A341213
a(n) is the smallest number m such that numbers m, m - 1, m - 2, ..., m - n + 1 have k, 2*k, 3*k, ..., n*k divisors respectively.
Original entry on oeis.org
1, 7, 47, 1019, 154379, 59423129, 3100501318, 126544656838
Offset: 1
a(3) = 47 because 45, 46 and 47 have 6, 4, and 2 divisors respectively and there is no smaller number having this property.
Cf.
A341214 (similar sequence with primes).
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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, 1 # corrected by Martin Ehrenstein, Apr 14 2021
while n > 0:
nn, t1 = 1, tau(a)
while nn < n and tau(a-nn) == (nn+1)*t1:
nn = nn+1
if nn == n:
print(n,a)
n = n+1
a = a+1 # A.H.M. Smeets, Feb 07 2021
A341212
Numbers m such that m, m - 1, m - 2, m - 3 and m - 4 have k, 2k, 3k, 4k and 5k divisors respectively.
Original entry on oeis.org
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
tau(154375) = 20, tau(154376) = 16, tau(154377) = 12, tau(154378) = 8, tau(154379) = 4.
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[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)]
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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 *)
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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
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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
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