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

Author

Antti Karttunen, Sep 24 2015

Keywords

• nonn

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

Crossrefs

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

Programs

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 *)

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

(define (A261100 n) (let loop ((k 1)) (if (> (A002182 k) n) (- k 1) (loop (+ 1 k)))))

Scheme

;; Requires Antti Karttunen's IntSeq-library.
(define A261100 (LEFTINV-LEASTMONO 1 1 A002182))

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)).

Extensions

Description clarified by Antti Karttunen, Jun 06 2017