A341212 Numbers m such that m, m - 1, m - 2, m - 3 and m - 4 have k, 2k, 3k, 4k and 5k divisors respectively.

154379, 1075198, 4211518, 4700758, 4745227, 5954379, 6036043, 6330235, 6485998, 6524878, 6851227, 7846798, 8536027, 8556358, 11718598, 12100027, 12126838, 13584838, 14869379, 15320587, 16934998, 17074379, 18154379, 18904027, 19013129, 19774379, 19779995

Offset: 1

Author

Jaroslav Krizek, Feb 07 2021

Keywords

• nonn

Comments

Numbers m such that tau(m) = tau(m - 1)/2 = tau(m - 2)/3 = tau(m - 3)/4 = tau(m - 4)/5, where tau(k) = the number of divisors of k (A000005).

Corresponding values of numbers k: 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, ...

First prime term 55414379 (= A341214(5)) of this sequence is the smallest prime p such that p, p - 1, p - 2, p - 3 and p - 4 have 2, 4, 6, 8 and 10 divisors respectively.

Examples

			tau(154375) = 20, tau(154376) = 16, tau(154377) = 12, tau(154378) = 8, tau(154379) = 4.
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..2590

Crossrefs

Cf. A000005, A340158, A340159, A341213, A341214.

Programs

Magma

[m: m in [5..10^6] | #Divisors(m - 1) eq 2*#Divisors(m) and #Divisors(m - 2) eq 3*#Divisors(m) and #Divisors(m - 3) eq 4*#Divisors(m) and #Divisors(m - 4) eq 5*#Divisors(m)]

Mathematica

seq[max_, n_] := Module[{d = DivisorSigma[0, Range[n]], s = {}}, Do[If[Length @ Union[d/Range[n, 1, -1]] == 1, AppendTo[s, k - 1]]; d = Join[Rest@d, {DivisorSigma[0, k]}], {k, n + 1, max}]; s]; seq[5*10^6, 5] (* Amiram Eldar, Feb 08 2021 *)

PARI

isok(m) = if (m>5, my(nb=numdiv(m)); (numdiv(m-1) == 2*nb) && (numdiv(m-2) == 3*nb) && (numdiv(m-3) == 4*nb) && (numdiv(m-4) == 5*nb)); \\ Michel Marcus, Apr 01 2021

Python

def tau(n): # A000005
    d, t = 1, 0
    while d*d < n:
        if n%d == 0:
            t = t+2
        d = d+1
    if d*d == n:
        t = t+1
    return t
n, a = 1, 2
while n <= 27:
    nn, t1 = 1, tau(a)
    while nn < 5 and tau(a-nn) == (nn+1)*t1:
        nn = nn+1
    if nn == 5:
        print(n,a)
        n = n+1
    a = a+1 # A.H.M. Smeets, Feb 07 2021