A343020 -id:A343020 - cp's OEIS frontend

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

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.

A350132 a(n) is the smallest number m such that tau(m-1) = tau(m+1) = tau(m)^n, where tau(m) is the number of divisors of m (A000005).

Original entry on oeis.org

34, 7, 41, 919, 18089, 446081, 27033161, 663929729, 74335064959, 6132592231039

Offset: 1

Jaroslav Krizek, Dec 15 2021

Triples of [tau(a(n) - 1), tau(a(n)), tau(a(n) + 1)] = [tau(a(n))^n, tau(a(n)), tau(a(n))^n]: [4, 4, 4], [4, 2, 4], [8, 2, 8], [16, 2, 16], [32, 2, 32], [64, 2, 64], [128, 2, 128], ...
Conjecture: a(n) is prime for all n >= 2, i.e., the sequence {a(n)} without the first term is the sequence of smallest primes p such that tau(p-1) = tau(p+1) = 2^n for n >= 2.
a(10) <= 6132592231039. - Jon E. Schoenfield, Jan 19 2022
From David A. Corneth, Jan 21 2022: (Start)
a(11) <= 864808145605249.
a(12) <= 246846832951283839.
a(13) <= 14552217960448488319. (End)

			34 is the 1st term of A169834, so a(1) = 34.

Cf. A000005, A169834, A343020, A343087.

Magma
```
Ax:=func; [Ax(n): n in [1..6]];
```

isok(m, n) = my(nd1=numdiv(m-1)); (nd1 == numdiv(m)^n) && (nd1 == numdiv(m+1));
a(n) = {my(m=2); while (!isok(m, n), m++); m;} \\ Michel Marcus, Dec 16 2021

a(8) from Jon E. Schoenfield and David A. Corneth, Dec 15 2021
a(9) from David A. Corneth and Martin Ehrenstein, Jan 14 2022
a(10) verified by Martin Ehrenstein, Jan 21 2022

A343087 a(n) is the smallest prime p such that tau(p-1) = 2^n.

Original entry on oeis.org

3, 7, 31, 211, 1321, 7561, 120121, 1580041, 24864841, 328648321, 7558911361, 162023621761, 5022732274561, 93163582512001, 4083134943888001, 151075992923856001, 5236072827921936001, 188391763176048432001, 8854412869274276304001, 469283882071536644112001, 29844457947060064452144001, 1917963226026370264485744001

Offset: 1

Jaroslav Krizek, Apr 04 2021 (following a suggestion of Vaclav Kotesovec)

nonn

tau(m) = the number of divisors of m (A000005).
Sequences of primes p such that tau(p-1) = 2^n for 2 <= n <= 5:
n = 2: 7, 11, 23, 47, 59, 83, 107, 167, 179, ... (A005385(k) for k >= 2).
n = 3: 31, 41, 43, 67, 71, 79, 89, 103, 131, 137, 139, 191, ...
n = 4: 211, 271, 281, 313, 331, 379, 409, 457, 463, 521, 547, ...
n = 5: 1321, 2281, 2311, 2377, 2689, 2731, 2857, 2971, 3001, ...
Conjecture: a(n) is also the smallest number m such that tau(m-1) = tau(m)^n.

			For n = 4; a(4) = 211 because 211 is the smallest prime p such that tau(p - 1) = 2^4; tau(210) = 16.

Bert Dobbelaere, Table of n, a(n) for n = 1..60

Cf. A000005, A000079, A037992, A343020.

Magma
```
Ax:=func; [Ax(n): n in [1..9]]
```

from sympy import isprime,nextprime
primes=[2]
def solve(v,k,i,j):
    global record,stack,primes
    if k==0:
        if isprime(v+1):
            record=v
        return
    while True:
        if i>=len(primes):
            primes.append(nextprime(primes[-1]))
        if jBert Dobbelaere, Apr 11 2021

a(11) from Vaclav Kotesovec, Apr 05 2021

More terms from Bert Dobbelaere, Apr 11 2021

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

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A350132 a(n) is the smallest number m such that tau(m-1) = tau(m+1) = tau(m)^n, where tau(m) is the number of divisors of m (A000005).

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A343087 a(n) is the smallest prime p such that tau(p-1) = 2^n.

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