2

%o     return bisection(f,n,n) # _Chai Wah Wu_, Sep 15 2024

%o (Python) # faster for initial segment of sequence

%o from sympy import isprime

%o from itertools import count, islice

%o def agen(): # generator of terms

%o     pic, prevc = 0, -1

%o     for i in count(4):

%o         if not isprime(i):

%o             if i == prevc + 1:

%o                 yield pic

%o             pic, prevc = pic+1, i

%o print(list(islice(agen(), 10000))) # _Michael S. Branicky_, Sep 17 2024

%Y Positions in A002808 of each element of A068780.

%Y The complement is A065890 shifted.

%Y First differences are A373403 (except first).

%Y The version for non-prime-powers is A375713, differences A373672.

%Y The version for prime-powers is A375734, differences A373671.

%Y The version for non-perfect-powers is A375740.

%Y The version for nonprime numbers is A375926.

%Y A000040 lists the prime numbers, differences A001223.

%Y A000961 lists prime-powers (inclusive), differences A057820.

%Y A002808 lists the composite numbers, differences A073783.

%Y A018252 lists the nonprime numbers, differences A065310.

%Y A046933 counts composite numbers between primes.

%Y Cf. A014689, A054546, A176246, A246655, A251092.

%K nonn

%O 1,1

%A _Gus Wiseman_, Sep 12 2024