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 1-6 of 6 results.

A220270 Number of biased numbers (A101550) less than 10^n.

Original entry on oeis.org

2, 43, 537, 5866, 61466, 630509, 6410996, 64836667, 653704646, 6578501544, 66115091613, 663868597795, 6661437722643, 66808391053328, 669760217536267, 6712237657325964, 67251395887426191
Offset: 1

Views

Author

Arkadiusz Wesolowski, Dec 11 2012

Keywords

Links

Crossrefs

Cf. A101550, A002162.

Programs

  • Mathematica
    Accumulate@Table[Length@Select[Range[10^(n - 1), 10^n - 1], FactorInteger[#][[-1, 1]] > 2*Sqrt[#] &], {n, 5}] (* or *)
lst = {}; s = -2; Do[Do[If[PrimeQ[i] || FactorInteger[i][[-1, 1]] > 2*Sqrt[i], s++], {i, 10^(n - 1), 10^n - 1}]; AppendTo[lst, s], {n, 5}]; lst

Formula

a(n) ~ 10^n*log(2) as n -> infinity.

Extensions

a(9)-a(11) from Donovan Johnson, Dec 12 2012
a(12)-a(17) from Hiroaki Yamanouchi, Aug 31 2014

A319802 Even numbers without middle divisors.

Original entry on oeis.org

10, 14, 22, 26, 34, 38, 44, 46, 52, 58, 62, 68, 74, 76, 78, 82, 86, 92, 94, 102, 106, 114, 116, 118, 122, 124, 134, 136, 138, 142, 146, 148, 152, 158, 164, 166, 172, 174, 178, 184, 186, 188, 194, 202, 206, 212, 214, 218, 222, 226, 230, 232, 236, 244, 246, 248, 250, 254, 258, 262, 268, 274, 278, 282, 284
Offset: 1

Views

Author

Omar E. Pol, Sep 28 2018

Keywords

Comments

Even numbers k such that the symmetric representation of sigma(k) has an even number of parts.
For the definition of middle divisors, see A067742.
For more information about the symmetric representation of sigma(k) see A237593.
Let p be a prime > 5. Then a(n) is a number of the form m*p where m is an even number < sqrt(p). - David A. Corneth, Sep 28 2018
First differs from A244894 at a(51) = 230. - R. J. Mathar, Oct 04 2018
Is this twice A101550? - Omar E. Pol, Oct 04 2018
This sequence is not twice A101550: first differs at a(57) = 250 != 254 = 2*A101550(57). - Michael S. Branicky, Oct 14 2021

Examples

			10 is in the sequence because it's an even number and the symmetric representation of sigma(10) = 18 has an even number of parts as shown below:
.
.     _ _ _ _ _ _ 9
.    |_ _ _ _ _  |
.              | |_
.              |_ _|_
.                  | |_ _ 9
.                  |_ _  |
.                      | |
.                      | |
.                      | |
.                      | |
.                      |_|
.

Links

Crossrefs

Intersection of A005843 and A071561.
Cf. A244894 (a subsequence).
Cf. A000203, A067742, A071090, A071562, A236104, A237270, A237271, A237593, A239932, A239934, A240542, A245092, A280919, A281007, A296508, A299761, A299777, A303297, A319529, A319796, A319801.

Programs

  • Python
    from sympy import divisors
def ok(n):
    if n < 2 or n%2 == 1: return False
    return not any(n//2 <= d*d < 2*n for d in divisors(n, generator=True))
print(list(filter(ok, range(285)))) # Michael S. Branicky, Oct 14 2021

A101549 Composite lopsided numbers: composite numbers n such that the largest prime factor > 2 sqrt(n).

Original entry on oeis.org

22, 26, 34, 38, 39, 46, 51, 57, 58, 62, 68, 69, 74, 76, 82, 86, 87, 92, 93, 94, 106, 111, 115, 116, 118, 122, 123, 124, 129, 134, 141, 142, 145, 146, 148, 155, 158, 159, 164, 166, 172, 174, 177, 178, 183, 185, 186, 188, 194, 201, 202, 203, 205, 206, 212, 213
Offset: 1

Views

Author

T. D. Noe, Dec 06 2004

Keywords

Comments

All primes > 3 are also lopsided. See A101550 for all lopsided numbers.

Links

Crossrefs

Cf. A063763 (composite n such that the largest prime factor > sqrt(n)), A064052 (n such that the largest prime factor > sqrt(n)).

Programs

  • Mathematica
    Select[Range[2, 300], !PrimeQ[ # ]&&FactorInteger[ # ][[ -1, 1]]>2Sqrt[ # ]&]

A320048 One half of composite numbers k with the property that the symmetric representation of sigma(k) has two parts.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 22, 23, 26, 29, 31, 34, 37, 38, 39, 41, 43, 46, 47, 51, 53, 57, 58, 59, 61, 62, 67, 68, 69, 71, 73, 74, 76, 79, 82, 83, 86, 87, 89, 92, 93, 94, 97, 101, 103, 106, 107, 109, 111, 113, 116, 118, 122, 123, 124, 127, 129, 131, 134, 137, 139, 141, 142, 146, 148, 149, 151, 157, 158, 159, 163, 164
Offset: 1

Views

Author

Omar E. Pol, Oct 04 2018

Keywords

Comments

Also, even numbers of A239929 divided by two.
First differs from A101550 at a(51). - R. J. Mathar, Oct 04 2018

Examples

			5 is in the sequence because 10 is a composite number, and the symmetric representation of sigma(10) = 18 has two parts (as shown below), and 10/2 = 5.
.
.     _ _ _ _ _ _ 9
.    |_ _ _ _ _  |
.              | |_
.              |_ _|_
.                  | |_ _ 9
.                  |_ _  |
.                      | |
.                      | |
.                      | |
.                      | |
.                      |_|
.

Crossrefs

Cf. A101550, A237271 (number of parts), A237270, A237593, A238443, A238524, A239929 (two parts), A239660, A239929, A239932, A239934, A240062 (k parts), A244894, A245092, A262626, A280107 (four parts).

Formula

a(n) = A244894(n)/2.

A374954 Positive integers k for which sqrt(k) < sqrt(p_1) + ... + sqrt(p_r), where p_1*...*p_r is the prime factorization of k.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 22, 24, 28, 32, 36, 40, 48, 64
Offset: 1

Views

Author

Felix Huber, Jul 29 2024

Keywords

Comments

This sequence is finite. Proof: First, let's assume that p_1 = ... = p_r = p, i.e. k = p^r. Then sqrt(p^r) < r*sqrt(p) or p < r^(2/(r-1)) respectively must apply. This inequality is satisfied for p = 2 and 2 <= r <= 6 as well as for p = 3 and r = 2. k can therefore contain at most r = 6 prime factors and is not a prime. By examining the individual ways for the highest value of k as a function of r, we find k = 2*2*2*2*2*2 = 64 for r = 6, k = 2*2*2*2*3 = 48 for r = 5, 2*2*2*5 = 40 for r = 4, 2*2*7 = 28 for r = 3 and 2*11 = 22 for r = 2. Therefore, this sequence is finite and its terms lie between 4 and 64.

Examples

			24 = 2*2*2*3 is in the sequence, because sqrt(24) < sqrt(2) + sqrt(2) + sqrt(2) + sqrt(3).

Crossrefs

Cf. A001414, A002808, A046343, A063538, A063539, A063762, A063763, A101550.

Programs

  • Maple
    A374954:=proc(k)
   local i,r,s,L;
   if not isprime(k) then
      L:=ifactors(k)[2];
      r:=numelems(L);
      s:=0;
      for i to r do
         s:=s+sqrt(L[i,1])*L[i,2]
      od;
      s:=evalf(s^2);
      if kA374954(k),k=4..64);

A348471 One half of the even numbers without middle divisors.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 22, 23, 26, 29, 31, 34, 37, 38, 39, 41, 43, 46, 47, 51, 53, 57, 58, 59, 61, 62, 67, 68, 69, 71, 73, 74, 76, 79, 82, 83, 86, 87, 89, 92, 93, 94, 97, 101, 103, 106, 107, 109, 111, 113, 115, 116, 118, 122, 123, 124, 125, 127, 129, 131, 134, 137, 139, 141
Offset: 1

Views

Author

Omar E. Pol, Oct 19 2021

Keywords

Comments

One half of the even numbers k such that the symmetric representation of sigma(k) has an even number of parts.

Crossrefs

First differs from A101550 at a(57).
A subsequence is A320048 from which first differs at a(51).
Cf. A071561, A237270, A237271, A237593, A244894, A319802.

Formula

a(n) = A319802(n)/2.
Showing 1-6 of 6 results.

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