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 1-4 of 4 results.

A285457 Least number k such that the absolute value of the difference between the number of divisors of k and k-1 is equal to n.

Original entry on oeis.org

3, 2, 6, 17, 12, 25, 24, 37, 48, 325, 60, 144, 120, 121, 168, 289, 180, 529, 240, 577, 481, 361, 360, 900, 960, 961, 721, 5185, 720, 841, 840, 2401, 1261, 17425, 1260, 14641, 1680, 1681, 2161, 8281, 2880, 3600, 6480, 7057, 2520, 6241, 2521, 82945, 6481, 225625, 7200
Offset: 0

Views

Author

Paolo P. Lava, Apr 26 2017

Keywords

Comments

Odd-indexed terms are equal to a square or to a square plus one. - Giovanni Resta, Apr 28 2017

Examples

			a(9) = 325 because 324 has 15 divisors (1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 81, 108, 162, 324), 325 has 6 divisors (1, 5, 13, 25, 65, 325) and 15 - 6 = 9.

Links

Crossrefs

Cf. A000005, A051950, A086550 (without abs), A285787 (with bigomega).

Programs

  • Maple
    with(numtheory): P:=proc(q) local a,b,k,v; v:=array(0..200);
for k from 0 to 200 do v[k]:=0; od; a:=1;
for k from 2 to q do b:=tau(k); if v[abs(b-a)]=0 then v[abs(b-a)]:=k; fi; a:=b; od; k:=0;
while v[k]>0 do print(v[k]); k:=k+1; od; print(); end: P(3*10^5);
  • Mathematica
    s = DivisorSigma[0, #] &@ Range[10^6]; 1 + First /@ Values@ KeySort@ PositionIndex@ Flatten@ Map[Abs@ Differences@ # &, Partition[s, 2, 1]] (* Michael De Vlieger, Apr 26 2017, Version 10 *)

Formula

Least solutions of the equation abs(A000005(k) - A000005(k-1)) = n.

A343018 a(n) is the smallest number m such that tau(m+1) = tau(m) + n.

Original entry on oeis.org

2, 1, 5, 49, 11, 35, 23, 399, 47, 1849, 59, 143, 119, 1599, 167, 575, 179, 1295, 239, 4355, 629, 2303, 359, 899, 959, 9215, 1007, 39999, 719, 20735, 839, 5183, 1799, 46655, 1259, 36863, 1679, 7055, 3023, 986049, 2879, 3599, 6479, 82943, 2519, 193599, 3359, 207935
Offset: 0

Views

Author

Jaroslav Krizek, Apr 02 2021

Keywords

Comments

tau(m) = the number of divisors of m (A000005).
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: 1, 3, 9, 15, 25, 63, 121, 195, 255, 361, 483, 729, ... (A055927).
n = 2: 5, 7, 13, 27, 37, 51, 61, 62, 73, 74, 91, 115, 123, ... (A230115).
n = 3: 49, 99, 1023, 1681, 1935, 2499, 8649, 9603, 20449, ... (A230653).
n = 4: 11, 17, 19, 31, 39, 43, 55, 65, 67, 69, 77, 87, 97, ... (A230654).
n = 5: 35, 169, 289, 529, 961, 1369, 2809, 3135, 4489, ... (A228453).

Examples

			For n = 3; a(3) = 49 because 49 is the smallest number such that tau(50) = 6 = tau(49) + 3 = 3 + 3.

Links

Crossrefs

Cf. A000005 (tau), A080371, A080372, A086550, A340799, A343019, A343020.

Programs

  • Magma
    Ax:=func; [Ax(n): n in [0..50]];
  • Maple
    N:= 60: # 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:= s - t;
  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, Jan 03 2025
  • Mathematica
    d = Differences @ Table[DivisorSigma[0, n], {n, 1, 10^6}]; 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 A343018(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

Formula

a(n) = A086550(n) - 1.

A086551 a(n) = smallest k such that tau(k)= n*tau(k-1) where tau(k) = number of divisors of k, or 0 if no such number exists.

Original entry on oeis.org

3, 2, 12, 24, 48, 60, 192, 168, 180, 240, 5120, 360, 20480, 2112, 720, 840, 983040, 1260, 786432, 3024, 2880, 15360, 62914560, 3360, 22032, 266240, 6300, 6720, 13690208256, 5040, 22548578304, 7560, 156672, 6881280, 25920, 10080
Offset: 1

Views

Author

Amarnath Murthy, Aug 28 2003

Keywords

Comments

Conjectures: (1) No term is zero. (2) a(n)-1 is a prime.
17 is the first n such that a(n)-1 is composite. a(17) = 2^16*3*5 and a(17)-1 is a product of two primes. - David Wasserman, Mar 24 2005

Examples

			a(6) = 60 as tau(60)/ tau(59) = 12/2 = 6.

Crossrefs

Cf. A086550.

Extensions

More terms from David Wasserman, Mar 24 2005

A086552 Numbers x such that tau(x)/tau(x-1) is an integer, where tau() is the number of divisors function.

Original entry on oeis.org

2, 3, 6, 8, 12, 14, 15, 18, 20, 22, 24, 27, 30, 32, 34, 35, 38, 39, 40, 42, 44, 45, 48, 50, 54, 56, 58, 60, 62, 66, 68, 70, 72, 74, 76, 78, 80, 84, 86, 87, 88, 90, 94, 95, 96, 98, 99, 102, 104, 105, 108, 110, 114, 117, 119, 120, 123, 126, 128, 130, 132, 134, 135, 136, 138
Offset: 1

Views

Author

Amarnath Murthy, Aug 28 2003

Keywords

Comments

Conjecture: (1) tau(x)/tau(x-1) = n has solutions for every n. (2) If x is the smallest number for a given n such that tau(x)/tau(x-1) = n > 1, then x-1 is a prime.

Examples

			12 is a member as tau(12)/tau(11) = 3, 15 is a member as tau(15)/tau(14) = 1.

Links

Crossrefs

Cf. A057922, A086550, A086551.

Programs

  • Maple
    with(numtheory): a:=proc(n) if type(tau(n)/tau(n-1), integer)=true then n else fi end: seq(a(n),n=2..150); # Emeric Deutsch, Mar 25 2005
  • PARI
    isok(n) = denominator(numdiv(n)/numdiv(n-1)) == 1; \\ Michel Marcus, Apr 12 2018

Extensions

More terms from David Wasserman and Emeric Deutsch, Mar 25 2005
Showing 1-4 of 4 results.

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