A230115 -id:A230115 - cp's OEIS frontend

A230653 Numbers k such that tau(k+1) - tau(k) = 3, where tau(k) = the number of divisors of k (A000005).

49, 99, 1023, 1681, 1935, 2499, 8649, 9603, 20449, 21903, 23715, 29583, 30975, 38024, 43263, 58563, 60515, 71824, 74528, 110223, 130321, 136899, 145924, 150543, 154449, 165649, 181475, 216224, 224675, 233288, 243049, 256035, 258063, 265225, 294849, 300303

Offset: 1

Jaroslav Krizek, Oct 27 2013

nonn

Numbers k such that A051950(k+1) = 3.
Numbers k such that A049820(k) - A049820(k+1) = 2.
k or k+1 is a perfect square. - David A. Corneth, Feb 16 2024

			99 is in the sequence because tau(100) - tau(99) = 9 - 6 = 3.

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 90 terms from Harvey P. Dale)

Cf. A055927 (numbers n such that tau(n+1) - tau(n) = 1), A230115 (numbers n such that tau(n+1) - tau(n) = 2), A000005.

Select[ Range[ 50000], DivisorSigma[0, # ] + 3 == DivisorSigma[0, # + 1] &]
Position[Differences[DivisorSigma[0,Range[300400]]],3]//Flatten (* Harvey P. Dale, Jun 30 2022 *)

isok(n) = numdiv(n+1) - numdiv(n) == 3; \\ Michel Marcus, Oct 27 2013

from sympy import divisor_count as tau
from itertools import count, islice
def agen(): # generator of terms, using comment by David A. Corneth
    for m in count(1):
        mm = m*m
        tmm = tau(mm)
        if tmm - tau(mm-1) == 3: yield mm-1
        if tau(mm+1) - tmm == 3: yield mm
print(list(islice(agen(), 36))) # Michael S. Branicky, Feb 16 2024

More terms from Michel Marcus, Oct 27 2013

A230654 Numbers n such that tau(n+1) - tau(n) = 4, where tau(n) = the number of divisors of n (A000005).

Original entry on oeis.org

11, 17, 19, 31, 39, 43, 55, 65, 67, 69, 77, 87, 97, 129, 134, 163, 175, 183, 185, 194, 207, 211, 221, 237, 241, 247, 249, 254, 265, 283, 295, 309, 321, 327, 331, 337, 343, 351, 365, 398, 404, 417, 437, 454, 458, 459, 469, 471, 473, 482, 493, 494, 497, 505, 517

Offset: 1

Jaroslav Krizek, Nov 03 2013

nonn

Numbers n such that A051950(n+1) = 4. Numbers n such that A049820(n) - A049820(n+1) = 3. Sequence of starts of first run of n (n>=2) consecutive integers m_1, m_2, ..., m_n such that tau(m_k) - tau(m_k-1) = 4, for all k=n...2: 11, 458, 3013, ... (a(5) > 100000); example for n=4: tau(3013) = 4, tau(3014) = 8, tau(3015) = 12, tau(3016) = 16.

			19 is in sequence because tau(20) - tau(19) = 6 - 2 = 4.

Jaroslav Krizek, Table of n, a(n) for n = 1..4000

Cf. A055927 (numbers n such that tau(n+1) - tau(n) = 1), A230115 (numbers n such that tau(n+1) - tau(n) = 2), A230653 (numbers n such that tau(n+1) - tau(n) = 3), A000005.

Select[ Range[ 50000], DivisorSigma[0, # ] + 4 == DivisorSigma[0, # + 1] &]

A228453 Numbers k such that tau(k+1) - tau(k) = 5, where tau(k) = the number of divisors of k (A000005).

