cp's OEIS Frontend

A375396 Numbers not divisible by the square of any prime factor except (possibly) the least. Hooklike numbers.

%I A375396 #25 May 10 2025 09:15:25
%S A375396 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,19,20,21,22,23,24,25,26,27,
%T A375396 28,29,30,31,32,33,34,35,37,38,39,40,41,42,43,44,45,46,47,48,49,51,52,
%U A375396 53,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71
%N A375396 Numbers not divisible by the square of any prime factor except (possibly) the least. Hooklike numbers.
%C A375396 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 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.
%C A375396 The complement is a superset of A036785 = products of a squarefree number and a prime power.
%C A375396 The asymptotic density of this sequence is (1/zeta(2)) * (1 + Sum_{p prime} (1/(p^2-p)) / Product_{primes q <= p} (1 + 1/q)) = 0.884855661165... . - _Amiram Eldar_, Oct 26 2024
%H A375396 Amiram Eldar, <a href="/A375396/b375396.txt">Table of n, a(n) for n = 1..10000</a>
%H A375396 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%F A375396 {a(n)} = {k >= 1 : A028234(k) is in A005117}. - _Peter Munn_, May 09 2025
%e A375396 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 not in the sequence.
%t A375396 Select[Range[100],SameQ@@Min /@ Split[Flatten[ConstantArray@@@FactorInteger[#]],UnsameQ]&]
%o A375396 (PARI) is(k) = if(k > 1, my(e = factor(k)[, 2]); vecprod(e) == e[1], 1); \\ _Amiram Eldar_, Oct 26 2024
%Y A375396 The complement is a superset of A036785.
%Y A375396 For maxima instead of minima we have A065200, counted by A034296.
%Y A375396 The complement for maxima is A065201, counted by A239955.
%Y A375396 Partitions of this type are counted by A115029.
%Y A375396 A version for compositions is A374519, counted by A374517.
%Y A375396 Also positions of identical rows in A375128, sums A374706, ranks A375400.
%Y A375396 The complement is A375397, counted by A375405.
%Y A375396 For distinct instead of identical minima we have A375398, counts A375134.
%Y A375396 The complement for distinct minima is A375399, counted by A375404.
%Y A375396 A000041 counts integer partitions, strict A000009.
%Y A375396 A003242 counts anti-run compositions, ranks A333489.
%Y A375396 A011782 comps counts compositions.
%Y A375396 A number's prime factors (A027746, reverse A238689) have sum A001414, min A020639, max A006530.
%Y A375396 A number's prime indices (A112798, reverse A296150) have sum A056239, min A055396, max A061395.
%Y A375396 Both have length A001222, distinct A001221.
%Y A375396 See the formula section for the relationships with A005117, A028234.
%Y A375396 Cf. A000005, A013661, A046660, A272919, A319066, A358905, A374686, A374704, A374742, A375133, A375136, A375401.
%K A375396 nonn
%O A375396 1,2
%A A375396 _Gus Wiseman_, Aug 16 2024