A343018 -id:A343018 - cp's OEIS frontend

A343020 a(n) is the smallest prime p such that tau(p+1) = 2^n.

2, 5, 23, 167, 839, 7559, 128519, 1081079, 20540519, 397837439, 8031343319, 188972783999, 3212537327999, 125568306863999, 2888071057871999, 190487121512687999, 4381203794791823999, 215961289494494543999, 13283916764437951631999, 540119185025730854543999, 26465840066260811872655999, 1356699703068812438127791999

Offset: 1

Jaroslav Krizek, Apr 02 2021

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: 5, 7, 13, 37, 61, 73, 157, 193, 277, 313, 397, 421, ...
n = 3: 23, 29, 41, 53, 101, 103, 109, 113, 127, 137, 151, ...
n = 4: 167, 263, 269, 311, 383, 389, 439, 461, 509, 569, ...
n = 5: 839, 1319, 1511, 1559, 1847, 1889, 2039, 2309, 2687, ...
Conjecture: a(n) is also the smallest number m such that tau(m+1) = tau(m)^n.

			a(4) = 167 because 167 is the smallest prime p such that tau(p+1) = 16 = 2^4.

David A. Corneth, Table of n, a(n) for n = 1..31

Cf. A037992, A080371, A080372, A340799, A343018, A343019.

Magma
```
Ax:=func; [Ax(n): n in [1..7]]
```

Do[p = 1; While[DivisorSigma[0, Prime[p] + 1] != 2^n, p++]; Print[n, " ", Prime[p]], {n, 1, 9}] (* Vaclav Kotesovec, Apr 03 2021 *)

a(n) = my(t=2^n); forprime(p=2, oo, if(numdiv(p+1)==t, return(p))); \\ Jinyuan Wang, Apr 02 2021

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 True
    sizeok=False
    cnt=True
    while cnt:
        if i>=len(primes):
            primes.append(nextprime(primes[-1]))
        if jBert Dobbelaere, Apr 11 2021

a(11) from Jinyuan Wang, Apr 02 2021

More terms from David A. Corneth, Apr 09 2021

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

Jaroslav Krizek, Apr 02 2021

nonn

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, ...

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

Robert Israel, Table of n, a(n) for n = 0..174

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

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

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

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 *)

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 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

cp's OEIS Frontend

A343020 a(n) is the smallest prime p such that tau(p+1) = 2^n.

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