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

%I A341212 #25 Jul 27 2024 23:17:37
%S A341212 154379,1075198,4211518,4700758,4745227,5954379,6036043,6330235,
%T A341212 6485998,6524878,6851227,7846798,8536027,8556358,11718598,12100027,
%U A341212 12126838,13584838,14869379,15320587,16934998,17074379,18154379,18904027,19013129,19774379,19779995
%N A341212 Numbers m such that m, m - 1, m - 2, m - 3 and m - 4 have k, 2k, 3k, 4k and 5k divisors respectively.
%C A341212 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).
%C A341212 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, ...
%C A341212 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.
%H A341212 Robert G. Wilson v, <a href="/A341212/b341212.txt">Table of n, a(n) for n = 1..2590</a>
%e A341212 tau(154375) = 20, tau(154376) = 16, tau(154377) = 12, tau(154378) = 8, tau(154379) = 4.
%t A341212 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 *)
%o A341212 (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)]
%o A341212 (Python)
%o A341212 def tau(n): # A000005
%o A341212     d, t = 1, 0
%o A341212     while d*d < n:
%o A341212         if n%d == 0:
%o A341212             t = t+2
%o A341212         d = d+1
%o A341212     if d*d == n:
%o A341212         t = t+1
%o A341212     return t
%o A341212 n, a = 1, 2
%o A341212 while n <= 27:
%o A341212     nn, t1 = 1, tau(a)
%o A341212     while nn < 5 and tau(a-nn) == (nn+1)*t1:
%o A341212         nn = nn+1
%o A341212     if nn == 5:
%o A341212         print(n,a)
%o A341212         n = n+1
%o A341212     a = a+1 # _A.H.M. Smeets_, Feb 07 2021
%o A341212 (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
%Y A341212 Cf. A000005, A340158, A340159, A341213, A341214.
%K A341212 nonn
%O A341212 1,1
%A A341212 _Jaroslav Krizek_, Feb 07 2021