A252758 -id:A252758 - cp's OEIS frontend

A251726 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).

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

Offset: 1

Antti Karttunen, Dec 17 2014

nonn

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).
Numbers n > 1 such that A252375(n) < 1 + A006530(n). Equally, one can substitute A251725 for A252375.
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

			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.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Ratio A251726(n)/A251727(n) (plotted with OEIS-server's Plot2-utility)

Complement: A251727. Subsequences: A251728, A000961 (after 1).
Characteristic function: A252372. Inverse function: A252373.
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)).
Sequence gives all n > 1 for which A284252(n) (equally: A284254) is 1, and A284256(n) (equally A284258) is 0, and also n > 1 such that A284260(n) = A006530(n).
Cf. A003961, A006530, A020639, A252375, A251725, A253784.
Related permutations: A252757-A252758.

pfQ[n_]:=Module[{f=FactorInteger[n]},f[[-1,1]]Harvey P. Dale, May 01 2015 *)

for(n=2, 150, if(vecmax(factor(n)[,1]) < vecmin(factor(n)[,1])^2, print1(n,", "))) \\ Indranil Ghosh, Mar 24 2017

from sympy import primefactors
print([n for n in range(2, 150) if max(primefactors(n))Indranil Ghosh, Mar 24 2017

Other identities. For all n >= 1:

A252373(a(n)) = n. [A252373 works as an inverse or ranking function for this sequence.]

A new simpler definition found Jan 01 2015 and the original definition moved to the Comments section.

A251727 Numbers n > 1 for which gpf(n) > spf(n)^2, where spf and gpf (smallest and greatest prime factor of n) are given by A020639(n) and A006530(n).

Original entry on oeis.org

10, 14, 20, 22, 26, 28, 30, 33, 34, 38, 39, 40, 42, 44, 46, 50, 51, 52, 56, 57, 58, 60, 62, 66, 68, 69, 70, 74, 76, 78, 80, 82, 84, 86, 87, 88, 90, 92, 93, 94, 98, 99, 100, 102, 104, 106, 110, 111, 112, 114, 116, 117, 118, 120, 122, 123, 124, 126, 129, 130, 132, 134, 136, 138, 140, 141, 142, 145, 146, 148, 150, 152

Offset: 1

Antti Karttunen, Dec 17 2014. A new simpler definition found Jan 01 2015 and the original definition moved to the Comments section

nonn

Numbers n > 1 for which the smallest r such that r^k <= spf(n) and gpf(n) < r^(k+1) [for some k >= 0] is gpf(n)+1. Here spf and gpf (smallest and greatest prime factor of n) are given by A020639(n) and A006530(n). (The original, equivalent definition of the sequence).
Numbers n > 1 such that A252375(n) = 1 + A006530(n). Equally, one can substitute A251725 for A252375.
Numbers n > 1 for which there doesn't exist any r <= gpf(n) such that r^k <= spf(n) and gpf(n) < r^(k+1), for some k >= 0, where spf and gpf (smallest and greatest prime factor of n) are given by A020639(n) and A006530(n).

Antti Karttunen, Table of n, a(n) for n = 1..10000

Complement: A251726. Subsequence: A138511.
Gives the positions of zeros in A252374 following its initial term.
Cf. A252371 (difference between the prime indices of gpf and spf of each a(n)).
Cf. A006530, A020639, A252375, A251725.
Related permutations: A252757-A252758.

A252757 Permutation of natural numbers: a(1)=1, and for n>1, if n is k-th number whose largest prime factor is less than the square of its smallest prime factor [i.e., n = A251726(k)], a(n) = 2a(k), otherwise, when n = A251727(k), a(n) = 1 + 2a(k).

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 128, 256, 3, 512, 6, 1024, 5, 12, 2048, 10, 24, 4096, 9, 20, 17, 48, 8192, 18, 33, 40, 65, 34, 129, 96, 16384, 257, 513, 36, 66, 80, 7, 1025, 13, 130, 2049, 68, 11, 258, 25, 192, 32768, 514, 4097, 21, 49, 1026, 72, 132, 8193, 19, 41, 160, 35, 14, 97, 2050, 26, 260, 16385, 4098, 37, 67, 81, 136, 22

Offset: 1

Antti Karttunen, Jan 02 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A252758.
Similar permutations: A243287, A135141, A237427.
Cf. A252372, A252373, A251726, A251727.

a(1)=1, and for n>1: if A252372(n) = 1 [i.e. the largest prime factor of n is less than the square of its smallest prime factor], a(n) = 2*a(A252373(k)), otherwise, a(n) = 1 + 2*a(n-A252373(n)-1).

cp's OEIS Frontend

A251726 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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A251727 Numbers n > 1 for which gpf(n) > spf(n)^2, where spf and gpf (smallest and greatest prime factor of n) are given by A020639(n) and A006530(n).

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A252757 Permutation of natural numbers: a(1)=1, and for n>1, if n is k-th number whose largest prime factor is less than the square of its smallest prime factor [i.e., n = A251726(k)], a(n) = 2a(k), otherwise, when n = A251727(k), a(n) = 1 + 2a(k).

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cp's OEIS Frontend

A251726 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).

Views

Author

Keywords

Comments

Examples

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Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A251727 Numbers n > 1 for which gpf(n) > spf(n)^2, where spf and gpf (smallest and greatest prime factor of n) are given by A020639(n) and A006530(n).

Views

Author

Keywords

Comments

Links

Crossrefs

A252757 Permutation of natural numbers: a(1)=1, and for n>1, if n is k-th number whose largest prime factor is less than the square of its smallest prime factor [i.e., n = A251726(k)], a(n) = 2*a(k), otherwise, when n = A251727(k), a(n) = 1 + 2*a(k).

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Author

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Formula

A252757 Permutation of natural numbers: a(1)=1, and for n>1, if n is k-th number whose largest prime factor is less than the square of its smallest prime factor [i.e., n = A251726(k)], a(n) = 2a(k), otherwise, when n = A251727(k), a(n) = 1 + 2a(k).