A262501 -id:A262501 - cp's OEIS frontend

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

Antti Karttunen, Sep 24 2015

nonn

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

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

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

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

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 *)

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

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

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

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)).

Description clarified by Antti Karttunen, Jun 06 2017

A054481 Highest common factor of successive highly composite numbers (1), A002182.

Original entry on oeis.org

1, 2, 2, 6, 12, 12, 12, 12, 60, 60, 60, 120, 360, 120, 420, 420, 840, 2520, 2520, 2520, 5040, 5040, 5040, 2520, 2520, 5040, 5040, 27720, 27720, 55440, 55440, 55440, 55440, 166320, 55440, 110880, 55440, 360360, 360360

Offset: 2

Henry Bottomley, Mar 31 2000

Not the same as the first differences of A002182. The latter are given by A262501, which differs from this sequence for the first time at n=25, where A262501(25) = 17640, while here the 25th term a(26) is 2520. The sequences differ next time at positions n = 52, 53, 54, 64, 67, 82, 83, 84, 85, 86, 87, 88, 90, 91, 96, 100, 106, ... (when one-based indexing as in A262501 is used). - Antti Karttunen, Sep 24 2015

			a(7)=12 because A002182(7)=36, A002182(6)=24 and GCD(36,24)=12.

Antti Karttunen, Table of n, a(n) for n = 2..1000 (computed from the b-file of A002182 provided by _T. D. Noe_)

Cf. A002182, A054482, A054483, A262501.

a(n) = GCD(A002182(n-1), A002182(n)) = A002182(n)/A054483(n) = A002182(n-1)/A054482(n).

Erroneous comment (wrong interpretation) removed by Antti Karttunen, Sep 25 2015

A333725 Number of primes between pairs of consecutive highly composite numbers (A002182).

Original entry on oeis.org

0, 1, 1, 2, 4, 2, 4, 2, 13, 11, 11, 20, 56, 18, 59, 58, 105, 307, 284, 278, 528, 515, 501, 241, 1684, 466, 456, 2491, 2403, 4676, 4561, 4459, 4396, 12839, 4202, 8317, 4111, 26274, 25673, 50073, 48866, 47998, 47441, 139491, 45881, 90692, 134351, 220465, 173831, 257677

Offset: 1

Robin Powell, Apr 03 2020

nonn

			There are no primes between HCN(1) and HCN(2), so a(1) = 0. The next term a(2) is equal to 1 as 3 is the only prime between HCN(2) and HCN(3); the prime 2 is not greater than HCN(2) and so is omitted here. The first gap to contain more than one prime occurs at a(4) = 2, which alludes to 7 and 11 being the only primes contained within HCN(4) and HCN(5).

Charles R Greathouse IV, Table of n, a(n) for n = 1..200

Cf. A002182, A000720, A262501.

Join[{0},PrimePi[#[[2]]]-PrimePi[#[[1]]]&/@Partition[DeleteDuplicates[Table[ {n,DivisorSigma[ 0,n]},{n,2,22*10^5}],GreaterEqual[#1[[2]],#2[[2]]]&][[All,1]],2,1]] (* Harvey P. Dale, Jan 09 2023 *)

a(n) = A000720(A002182(n+1)) - A000720(A002182(n)) for n > 1. - Amiram Eldar, Apr 26 2020

More terms from Giovanni Resta, Apr 04 2020

cp's OEIS Frontend

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

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A054481 Highest common factor of successive highly composite numbers (1), A002182.

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A333725 Number of primes between pairs of consecutive highly composite numbers (A002182).

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Mathematica

Formula

Extensions