cp's OEIS Frontend

A375704 Maximum of the n-th maximal run of adjacent (increasing by one at a time) non-perfect-powers.

%I A375704 #9 Aug 29 2024 23:33:43
%S A375704 3,7,15,24,26,31,35,48,63,80,99,120,124,127,143,168,195,215,224,242,
%T A375704 255,288,323,342,360,399,440,483,511,528,575,624,675,728,783,840,899,
%U A375704 960,999,1023,1088,1155,1224,1295,1330,1368,1443,1520,1599,1680,1727,1763
%N A375704 Maximum of the n-th maximal run of adjacent (increasing by one at a time) non-perfect-powers.
%C A375704 Non-perfect-powers (A007916) are numbers with no proper integer roots.
%C A375704 Also numbers k > 0 such that k is a perfect power (A001597) but k+1 is not.
%F A375704 For n > 2 we have a(n) = A045542(n+1).
%e A375704 The list of all non-perfect-powers, split into runs, begins:
%e A375704    2   3
%e A375704    5   6   7
%e A375704   10  11  12  13  14  15
%e A375704   17  18  19  20  21  22  23  24
%e A375704   26
%e A375704   28  29  30  31
%e A375704   33  34  35
%e A375704   37  38  39  40  41  42  43  44  45  46  47  48
%e A375704 Row n begins with A375703(n), ends with a(n), adds up to A375705(n), and has length A375702(n).
%t A375704 radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
%t A375704 Max/@Split[Select[Range[100],radQ],#1+1==#2&]//Most
%t A375704 - or -
%t A375704 radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
%t A375704 Select[Range[100],radQ[#]&&!radQ[#+1]&]
%Y A375704 For nonprime numbers: A006093, min A055670, anti-runs A068780, min A005381.
%Y A375704 For prime numbers we have A045344.
%Y A375704 Inserting 8 after 7 gives A045542.
%Y A375704 For nonsquarefree numbers we have A072284(n) + 1, anti-runs A068781.
%Y A375704 For squarefree numbers we have A373415, anti-runs A007674.
%Y A375704 For prime-powers we have A373674 (min A373673), anti-runs A006549 (A120430).
%Y A375704 Non-prime-powers: A373677 (min A373676), anti-runs A255346 (min A373575).
%Y A375704 The anti-run version is A375739.
%Y A375704 A001597 lists perfect-powers, differences A053289.
%Y A375704 A046933 counts composite numbers between primes.
%Y A375704 A375736 gives lengths of anti-runs of non-prime-powers, sums A375737.
%Y A375704 For runs of non-perfect-powers (A007916):
%Y A375704 - length: A375702 = A053289(n+1) - 1
%Y A375704 - first: A375703 (same as A216765 with 2 exceptions)
%Y A375704 - last: A375704 (this) (same as A045542 with 8 removed)
%Y A375704 - sum: A375705
%Y A375704 Cf. A053797, A053806, A061398, A061399, A251092, A373408, A375708, A375714.
%K A375704 nonn
%O A375704 1,1
%A A375704 _Gus Wiseman_, Aug 29 2024