cp's OEIS Frontend

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.

Showing 1-3 of 3 results.

A343019 a(n) is the smallest number m such that tau(m+1) = tau(m) - n.

Original entry on oeis.org

2, 4, 6, 16, 12, 24, 30, 36, 84, 324, 60, 144, 192, 120, 210, 288, 180, 528, 240, 576, 480, 360, 420, 900, 1344, 960, 720, 5184, 1008, 840, 1320, 2400, 1260, 17424, 1800, 14640, 2640, 1680, 2160, 8280, 4800, 3600, 11220, 7056, 3780, 6240, 2520, 82944, 6480
Offset: 0

Views

Author

Jaroslav Krizek, Apr 02 2021

Keywords

Comments

tau(m) = the number of divisors of m (A000005).
A greedy inverse of A051950.
Sequences of numbers m such that tau(m+1) = tau(m) - n for 0 <= n <= 5:
n = 0: 2, 14, 21, 26, 33, 34, 38, 44, 57, 75, 85, 86, 93, ... (A005237).
n = 1: 4, 8, 81, 441, 625, 1089, 2024, 2401, 3025, 3968, ... (A068208).
n = 2: 6, 10, 20, 22, 32, 45, 46, 50, 58, 68, 76, 82, 92, ... (A227874).
n = 3: 16, 64, 224, 675, 1444, 2115, 3843, 5475, 6724, 9801, ...
n = 4: 12, 18, 28, 52, 54, 56, 105, 110, 114, 128, 148, 154, ...
n = 5: 24, 80, 225, 484, 1024, 1088, 1156, 1225, 1521, 2116, ...

Examples

			For n = 3; a(3) = 16 because 16 is the smallest number such that tau(17) = 2 = tau(16) - 3 = 5 - 3.

Links

Crossrefs

Cf. A000005 (tau), A051950, A080371, A080372, A343018.

Programs

  • Magma
    Ax:=func; [Ax(n): n in [0..50]]
  • Maple
    N:= 50: # for a(0)..a(N)
V:= Array(0..N): count:=0: t:= numtheory:-tau(1):
for m from 1 while count < N+1 do
  s:= numtheory:-tau(m+1); v:= t - s;
  if v >= 0 and v <= N and V[v] = 0 then
    count:= count+1; V[v]:= m;
  fi;
  t:= s;
od:
convert(V,list); # Robert Israel, Jul 03 2024
  • Mathematica
    d = Differences @ Table[DivisorSigma[0, n], {n, 1, 10^5}]; a[n_] := If[(p = Position[d, -n]) != {}, p[[1, 1]], 0]; s = {}; n = 0; While[(a1 = a[n]) > 0, AppendTo[s, a1]; n++]; s (* Amiram Eldar, Apr 03 2021 *)
  • PARI
    a(n) = my(m=1); while (numdiv(m+1) != numdiv(m) - n, m++); m; \\ Michel Marcus, Apr 03 2021
  • Python
    from itertools import count, pairwise
from sympy import divisor_count
def A343019(n): return next(m+1 for m, t in enumerate(pairwise(map(divisor_count,count(1)))) if t[1] == t[0]-n) # Chai Wah Wu, Jul 25 2022

A339777 Numbers m such that tau(m) = tau(m + 1) + 1 = tau(m + 2), where tau(k) = the number of divisors of k (A000005).

Original entry on oeis.org

8, 110888, 149768, 1119363, 1172888, 2676495, 3143528, 4782968, 5895183, 8596623, 9168783, 15896168, 19114383, 28174863, 48052623, 50523663, 58186383, 72641528, 82664463, 98168463, 113465103, 139523343, 178810383, 208860303, 223681935, 230675343, 248755983, 249260943
Offset: 1

Views

Author

Jaroslav Krizek, Dec 16 2020

Keywords

Comments

Corresponding values of tau(a(n)): 4, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, ...
Triplets of [tau(a(n)), tau(a(n) + 1), tau(a(n) + 2)] = [tau(a(n)), tau(a(n)) - 1, tau(a(n))]: [4, 3, 4], [16, 15, 16], [16, 15, 16], [16, 15, 16], [16, 15, 16], [16, 15, 16], [16, 15, 16], [16, 15, 16], [16, 15, 16], ...
a(n) is one less than a perfect square. - David A. Corneth, Dec 29 2020

Examples

			tau(8) = 4, tau(9) = 3, tau(10) = 4.

