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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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.

Original entry on oeis.org

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

Views

Author

Fred Schneider, Mar 29 2017

Keywords

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
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

Examples

			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.

Links

Crossrefs

Cf. A000005, A000203, A006558, A013632, A028965, A075046, A285893.
See also A286287, A286288, A286289, A303577, A303578.

Programs

  • 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 *)
  • 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
  • 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
  • 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

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

A286287 Least number to start a run of exactly n nondecreasing values of little omega (A001221).

Original entry on oeis.org

106, 11, 13, 7, 512, 1, 1941, 141, 6847, 211, 195031, 82321, 808083, 534077, 3355906, 526093, 526889774, 127890361, 22529949392, 118968284927, 164159173895, 244022049199, 3022058317713, 585927201061
Offset: 1

Views

Author

N. J. A. Sloane, May 16 2017

Keywords

Comments

a(17) > 10^7.
a(18) = 127890361; a(n) > 4*10^8 for n=17 and for n >= 19. - Jon E. Schoenfield, Jul 16 2017
a(n) > 6*10^12 for n >= 25. - Giovanni Resta, Jul 16 2019

Examples

			We have omega(10) = 2, omega(11) = 1, omega(12) = 2, omega(13) = 1. Therefore 11 starts a run of exactly 2 consecutive integers (11, 12) which have nondecreasing (here: strictly increasing) values of omega.
The 6 numbers from 1 through 6 yield values (0, 1, ..., 1, 2) for omega, therefore a(6) = 1. The 4 numbers from 7 through 10 yield values (1, 1, 1, 2) for omega, therefore a(4) = 7.
A run of length 1 is a single number n such that omega(n-1) > omega(n) > omega(n+1). (If we had "<=" in one of the cases, it would be part of a run of at least 2 numbers with nondecreasing omega.) This first happens for a(1) = 106.

References

  • M. F. Hasler, Posting to Sequence Fans Mailing List, May 06 2017

Crossrefs

Cf. A001221, A284597 (analog for tau = A000005), A285893 (analog for sigma = A000203).
See also A286288, A286289.

Programs

  • Mathematica
    Prepend[#, Module[{k = 2}, While[Sign@ Differences@ PrimeNu[k + {-1, 0, 1}] != {-1, -1}, k++]; k]] &@ Function[s, Function[r, If[Length@ # > 0, #[[1, 1]], -1] &@ Select[s, Length@ # == r &]] /@ Range@ Max@ Map[Length, s]]@ DeleteCases[SplitBy[MapIndexed[Function[k, (2 Boole[#1 <= #2] - 1) k & @@ #1]@ First@ #2 &, Partition[Array[PrimeNu, 10^7], 2, 1]], Sign], w_ /; First@ w < 0] (* Michael De Vlieger, May 19 2017 *)
  • PARI
    alias("A","A286287"); A=vector(19); apply(scan(N, s=1, t=omega(s))=for(k=s+1, N, t>(t=omega(k))||next; k-s>#A||A[k-s]||printf(" a(%d)=%d, ", k-s, s)||A[k-s]=s; s=k); done, [4e6]) \\ Then the search may be extended using scan(END,START). - M. F. Hasler, May 16 2017

Extensions

Edited by M. F. Hasler, May 16 2017
a(17)-a(24) from Giovanni Resta, Jul 16 2019

A286288 Least number to start a run of exactly n nondecreasing values of (big) Omega (A001222).

Original entry on oeis.org

46, 5, 43, 1, 2021, 121, 25202, 2521, 162121, 460801, 27268546, 23553169, 244068841, 913535283, 3195380866, 2088087121, 5988790769809, 2601212829601
Offset: 1

Views

Author

N. J. A. Sloane, May 16 2017

Keywords

Comments

a(11) > 10^7. - M. F. Hasler, May 16 2017
a(19) > 7.5*10^12. - Giovanni Resta, Nov 12 2019

Examples

			Omega(1..5) = (0, 1, 1, 2, 1), therefore the first run of 4 numbers with nondecreasing Omega (= A001222) starts at a(4) = 1.
Omega(4..7) = (2, 1, 2, 1), so the first run of 2 numbers with nondecreasing Omega starts at a(2) = 5.
A run of subsequent numbers with nondecreasing Omega is of length 1 if it consists of a single number n with Omega(n-1) > Omega(n) > Omega(n+1) (else n belongs to a run of length >= 2). This happens first for a(1) = 46.

References

  • M. F. Hasler, Posting to Sequence Fans Mailing List, May 06 2017

Crossrefs

Cf. A001222, A284597 (analog for sigma_0 = A000005), A285893 (analog for sigma = A000203), A286287 (analog for omega = A001221).
See also A286289.

Programs

  • Mathematica
    Prepend[#, Module[{k = 2}, While[Sign@ Differences@ PrimeOmega[k + {-1, 0, 1}] != {-1, -1}, k++]; k]] &@ Function[s, Function[r, If[Length@ # > 0, #[[1, 1]], -1] &@ Select[s, Length@ # == r &]] /@ Range@ Max@ Map[Length, s]]@ DeleteCases[SplitBy[MapIndexed[Function[k, (2 Boole[#1 <= #2] - 1) k & @@ #1]@ First@ #2 &, Partition[Array[PrimeOmega, 10^7], 2, 1]], Sign], w_ /; First@ w < 0] (* Michael De Vlieger, May 19 2017 *)
  • PARI
    alias('A, 'A286288);\\ A bug in PARI 2.9.2 requires the alias() to be issued on a line on itself.
A=vector(19); apply(scan(N, s=1, t=bigomega(s))=for(k=s+1, N, t>(t=bigomega(k))||next; k-s>#A||A[k-s]||printf(" a(%d)=%d, ", k-s, s)||A[k-s]=s; s=k); done, [1e7]) \\ Then the search may be extended using scan(END, START).
\\ M. F. Hasler, May 16 2017

Extensions

Edited by M. F. Hasler, May 16 2017
a(11)-a(13) from Jon E. Schoenfield, Jul 16 2017
a(14)-a(18) from Giovanni Resta, Nov 12 2019
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