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

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, 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, 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

Offset

1,2

Comments

Each n occurs A262501(n) times.

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

Links

Antti Karttunen, Table of n, a(n) for n = 1..10080

Formula

a(n) = the least k for which A002182(k+1) > n.

Other identities. For all n >= 1:

a(A002182(n)) = n. [The least monotonic sequence satisfying this condition.]

A070319(n) = A002183(a(n)).

Maple

with(numtheory):
A261100_list := proc(len) local n, k, j, b, A, tn: A := NULL; k := 0;
for n from 1 to len do
    b := true; tn := tau(n);
    for j from 1 to n-1 while b do b := b and tau(j) < tn od:
    if b then k := k + 1 fi;
    A := A, k
od: A end: A261100_list(120); # Peter Luschny, Jun 06 2017

Mathematica

A002182 = Import["https://oeis.org/A002182/b002182.txt", "Table"];
inter = Interpolation[Reverse /@ A002182, InterpolationOrder -> 0];
A261100 = Rest[inter /@ Range[200]] - 1 (* Jean-François Alcover, Oct 25 2019 *)

Prog

(PARI)
v002182 = vector(1000); v002182[1] = 1; \\ For memoization.
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); };
A261100(n) = { my(k=1); while(A002182(k)<=n, k=k+1); (k-1); } \\ Antti Karttunen, Jun 06 2017
(Scheme, two variants, the other one requiring Antti Karttunen's IntSeq-library)
(define (A261100 n) (let loop ((k 1)) (if (> (A002182 k) n) (- k 1) (loop (+ 1 k)))))
(define A261100 (LEFTINV-LEASTMONO 1 1 A002182))

Crossrefs

Cf. A000005, A002182, A002183, A060990, A070319, A262501.

Sequence in context: A341155 A341166 A274102 * A130249 A286105 A061071

Adjacent sequences: A261097 A261098 A261099 * A261101 A261102 A261103

Keyword

nonn

Author

Antti Karttunen, Sep 24 2015

Extensions

Description clarified by Antti Karttunen, Jun 06 2017

Status

approved