Links

Crossrefs

Cf. A000005, A339776.
Subsequence of A005563.
Intersection of A062832 and A068208.

Programs

  • Magma
    [m: m in [2..10^6] | #Divisors(m + 1) + 1 eq #Divisors(m) and #Divisors(m + 2) eq #Divisors(m)]
  • Mathematica
    d1 = 1; d2 = 2; s = {}; Do[d3 = DivisorSigma[0, n]; If[Equal @@ {d1, d2 + 1, d3}, AppendTo[s, n - 2]]; d1 = d2; d2 = d3, {n, 3, 10^7}]; s (* Amiram Eldar, Dec 17 2020 *)
Position[Partition[DivisorSigma[0,Range[59*10^5]],3,1],?(#[[1]]==#[[2]]+1==#[[3]]&),1,Heads->False]//Flatten (* _Harvey P. Dale, May 25 2023 *)
  • PARI
    isok(m) =  my(nb = numdiv(m)); (numdiv(m+2) == nb) && (numdiv(m+1) == nb-1); \\ Michel Marcus, Dec 18 2020

Extensions

More terms from Amiram Eldar, Dec 16 2020

A353032 a(n) is the smallest number m with n divisors such that m+1 has n-1 divisors, or 0 if no such number exists.

Original entry on oeis.org

0, 0, 4, 8, 81, 0, 0, 0, 441, 6723, 0, 0, 0, 0, 767495140624, 2024, 665416609183179841, 0, 0, 0, 2050624, 263168, 0, 0, 670801950625, 0, 10871232294189453124, 532899, 0, 0, 0, 0, 67634176, 0, 55471075527984793933106579132930662929175947116953798971172816083061185149078369140624
Offset: 1

Views

Author

Jaroslav Krizek, Apr 18 2022

Keywords

Comments

For n > 33, a(64) = 6890624 is the only positive term <= 10^8.
There is no number m <= 10^10 that is the first start of run of 3 consecutive integers m, m+1 and m+2 with triplet [tau(m), tau(m+1), tau(m+2)] = [tau(m), tau(m) - 1, tau(m) - 2].
If a(11) > 0 then a(11) > 10^100. - Charles R Greathouse IV, Apr 20 2022
a(36) = 1626347583, a(40) = 1173953168, a(49) = 304006671424, a(65) = 25221297570561, a(81) = 15579533124, a(96) = 68195356770303, a(100) = 1698353697680, a(136) = 28528257204224, a(256) = 334435516415. - Jon E. Schoenfield, Apr 24 2022
From Jon E. Schoenfield, May 01 2022: (Start)
a(35) is the smallest m such that m = 16*p^6 = q^16*r - 1 where p, q, and r are odd primes; a(35) <= 16*123024356097427^6 (an 86-digit number).
a(37) = a(38) = 0;
a(39) <= 1134572901070399771884918212890624;
a(41) <= 350847983^40 (a 342-digit number). (End)

Examples

			For n = 5; a(5) = 81 because 81 is the smallest number m such that tau(m) = tau(81) = 5 and tau(82) = tau(m) - 1 = 4.

Crossrefs

Cf. A000005, A068208, A343144.

Programs

  • Magma
    Ax:=func; [Ax(n): n in [1..5]]

Extensions

a(6)-a(8) from Jon E. Schoenfield, Apr 20 2022
a(9)-a(10), a(16), a(21)-a(22), a(28), a(33) from Jaroslav Krizek, Apr 20 2022
Remaining terms through a(34) from Jon E. Schoenfield, Apr 30 2022
a(35) from Jinyuan Wang, May 21 2022
Showing 1-3 of 3 results.

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