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

1,1

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.

Links

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

Example

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

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 *)

Prog

(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)]
(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
(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

Crossrefs

Cf. A000005, A340158, A340159, A341213, A314214.

Sequence in context: A339528 A234553 A073086 * A255040 A255033 A256951

Adjacent sequences: A341209 A341210 A341211 * A341213 A341214 A341215

Keyword

nonn

Author

Jaroslav Krizek, Feb 07 2021

Status

approved