A075046 -id:A075046 - cp's OEIS frontend

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.

46, 5, 43, 1, 1613, 241, 17011, 12853, 234613, 376741, 78312721, 125938261, 4019167441, 16586155153, 35237422882, 1296230533473, 42301168491121, 61118966262061

Offset: 1

Fred Schneider, Mar 29 2017

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
Conjecture follows from Dickson's conjecture (see link). - Robert Israel, Mar 30 2020
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

			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.

Robert Israel, Dickson's conjecture implies all a(n)>0

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

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

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 *)

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

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

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

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

A075031 a(n) is the smallest number k such that the number of divisors of the n numbers from k through k+n-1 are nonincreasing.

Original entry on oeis.org

1, 2, 20, 20, 714, 714, 25550, 90180, 142803, 809300, 27195648, 27195648, 6973441007, 37962822225, 37962822225, 114296059262, 265228019405583, 394047434860662, 2493689139940250

Offset: 1

Amarnath Murthy, Sep 02 2002

tau(k) >= tau(k+1) >= ... >= tau(k+n-1).
Next term is > 2000000. - David Wasserman, May 06 2005
a(17) > 10^12. [Donovan Johnson, Oct 13 2009]
a(20) > 2.64x10^15. - Jud McCranie, Mar 27 2019

			a(3)=a(4) = 20 as tau(20) > tau(21) = tau(22) > tau(23).

Cf. A075028, A075029, A075046.

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 19 2003
a(11)-a(16) from Donovan Johnson, Oct 13 2009
a(17)-a(19) from Jud McCranie, Mar 27 2019

A364804 a(n) is the smallest number k such that the number of prime divisors (counted with multiplicity) of the n numbers from k through k+n-1 are in nondescending order.

Original entry on oeis.org

1, 1, 1, 1, 121, 121, 2521, 2521, 162121, 460801, 23553169, 23553169, 244068841, 913535283

Offset: 1

Ilya Gutkovskiy, Aug 08 2023

Smallest initial number k of n consecutive numbers satisfying bigomega(k) <= bigomega(k+1) <= ... <= bigomega(k+n-1).

			a(5) = 121 = a(6) as bigomega(121) = bigomega(122) = bigomega(123) = 2 < bigomega(124) = bigomega(125) = 3 < bigomega(126) = 4.

Cf. A001222, A075046, A286288, A364805.

k = 1; Do[While[t = Table[PrimeOmega[i], {i, k, k + n - 1}]; t != Sort[t], k++]; Print[k], {n, 1, 10}]

a(n) = my(k=1, list=List(vector(n, i, bigomega(i)))); while (vecsort(list) != list, listpop(list, 1); k++; listput(list, bigomega(k+n-1))); k; \\ Michel Marcus, Aug 14 2023

a(11)-a(14) from Michel Marcus, Aug 14 2023

A364805 a(n) is the smallest number k such that the number of distinct prime divisors of the n numbers from k through k+n-1 are in nondescending order.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 141, 141, 211, 211, 82321, 82321, 526093, 526093, 526093, 526093, 127890361, 127890361

Offset: 1

Ilya Gutkovskiy, Aug 08 2023

Smallest initial number k of n consecutive numbers satisfying omega(k) <= omega(k+1) <= ... <= omega(k+n-1).

			a(9) = 211 = a(10) as omega(211) = 1 < omega(212) = omega(213) = omega(214) = omega(215) = omega(216) = omega(217) = omega(218) = omega(219) = 2 < omega(220) = 3.

Cf. A001221, A075046, A286287, A364804.

k = 1; Do[While[t = Table[PrimeNu[i], {i, k, k + n - 1}]; t != Sort[t], k++]; Print[k], {n, 1, 16}]

a(n) = my(k=1, list=List(vector(n, i, omega(i)))); while (vecsort(list) != list, listpop(list, 1); k++; listput(list, omega(k+n-1))); k; \\ Michel Marcus, Aug 14 2023

a(17)-a(18) from Michel Marcus, Aug 14 2023

cp's OEIS Frontend

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.

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A075031 a(n) is the smallest number k such that the number of divisors of the n numbers from k through k+n-1 are nonincreasing.

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A364804 a(n) is the smallest number k such that the number of prime divisors (counted with multiplicity) of the n numbers from k through k+n-1 are in nondescending order.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A364805 a(n) is the smallest number k such that the number of distinct prime divisors of the n numbers from k through k+n-1 are in nondescending order.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions