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A261100 a(n) is the greatest m for which A002182(m) <= n; the least monotonic left inverse for highly composite numbers A002182.

%I A261100 #33 Feb 26 2025 18:12:19
%S A261100 1,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,
%T A261100 6,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,
%U A261100 9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10
%N A261100 a(n) is the greatest m for which A002182(m) <= n; the least monotonic left inverse for highly composite numbers A002182.
%C A261100 Each n occurs A262501(n) times.
%C A261100 This is the only sequence w, which (1) satisfies w(A002182(n)) = n for all n >= 1 (thus is a left inverse of A002182), which (2) is monotonic (by necessity growing, although not strictly so), and which (3) is the lexicographically least of all sequences satisfying both (1) and (2). In other words, the largest number m for which A002182(m) <= n. - _Antti Karttunen_, Jun 06 2017
%H A261100 Antti Karttunen, <a href="/A261100/b261100.txt">Table of n, a(n) for n = 1..10080</a>
%F A261100 a(n) = the least k for which A002182(k+1) > n.
%F A261100 Other identities. For all n >= 1:
%F A261100 a(A002182(n)) = n. [The least monotonic sequence satisfying this condition.]
%F A261100 A070319(n) = A002183(a(n)).
%p A261100 with(numtheory):
%p A261100 A261100_list := proc(len) local n, k, j, b, A, tn: A := NULL; k := 0;
%p A261100 for n from 1 to len do
%p A261100     b := true; tn := tau(n);
%p A261100     for j from 1 to n-1 while b do b := b and tau(j) < tn od:
%p A261100     if b then k := k + 1 fi;
%p A261100     A := A,k
%p A261100 od: A end: A261100_list(120); # _Peter Luschny_, Jun 06 2017
%t A261100 A002182 = Import["https://oeis.org/A002182/b002182.txt", "Table"];
%t A261100 inter = Interpolation[Reverse /@ A002182, InterpolationOrder -> 0];
%t A261100 A261100 = Rest[inter /@ Range[200]] - 1 (* _Jean-François Alcover_, Oct 25 2019 *)
%o A261100 (PARI)
%o A261100 v002182 = vector(1000); v002182[1] = 1; \\ For memoization.
%o A261100 A002182(n) = { my(d,k); if(v002182[n],v002182[n], k = A002182(n-1); d = numdiv(k); while(numdiv(k) <= d, k=k+1); v002182[n] = k; k); };
%o A261100 A261100(n) = { my(k=1); while(A002182(k)<=n,k=k+1); (k-1); } \\ _Antti Karttunen_, Jun 06 2017
%o A261100 (Scheme)
%o A261100 (define (A261100 n) (let loop ((k 1)) (if (> (A002182 k) n) (- k 1) (loop (+ 1 k)))))
%o A261100 (Scheme)
%o A261100 ;; Requires _Antti Karttunen_'s IntSeq-library.
%o A261100 (define A261100 (LEFTINV-LEASTMONO 1 1 A002182))
%Y A261100 Cf. A000005, A002182, A002183, A060990, A070319, A262501.
%K A261100 nonn
%O A261100 1,2
%A A261100 _Antti Karttunen_, Sep 24 2015
%E A261100 Description clarified by _Antti Karttunen_, Jun 06 2017