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A251726
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Numbers n > 1 for which gpf(n) < lpf(n)^2, where lpf and gpf (least and greatest prime factor of n) are given by A020639(n) and A006530(n).
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17
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2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 21, 23, 24, 25, 27, 29, 31, 32, 35, 36, 37, 41, 43, 45, 47, 48, 49, 53, 54, 55, 59, 61, 63, 64, 65, 67, 71, 72, 73, 75, 77, 79, 81, 83, 85, 89, 91, 95, 96, 97, 101, 103, 105, 107, 108, 109, 113, 115, 119, 121, 125, 127, 128, 131, 133, 135, 137, 139, 143, 144
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OFFSET
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1,1
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COMMENTS
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Numbers n > 1 for which there exists r <= gpf(n) such that r^k <= lpf(n) and gpf(n) < r^(k+1) for some k >= 0, where lpf and gpf (least and greatest prime factor of n) are given by A020639(n) and A006530(n) (the original, equivalent definition of the sequence).
These are numbers n all of whose prime factors "fit between" two consecutive powers of some positive integer which itself is <= the largest prime factor of n.
Conjecture: If any n is in the sequence, then so is A003961(n).
Note: if Legendre's or Brocard's conjecture is true, then the above conjecture is true as well. See my comments at A251728. - Antti Karttunen, Jan 01 2015
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LINKS
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FORMULA
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Other identities. For all n >= 1:
A252373(a(n)) = n. [A252373 works as an inverse or ranking function for this sequence.]
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EXAMPLE
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For 35 = 5*7, 7 is less than 5^2, thus 35 is included.
For 90 = 2*3*3*5, 5 is not less than 2^2, thus 90 is NOT included.
For 105 = 3*5*7, 7 is less than 3^2, thus 105 is included.
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MATHEMATICA
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pfQ[n_]:=Module[{f=FactorInteger[n]}, f[[-1, 1]]<f[[1, 1]]^2]; Select[ Range[ 200], pfQ] (* Harvey P. Dale, May 01 2015 *)
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PROG
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(PARI) for(n=2, 150, if(vecmax(factor(n)[, 1]) < vecmin(factor(n)[, 1])^2, print1(n, ", "))) \\ Indranil Ghosh, Mar 24 2017
(Python)
from sympy import primefactors
print([n for n in range(2, 150) if max(primefactors(n))<min(primefactors(n))**2]) # Indranil Ghosh, Mar 24 2017
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CROSSREFS
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Gives the positions of zeros in A252459 (after its initial zero), cf. also A284261.
Cf. A252370 (gives the difference between the prime indices of gpf and lpf for each a(n)).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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A new simpler definition found Jan 01 2015 and the original definition moved to the Comments section.
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STATUS
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approved
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