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A284597 a(n) is the least number that begins a run of exactly n consecutive numbers with a nondecreasing number of divisors, or -1 if no such number exists. 7
46, 5, 43, 1, 1613, 241, 17011, 12853, 234613, 376741, 78312721, 125938261, 4019167441, 16586155153, 35237422882, 1296230533473, 42301168491121, 61118966262061 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The words "begins" and "exactly" in the definition are crucial. The initial values of tau (number of divisors function, A000005) can be partitioned into nondecreasing runs as follows: {1, 2, 2, 3}, {2, 4}, {2, 4}, {3, 4}, {2, 6}, {2, 4, 4, 5}, {2, 6}, {2, 6}, {4, 4}, {2, 8}, {3, 4, 4, 6}, {2, 8}, {2, 6}, {4, 4, 4, 9}, {2, 4, 4, 8}, {2, 8}, {2, 6, 6}, {4}, {2, 10}, ... From this we can see that a(1) = 46 (the first singleton), a(2)=5 (the first pair), a(3)=43 (the first triple), a(4)=1, etc. - Bill McEachen and Giovanni Resta, Apr 26 2017. (see also A303577 and A303578 - N. J. A. Sloane, Apr 29 2018

Initial values computed with a brute force C++ program.

It seems very likely that one can always find a(n) and that we never need to take a(n) = -1. But this is at present only a conjecture. - N. J. A. Sloane, May 04 2017

If a(n) > 1, then A013632(a(n)) >= n. Might be useful to help speed up brute force search. - Chai Wah Wu, May 04 2017

The analog sequence for sigma (sum of divisors) instead of tau (number of divisors) is A285893 (see also A028965). - M. F. Hasler, May 06 2017

a(n) > 3.37*10^14 for n > 18. - Robert Gerbicz, May 14 2017

LINKS

Table of n, a(n) for n=1..18.

EXAMPLE

241 = 241^1 => 2 divisors

242 = 2^1 * 11^2 => 6 divisors

243 = 3^5 => 6 divisors

244 = 2^2 * 61^1 => 6 divisors

245 = 5^1 * 7^2 => 6 divisors

246 = 2^1 * 3^1 * 41^1 => 8 divisors

247 = 13^1 * 19^1 => 4 divisors

So, 247 breaks the chain. 241 is the lowest number that is the beginning of exactly 6 consecutive numbers with a nondecreasing number of divisors. So it is the 6th term in the sequence.

Note also that a(5) is not 242, even though tau evaluated at 242, 243,..., 246 gives 5 nondecreasing values, because here we deal with full runs and 242 belongs to the run of 6 values starting at 241.

MATHEMATICA

Function[s, {46}~Join~Map[Function[r, Select[s, Last@ # == r &][[1, 1]]], Range[2, Max[s[[All, -1]] ] ]]]@ Map[{#[[1, 1]], Length@ # + 1} &, DeleteCases[SplitBy[#, #[[-1]] >= 0 &], k_ /; k[[1, -1]] < 0]] &@ MapIndexed[{First@ #2, #1} &, Differences@ Array[DivisorSigma[0, #] &, 10^6]] (* Michael De Vlieger, May 06 2017 *)

PROG

(PARI) genit()={for(n=1, 20, q=0; ibgn=1; for(m=ibgn, 9E99, mark1=q; q=numdiv(m); if(mark1==0, summ=0; dun=0; mark2=m); if(q>=mark1, summ+=1, dun=1); if(dun>0&&summ==n, print(n, " ", mark2); break); if(dun>0&&summ!=n, q=0; m-=1))); } \\ Bill McEachen, Apr 25 2017

(Python)

from sympy import divisor_count

def A284597(n):

    count, starti, s, i = 0, 1, 0, 1

    while True:

        d = divisor_count(i)

        if d < s:

            if count == n:

                return starti

            starti = i

            count = 0

        s = d

        i += 1

        count += 1 # Chai Wah Wu, May 04 2017

(PARI) A284597=vector(19); apply(scan(N, s=1, t=numdiv(s))=for(k=s+1, N, t>(t=numdiv(k))||next; k-s>#A284597||A284597[k-s]||printf(" a(%d)=%d, ", k-s, s)||A284597[k-s]=s; s=k); done, [10^6]) \\ Finds a(1..10) in ~ 1 sec, but would take 100 times longer to get one more term with scan(10^8). You may extend the search using scan(END, START). - M. F. Hasler, May 06 2017

CROSSREFS

Cf. A000005, A000203, A006558, A013632, A028965, A075046, A285893.

See also A286287, A286288, A286289, A303577, A303578.

Sequence in context: A261513 A036204 A270814 * A286288 A051161 A260512

Adjacent sequences:  A284594 A284595 A284596 * A284598 A284599 A284600

KEYWORD

nonn,hard,more

AUTHOR

Fred Schneider, Mar 29 2017

EXTENSIONS

a(1), a(2), a(4) corrected by Bill McEachen and Giovanni Resta, Apr 26 2017

a(17)-a(18) from Robert Gerbicz, May 14 2017

STATUS

approved

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Last modified June 22 14:29 EDT 2018. Contains 305671 sequences. (Running on oeis4.)