Gerhard Palme - cp's OEIS frontend

User: Gerhard Palme

Gerhard Palme has authored 2 sequences.

A309355 Even numbers k such that k! is divisible by k*(k+1)/2.

8, 14, 20, 24, 26, 32, 34, 38, 44, 48, 50, 54, 56, 62, 64, 68, 74, 76, 80, 84, 86, 90, 92, 94, 98, 104, 110, 114, 116, 118, 120, 122, 124, 128, 132, 134, 140, 142, 144, 146, 152, 154, 158, 160, 164, 168, 170, 174, 176, 182, 184, 186, 188, 194, 200, 202, 204, 206

Offset: 1

Gerhard Palme, Jul 25 2019

nonn

Even terms in A060462.
And A071904 are the successors of a(n).
Even numbers that are not a prime - 1. That is, even numbers not in A006093. - Terry D. Grant, Oct 31 2020

			8! = 40320 is divisible by 8*9/2 = 36.
14! is divisible by 14*15/2.

J. D. E. Konhauser et al., Which Way Did The Bicycle Go?, Problem 98, pp. 29; 145-146, MAA Washington DC, 1996.
Die WURZEL - Zeitschrift für Mathematik, 53. Jahrgang, Juli 2019, S. 171, WURZEL-Aufgabe 2019-36 von Gerhard Dietel, Regensburg.

Cf. A060462, A071904.
Essentially the same as A186193.
Cf. A006093.

[k: k in [2..250]|IsEven(k) and Factorial(k) mod Binomial(k+1,2) eq 0]; // Marius A. Burtea, Jul 28 2019

Complement[Table[2 n, {n, 1, 103}], Table[EulerPhi[Prime[n]], {n, 1, 103}]] (* Terry D. Grant, Oct 31 2020 *)

forcomposite(c=4,10^3,if(c%2==1,print1(c-1,", "))); \\ Joerg Arndt, Jul 25 2019

from sympy import primepi
def A309355(n):
    if n == 1: return 8
    m, k = n, primepi(n) + n + (n>>1)
    while m != k:
        m, k = k, primepi(k) + n + (k>>1)
    return m-1 # Chai Wah Wu, Aug 02 2024

a(n) = A071904(n) - 1.

A152826 Numbers that are divisible by the product of the digit-sums of their neighbors.

Original entry on oeis.org

105, 672, 1200, 1530, 1560, 2145, 2907, 3060, 3432, 4704, 5814, 6006, 6120, 6240, 8721, 9570, 10710, 10752, 10920, 11154, 11628, 11700, 12240, 12441, 13260, 14535, 15015, 16302, 17442, 19656, 20163, 20280, 20832, 21420, 22620, 23256, 23400

Offset: 1

Gerhard Palme (GerhardPalme(AT)gmx.de), Dec 13 2008

All terms are multiples of 3. Up to 10^8 there are exactly 6865 odd and 25055 even terms. - Zak Seidov, May 11 2013
Note the linear patterns in my jpg file. - Zak Seidov, May 11 2013
Subsequences include 10^(2+6k)+5, 10^(3+16k)+530, 10^(3+6k)+560, 2*10^(3+6k)+145, 2*10^(3+144k)+907, etc. - Robert Israel, Jan 05 2016

Die WURZEL - Zeitschrift für Mathematik, 42. Jahrgang, Dezember 2008, Seite 287, WURZEL-Aufgabe sigma55 von Reiner Moewald, Germersheim.

Robert Israel, Table of n, a(n) for n = 1..1000
Zak Seidov, plot of a(n)/3/A007953 (a(n) - 1)/A007953 (a(n) + 1) for n=1..31920.

Cf. A007953.

class Program { static void Main(string[] args) { for (int i = 1; i < 1000000; i++) { int q1 = quersumme(i - 1); int q2 = quersumme(i + 1); if ((double)i % ((double)q1 * (double)q2) == 0) { Console.WriteLine(i); } } Console.ReadLine(); } static int quersumme(int num) { string n = num.ToString(); int retval = 0; int i = n.Length - 1; while (i >= 0) { retval += Int32.Parse(n.Substring(i, 1)); i--; } return retval; } }

ds:= n -> convert(convert(n,base,10),`+`):
filter:= n -> n mod (ds(n-1)*ds(n+1))=0:
select(filter, 3*[$1..10000]); # Robert Israel, Jan 05 2016

Select[Range[50000], Divisible[#,Total[IntegerDigits[#-1]] * Total[IntegerDigits[#+1]]]&] (* Harvey P. Dale, Dec 11 2010 *)

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User: Gerhard Palme

A309355 Even numbers k such that k! is divisible by k*(k+1)/2.

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