OFFSET
1,1
COMMENTS
Also numbers k such that the minima of the maximal anti-runs in the weakly increasing sequence of prime factors of k (with multiplicity) are not identical. Here, an anti-run is a sequence with no adjacent equal parts, and the minima of the maximal anti-runs in a sequence are obtained by splitting it into maximal anti-run subsequences and taking the least term of each. Note the prime factors can alternatively be taken in weakly decreasing order.
Includes all terms of A036785 = non-products of a squarefree number and a prime power.
The asymptotic density of this sequence is 1 - (1/zeta(2)) * (1 + Sum_{p prime} (1/(p^2-p)) / Product_{primes q <= p} (1 + 1/q) = 0.11514433883... . - Amiram Eldar, Oct 26 2024
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
EXAMPLE
The prime factors of 300 are {2,2,3,5,5}, with maximal anti-runs ((2),(2,3,5),(5)), with minima (2,2,5), so 300 is in the sequence.
The terms together with their prime indices begin:
18: {1,2,2}
36: {1,1,2,2}
50: {1,3,3}
54: {1,2,2,2}
72: {1,1,1,2,2}
75: {2,3,3}
90: {1,2,2,3}
98: {1,4,4}
100: {1,1,3,3}
108: {1,1,2,2,2}
126: {1,2,2,4}
144: {1,1,1,1,2,2}
MATHEMATICA
Select[Range[100], !SameQ@@Min /@ Split[Flatten[ConstantArray@@@FactorInteger[#]], UnsameQ]&]
PROG
(PARI) is(k) = if(k > 1, my(e = factor(k)[, 2]); vecprod(e) > e[1], 0); \\ Amiram Eldar, Oct 26 2024
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Gus Wiseman, Aug 16 2024
STATUS
approved