Original entry on oeis.org

35, 169, 289, 529, 961, 1369, 2809, 3135, 4489, 7921, 9409, 10609, 10815, 11881, 12769, 16129, 18495, 18769, 22201, 22801, 26569, 27889, 32041, 33855, 38809, 44521, 49729, 51529, 52441, 53823, 58081, 61503, 69169, 72361, 76729, 78961, 80089, 96721

Offset: 1

Jaroslav Krizek, Nov 03 2013

nonn

Numbers k such that A051950(k+1) = 5.
Numbers k such that A049820(k) - A049820(k+1) = 4.
Either k or k+1 is a square. - Amiram Eldar, Apr 17 2024

			35 is in sequence because tau(36) - tau(35) = 9 - 4 = 5.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A049820, A051950.

Numbers k such that tau(k+1) - tau(k) = m: A055927 (m = 1), A230115 (m = 2), A230653 (m = 3), A230654 (m = 4), this sequence (m = 5).

Select[ Range[ 50000], DivisorSigma[0, # ] + 5 == DivisorSigma[0, # + 1] &]

lista(kmax) = {my(d); for(k = 2, kmax, d = numdiv(k^2); if(d == numdiv(k^2-1) + 5, print1(k^2-1, ", ")); if(d == numdiv(k^2+1) - 5, print1(k^2, ", ")));} \\ Amiram Eldar, Apr 17 2024

A227874 Numbers n such that tau(n+1) - tau(n) = -2, where tau(n) = the number of divisors of n (A000005).

Original entry on oeis.org

6, 10, 20, 22, 32, 45, 46, 50, 58, 68, 76, 82, 92, 106, 117, 124, 152, 166, 170, 174, 178, 212, 226, 236, 261, 262, 272, 325, 333, 338, 346, 358, 382, 405, 412, 424, 435, 436, 452, 464, 466, 474, 477, 478, 495, 502, 506, 512, 530, 555, 562, 567, 574, 578, 586

Offset: 1

Jaroslav Krizek, Nov 03 2013

nonn

Numbers n such that tau(n) - tau(n+1) = 2. Numbers n such that A051950(n+1) = -2. Numbers n such that A049820(n) - A049820(n+1) = -3.

Sequence of starts of first run of n (n>=2) consecutive integers m_1, m_2, ..., m_n such that tau(m_k) - tau(m_k-1) = -2, for all k=n...2: 6, 45, 1016, ... (a(5) > 100000); example for n=4: tau(1016) = 8, tau(1017) = 6, tau(1018) = 4, tau(1019) = 2.

			45 is in sequence because tau(46) - tau(45) = 4 - 6 = -2.

Jaroslav Krizek, Table of n, a(n) for n = 1..2000

Cf. A000005.
Cf. A055927 (numbers n such that tau(n+1) - tau(n) = 1).
Cf. A230115 (numbers n such that tau(n+1) - tau(n) = 2).
Cf. A230653 (numbers n such that tau(n+1) - tau(n) = 3).
Cf. A230654 (numbers n such that tau(n+1) - tau(n) = 4).
Cf. A228453 (numbers n such that tau(n+1) - tau(n) = 5).

Select[ Range[ 50000], DivisorSigma[0, # ] - 2 == DivisorSigma[0, # + 1] &]

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

Jaroslav Krizek, Apr 02 2021

nonn

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).

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

Robert Israel, Table of n, a(n) for n = 0..188

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

Magma
```
Ax:=func; [Ax(n): n in [0..50]];
```

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

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 *)

a(n) = my(m=1); while (numdiv(m+1) != numdiv(m) + n, m++); m; \\ Michel Marcus, Apr 03 2021

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

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

cp's OEIS Frontend

A230653 Numbers k such that tau(k+1) - tau(k) = 3, where tau(k) = the number of divisors of k (A000005).

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A230654 Numbers n such that tau(n+1) - tau(n) = 4, where tau(n) = the number of divisors of n (A000005).

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A228453 Numbers k such that tau(k+1) - tau(k) = 5, where tau(k) = the number of divisors of k (A000005).

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A227874 Numbers n such that tau(n+1) - tau(n) = -2, where tau(n) = the number of divisors of n (A000005).

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

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