A375703 - OEIS

%I #10 Aug 29 2024 17:19:19

%S 2,5,10,17,26,28,33,37,50,65,82,101,122,126,129,145,170,197,217,226,

%T 244,257,290,325,344,362,401,442,485,513,530,577,626,677,730,785,842,

%U 901,962,1001,1025,1090,1157,1226,1297,1332,1370,1445,1522,1601,1682,1729

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

%C Non-perfect-powers A007916 are numbers without a proper integer root.

%F Numbers k > 0 such that k-1 is a perfect power (A001597) but k is not.

%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 has length A375702, first a(n), last A375704, sum A375705.

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

%t Min/@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 prime numbers we have A045344.

%Y For nonsquarefree numbers we have A053806, anti-runs A373410.

%Y For nonprime numbers we have A055670, anti-runs A005381.

%Y For squarefree numbers we have A072284, anti-runs A373408.

%Y The anti-run version is A216765 (same as A375703 with 2 exceptions).

%Y For non-prime-powers we have A373673, anti-runs A120430.

%Y For prime-powers we have A373676, anti-runs A373575.

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

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

%Y - first: A375703 (this)

%Y - last: A375704

%Y - sum: A375705

%Y A001597 lists perfect-powers, differences A053289.

%Y A007916 lists non-perfect-powers, differences A375706.

%Y A046933 counts composite numbers between primes.

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

%Y Cf. A007674, A045542, A375708, A375713, A375714.

%K nonn

%O 1,1

%A _Gus Wiseman_, Aug 28 2024