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A067128
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Ramanujan's largely composite numbers, defined to be numbers m such that d(m) >= d(k) for k = 1 to m-1.
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40
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1, 2, 3, 4, 6, 8, 10, 12, 18, 20, 24, 30, 36, 48, 60, 72, 84, 90, 96, 108, 120, 168, 180, 240, 336, 360, 420, 480, 504, 540, 600, 630, 660, 672, 720, 840, 1080, 1260, 1440, 1680, 2160, 2520, 3360, 3780, 3960, 4200, 4320, 4620, 4680, 5040, 7560, 9240
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OFFSET
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1,2
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COMMENTS
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This sequence is a subsequence of A034287; are they identical? They match for m up to 1500000.
Identical to A034287 for the 105834 terms less than 10^150.
Every subsequence of terms, having the fixed greatest prime divisor prime(k), k=1,2,..., is finite. For a proof see A273015. The list of these subsequences begins {2,4,8}, {3,6,12,18,24,36,48,72,96,108}, ... - Vladimir Shevelev, May 13 2016
By a result of Erdős (1944), a(n+1) <= 2*a(n): see Erdős link. - David A. Corneth, May 20, 2016
Out of the first 10000 terms of this sequence, 1766 are adjacent to a prime. - Dmitry Kamenetsky, Jul 02 2024
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LINKS
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EXAMPLE
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8 is a term as d(8) = 4 and d(k) <= 4 for k = 1,...,7.
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MAPLE
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isA067128 := proc(n)
local nd, k ;
nd := numtheory[tau](n) ;
for k from 1 to n-1 do
if numtheory[tau](k) > nd then
return false ;
end if;
end do:
true ;
end proc:
option remember;
if n = 1 then
1;
else
for a from procname(n-1)+1 do
if isA067128(a) then
return a;
end if;
end do:
end if;
end proc:
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MATHEMATICA
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For[n=1; max=0, True, n++, If[(d=DivisorSigma[0, n])>=max, Print[n]; max=d]]
NestList[Function[last,
NestWhile[# + 1 &, last + 1,
DivisorSigma[0, #] < DivisorSigma[0, last] &]], 1, 70] (* Steven Lu, Nov 28 2022 *)
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PROG
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(PARI) is(n) = my(nd=numdiv(n)); for(k=1, n-1, if(numdiv(k) > nd, return(0))); return(1) \\ Felix Fröhlich, May 22 2016
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CROSSREFS
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Number of divisors of a(n): A273353.
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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