cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Dec 17 2014

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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.

Programs

  • Mathematica
    pfQ[n_]:=Module[{f=FactorInteger[n]},f[[-1,1]]Harvey P. Dale, May 01 2015 *)
  • 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))Indranil Ghosh, Mar 24 2017

Formula

Other identities. For all n >= 1:
A252373(a(n)) = n. [A252373 works as an inverse or ranking function for this sequence.]

Extensions

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

Views

Author

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

Keywords

Comments

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).

Links

Crossrefs

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.

A138510 Smallest number b such that in base b the prime factors of the n-th semiprime (A001358) have equal lengths.

Original entry on oeis.org

1, 2, 1, 6, 8, 3, 3, 12, 1, 14, 12, 18, 2, 20, 14, 24, 1, 18, 4, 20, 30, 32, 4, 24, 38, 4, 42, 5, 44, 30, 4, 32, 48, 5, 54, 38, 5, 60, 5, 1, 62, 42, 44, 5, 68, 48, 72, 2, 30, 74, 32, 80, 54, 5, 84, 1, 60, 90, 62, 38, 3, 98, 68, 102, 6, 42, 104, 3, 72, 108, 44, 6, 110, 74, 3, 114, 48, 80
Offset: 1

Views

Author

Reinhard Zumkeller, Mar 21 2008

Keywords

Comments

a(n) = 1 iff A001358(n) is the square of a prime (A001248);
a(A174956(A138511(n))) = A084127(A174956(A138511(n))) + 1.
Equally, 1 if A001358(n) = p^2, otherwise, if A001358(n) = p*q (p, q primes, p < q), then a(n) = A252375(n) = the least r such that r^k <= p < q < r^(k+1), for some k >= 0. - Antti Karttunen, Dec 16 2014
a(A174956(A085721(n))) <= 2. - Reinhard Zumkeller, Dec 19 2014

Examples

			For n=31, the n-th semiprime is A001358(31) = 91 = 7*13;
     7 =  111_2 =  21_3 = 13_4
and 13 = 1101_2 = 111_3 = 31_4, so a(31) = 4. [corrected by _Jon E. Schoenfield_, Sep 23 2018]
.
Illustration of initial terms, n <= 25:
.   n | A001358(n) =  p * q |  b = a(n) | p and q in base b
. ----+---------------------+-----------+-------------------
.   1 |       4       2   2 |      1    |     [1]        [1]
.   2 |       6       2   3 |      2    |   [1,0]      [1,1]
.   3 |       9       3   3 |      1    | [1,1,1]    [1,1,1]
.   4 |  **  10       2   5 |      6    |     [2]        [5]
.   5 |  **  14       2   7 |      8    |     [2]        [7]
.   6 |      15       3   5 |      3    |   [1,0]      [1,2]
.   7 |      21       3   7 |      3    |   [1,0]      [2,1]
.   8 |  **  22       2  11 |     12    |     [2]       [11]
.   9 |      25       5   5 |      1    |   [1]^5      [1]^5
.  10 |  **  26       2  13 |     14    |     [2]       [13]
.  11 |  **  33       3  11 |     12    |     [3]       [11]
.  12 |  **  34       2  17 |     18    |     [2]       [17]
.  13 |      35       5   7 |      2    | [1,0,1]    [1,1,1]
.  14 |  **  38       2  19 |     20    |     [2]       [19]
.  15 |  **  39       3  13 |     14    |     [3]       [13]
.  16 |  **  46       2  23 |     24    |     [2]       [23]
.  17 |      49       7   7 |      1    |   [1]^7      [1]^7
.  18 |  **  51       3  17 |     18    |     [3]       [17]
.  19 |      55       5  11 |      4    |   [1,1]      [2,3]
.  20 |  **  57       3  19 |     20    |     [3]       [19]
.  21 |  **  58       2  29 |     30    |     [2]       [29]
.  22 |  **  62       2  31 |     32    |     [2]       [31]
.  23 |      65       5  13 |      4    |   [1,1]      [3,1]
.  24 |  **  69       3  23 |     24    |     [3]       [23]
.  25 |  **  74       2  37 |     38    |     [2]       [37]
where p = A084126(n) and q = A084127(n),
semiprimes marked with ** indicate terms of A138511, i.e. b = q + 1.

Links

Crossrefs

Cf. A078972, A085721.
Cf. A138511, A251728, A252375, A084127, A020639, A162319, A174956, A001358.

Programs

  • Haskell
    import Data.List (genericIndex, unfoldr); import Data.Tuple (swap)
import Data.Maybe (mapMaybe)
a138510 n = genericIndex a138510_list (n - 1)
a138510_list = mapMaybe f [1..] where
  f x | a010051' q == 0 = Nothing
      | q == p          = Just 1
      | otherwise       = Just $
        head [b | b <- [2..], length (d b p) == length (d b q)]
      where q = div x p; p = a020639 x
  d b = unfoldr (\z -> if z == 0 then Nothing else Just $ swap $ divMod z b)
-- Reinhard Zumkeller, Dec 16 2014
  • Scheme
    (define (A138510 n) (A251725 (A001358 n))) ;; Antti Karttunen, Dec 16 2014

Formula

a(n) = A251725(A001358(n)). - Antti Karttunen, Dec 16 2014

Extensions

Wrong comment corrected by Reinhard Zumkeller, Dec 16 2014

A252375 a(n) = smallest r 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).

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 2, 2, 2, 8, 3, 2, 2, 2, 2, 6, 3, 12, 2, 2, 2, 14, 2, 8, 2, 6, 2, 2, 12, 18, 2, 2, 2, 20, 14, 6, 2, 8, 2, 12, 3, 24, 2, 2, 2, 6, 18, 14, 2, 2, 4, 8, 20, 30, 2, 6, 2, 32, 3, 2, 4, 12, 2, 18, 24, 8, 2, 2, 2, 38, 3, 20, 4, 14, 2, 6, 2, 42, 2, 8, 5, 44, 30, 12, 2, 6, 4, 24, 32
Offset: 1

Views

Author

Antti Karttunen, Dec 17 2014

Keywords

Links

Crossrefs

A252374 gives the corresponding exponents.
Cf. A251726 (those n for which a(n) <= A006530(n)).
Cf. A251727 (those n > 1 for which a(n) = A006530(n)+1).
Cf. A006530, A020639, A066048, A138510, A251725.

Programs

  • Scheme
    (define (A252375 n) (let ((spf (A020639 n)) (gpf (A006530 n))) (let outerloop ((r 2)) (let innerloop ((rx 1)) (cond ((and (<= rx spf) (< gpf (* r rx))) r) ((<= rx spf) (innerloop (* r rx))) (else (outerloop (+ 1 r))))))))
(define (A252375 n) (let ((x (A251725 n))) (if (= 1 x) 2 x))) ;; Alternatively, using the implementation of A251725.

Formula

If A251725(n) = 1, a(n) = 2, otherwise a(n) = A251725(n).
Other identities. For all n >= 1:
a(n) = a(A066048(n)). [The result depends only on the smallest and the largest prime factor of n.]
Showing 1-4 of 4 results.

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