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.

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

A320537 Square array read by antidiagonals in which T(n,k) is the n-th even number j with the property that the symmetric representation of sigma(j) has k parts.

Original entry on oeis.org

2, 4, 10, 6, 14, 50, 8, 22, 70, 230, 12, 26, 98, 250, 1150, 16, 34, 110, 290, 1250, 5050, 18, 38, 130, 310, 1450, 5150, 22310, 20, 44, 154, 370, 1550, 5290, 23230, 106030, 24, 46, 170, 406, 1850, 5350, 23690, 106490, 510050, 28, 52, 182, 410, 2030, 5450, 24610, 107410, 513130, 2065450
Offset: 1

Views

Author

Omar E. Pol, Oct 15 2018

Keywords

Comments

This is a permutation of the positive even numbers (A299174).
The union of all odd-indexed columns gives A319796, the even numbers in A071562.
The union of all even-indexed columns gives A319802, the even numbers in A071561.

Examples

			From _Hartmut F. W. Hoft_, Oct 06 2021: (Start)
The 10x10 section of table T(n,k):
(Table with first 20 terms from _Omar E. Pol_)
------------------------------------------------------------------
n\k | 1   2   3    4    5     6     7      8       9       10  ...
------------------------------------------------------------------
  1 | 2   10  50   230  1150  5050  22310  106030  510050  2065450
  2 | 4   14  70   250  1250  5150  23230  106490  513130  2115950
  3 | 6   22  98   290  1450  5290  23690  107410  520150  2126050
  4 | 8   26  110  310  1550  5350  24610  110170  530150  2157850
  5 | 12  34  130  370  1850  5450  25070  112010  530450  2164070
  6 | 16  38  154  406  2030  5650  25250  112930  532450  2168150
  7 | 18  44  170  410  2050  5750  25750  114770  534290  2176550
  8 | 20  46  182  430  2150  6250  25990  115690  537050  2186650
  9 | 24  52  190  434  2170  6350  26450  116150  540350  2216950
  10| 28  58  238  470  2350  6550  26750  117070  544870  2219650
   ... (End)

Crossrefs

Row 1 is A320521.
Column 1 gives A174973 = A238443, without the 1.
Column 2 gives A244894.
Cf. A000203, A071561, A071562, A236104, A237270, A237271, A237593, A238443, A239663, A239665, A239929, A240062, A245092, A262626, A299174, A319796, A319802, A341969, A341970, A341971, A346969, A348171.

Programs

  • Mathematica
    (* function a341969 is defined in A341969 *)
sArray[b_, pMax_] := Module[{list=Table[{}, pMax], i, p}, For[i=2, i<=b, i+=2, p=Length[Select[SplitBy[a341969[i], #!=0&], #[[1]]!=0&]]; If[p<=pMax&&Length[list[[p]]]Hartmut F. W. Hoft, Oct 06 2021 *)

Extensions

Terms a(21) and beyond from Hartmut F. W. Hoft, Oct 06 2021

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.

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 1-4 of 4 results.

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