A375704 - OEIS

%I #9 Aug 29 2024 23:33:43

%S 3,7,15,24,26,31,35,48,63,80,99,120,124,127,143,168,195,215,224,242,

%T 255,288,323,342,360,399,440,483,511,528,575,624,675,728,783,840,899,

%U 960,999,1023,1088,1155,1224,1295,1330,1368,1443,1520,1599,1680,1727,1763

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

%C Non-perfect-powers (A007916) are numbers with no proper integer roots.

%C Also numbers k > 0 such that k is a perfect power (A001597) but k+1 is not.

%F For n > 2 we have a(n) = A045542(n+1).

%e The list of all non-perfect-powers, split into runs, begins:

%e    2   3

%e    5   6   7

%e   10  11  12  13  14  15

%e   17  18  19  20  21  22  23  24

%e   26

%e   28  29  30  31

%e   33  34  35

%e   37  38  39  40  41  42  43  44  45  46  47  48

%e Row n begins with A375703(n), ends with a(n), adds up to A375705(n), and has length A375702(n).

%t radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;

%t Max/@Split[Select[Range[100],radQ],#1+1==#2&]//Most

%t - or -

%t radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;

%t Select[Range[100],radQ[#]&&!radQ[#+1]&]

%Y For nonprime numbers: A006093, min A055670, anti-runs A068780, min A005381.

%Y For prime numbers we have A045344.

%Y Inserting 8 after 7 gives A045542.

%Y For nonsquarefree numbers we have A072284(n) + 1, anti-runs A068781.

%Y For squarefree numbers we have A373415, anti-runs A007674.

%Y For prime-powers we have A373674 (min A373673), anti-runs A006549 (A120430).

%Y Non-prime-powers: A373677 (min A373676), anti-runs A255346 (min A373575).

%Y The anti-run version is A375739.

%Y A001597 lists perfect-powers, differences A053289.

%Y A046933 counts composite numbers between primes.

%Y A375736 gives lengths of anti-runs of non-prime-powers, sums A375737.

%Y For runs of non-perfect-powers (A007916):

%Y - length: A375702 = A053289(n+1) - 1

%Y - first: A375703 (same as A216765 with 2 exceptions)

%Y - last: A375704 (this) (same as A045542 with 8 removed)

%Y - sum: A375705

%Y Cf. A053797, A053806, A061398, A061399, A251092, A373408, A375708, A375714.

%K nonn

%O 1,1

%A _Gus Wiseman_, Aug 29 2024