This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A376593 #10 Oct 02 2024 09:32:54
%S A376593 -3,2,1,-2,0,2,-3,1,-1,3,0,0,0,-3,2,-2,0,1,0,0,2,-1,-2,3,0,-1,-2,3,-3,
%T A376593 2,1,-2,0,2,-2,-1,0,3,0,0,0,-3,2,-2,2,-2,0,1,2,-1,-2,3,0,-1,-2,1,0,-1,
%U A376593 2,1,-2,0,2,-3,1,-1,2,-2,3,0,0,-3,2,1,-2,0,2
%N A376593 Second differences of consecutive nonsquarefree numbers (A013929). First differences of A078147.
%C A376593 The range is {-3, -2, -1, 0, 1, 2, 3}.
%e A376593 The nonsquarefree numbers (A013929) are:
%e A376593 4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, 52, ...
%e A376593 with first differences (A078147):
%e A376593 4, 1, 3, 4, 2, 2, 4, 1, 2, 1, 4, 4, 4, 4, 1, 3, 1, 1, 2, 2, 2, 4, 3, 1, 4, 4, ...
%e A376593 with first differences (A376593):
%e A376593 -3, 2, 1, -2, 0, 2, -3, 1, -1, 3, 0, 0, 0, -3, 2, -2, 0, 1, 0, 0, 2, -1, -2, ...
%t A376593 Differences[Select[Range[100],!SquareFreeQ[#]&],2]
%o A376593 (Python)
%o A376593 from math import isqrt
%o A376593 from sympy import mobius, factorint
%o A376593 def A376593(n):
%o A376593 def f(x): return n+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
%o A376593 m, k = n, f(n)
%o A376593 while m != k: m, k = k, f(k)
%o A376593 k = next(i for i in range(1,5) if any(d>1 for d in factorint(m+i).values()))
%o A376593 return next(i for i in range(1-k,5-k) if any(d>1 for d in factorint(m+(k<<1)+i).values())) # _Chai Wah Wu_, Oct 02 2024
%Y A376593 The version for A000002 is A376604, first differences of A054354.
%Y A376593 The first differences were A078147.
%Y A376593 Zeros are A376594, complement A376595.
%Y A376593 A000040 lists the prime numbers, differences A001223.
%Y A376593 A005117 lists squarefree numbers, differences A076259.
%Y A376593 A064113 lists positions of adjacent equal prime gaps.
%Y A376593 A114374 counts partitions into nonsquarefree numbers.
%Y A376593 A246655 lists prime-powers exclusive, inclusive A000961.
%Y A376593 A333254 lists run-lengths of differences between consecutive primes.
%Y A376593 For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376596 (prime-power inclusive), A376599 (non-prime-power inclusive).
%Y A376593 For nonsquarefree numbers: A013929 (terms), A078147 (first differences), A376594 (inflections and undulations), A376595 (nonzero curvature).
%Y A376593 Cf. A061398, A053797, A053806, A120992, A182853, A251092, A373198, A375707, A376305, A376306, A376311, A376312, A376655.
%K A376593 sign
%O A376593 1,1
%A A376593 _Gus Wiseman_, Oct 01 2024