login
A375703
Minimum of the n-th maximal run of adjacent (increasing by one at a time) non-perfect-powers.
13
2, 5, 10, 17, 26, 28, 33, 37, 50, 65, 82, 101, 122, 126, 129, 145, 170, 197, 217, 226, 244, 257, 290, 325, 344, 362, 401, 442, 485, 513, 530, 577, 626, 677, 730, 785, 842, 901, 962, 1001, 1025, 1090, 1157, 1226, 1297, 1332, 1370, 1445, 1522, 1601, 1682, 1729
OFFSET
1,1
COMMENTS
Non-perfect-powers A007916 are numbers without a proper integer root.
FORMULA
Numbers k > 0 such that k-1 is a perfect power (A001597) but k is not.
EXAMPLE
The list of all non-perfect-powers, split into runs, begins:
2 3
5 6 7
10 11 12 13 14 15
17 18 19 20 21 22 23 24
26
28 29 30 31
33 34 35
37 38 39 40 41 42 43 44 45 46 47 48
Row n has length A375702, first a(n), last A375704, sum A375705.
MATHEMATICA
radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Min/@Split[Select[Range[100], radQ], #1+1==#2&]//Most
- or -
radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Select[Range[100], radQ[#]&&!radQ[#-1]&]
CROSSREFS
For prime numbers we have A045344.
For nonsquarefree numbers we have A053806, anti-runs A373410.
For nonprime numbers we have A055670, anti-runs A005381.
For squarefree numbers we have A072284, anti-runs A373408.
The anti-run version is A216765 (same as A375703 with 2 exceptions).
For non-prime-powers we have A373673, anti-runs A120430.
For prime-powers we have A373676, anti-runs A373575.
For runs of non-perfect-powers (A007916):
- length: A375702 = A053289(n+1) - 1.
- first: A375703 (this)
- last: A375704
- sum: A375705
A001597 lists perfect-powers, differences A053289.
A007916 lists non-perfect-powers, differences A375706.
A046933 counts composite numbers between primes.
A375736 gives lengths of anti-runs of non-prime-powers, sums A375737.
Sequence in context: A340039 A003192 A018682 * A078393 A340045 A100292
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 28 2024
STATUS
approved