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A377782 First-differences of A031218(n) = greatest number <= n that is 1 or a prime-power.

%I A377782 #14 Nov 17 2024 20:35:32
%S A377782 1,1,1,1,0,2,1,1,0,2,0,2,0,0,3,1,0,2,0,0,0,4,0,2,0,2,0,2,0,2,1,0,0,0,
%T A377782 0,5,0,0,0,4,0,2,0,0,0,4,0,2,0,0,0,4,0,0,0,0,0,6,0,2,0,0,3,0,0,3,0,0,
%U A377782 0,4,0,2,0,0,0,0,0,6,0,2,0,2,0,0,0,0,0
%N A377782 First-differences of A031218(n) = greatest number <= n that is 1 or a prime-power.
%C A377782 Note 1 is a power of a prime (A000961) but not a prime-power (A246655).
%t A377782 Differences[Table[NestWhile[#-1&,n,#>1&&!PrimePowerQ[#]&],{n,100}]]
%Y A377782 Positions of 1 are A006549.
%Y A377782 Positions of 0 are A080765 = A024619 - 1, complement A181062 = A000961 - 1.
%Y A377782 Positions of 2 are A120432 (except initial terms).
%Y A377782 Sorted positions of first appearances appear to include A167236 - 1.
%Y A377782 Positions of terms > 1 are A373677.
%Y A377782 The restriction to primes minus 1 is A377289.
%Y A377782 Below, A (B) indicates that A is the first-differences of B:
%Y A377782 - This sequence is A377782 (A031218), which has restriction to primes A065514 (A377781).
%Y A377782 - The opposite is A377780 (A000015), restriction A377703 (A345531).
%Y A377782 - For nonsquarefree we have A378036 (A378033), opposite A378039 (A120327).
%Y A377782 - For squarefree we have A378085 (A112925), restriction A378038 (A070321).
%Y A377782 A000040 lists the primes, differences A001223.
%Y A377782 A000961 and A246655 list prime-powers, differences A057820.
%Y A377782 A024619 lists the non-prime-powers, differences A375735, seconds A376599.
%Y A377782 A361102 lists the non-powers of primes, differences A375708.
%Y A377782 A378034 gives differences of A378032 (restriction of A378033).
%Y A377782 Prime-powers between primes: A053607, A080101, A366833, A377057, A377286, A377287.
%Y A377782 Cf. A053289, A343249, A375706, A377281, A377282.
%K A377782 nonn
%O A377782 1,6
%A A377782 _Gus Wiseman_, Nov 16 2024