Michael Engling - cp's OEIS frontend

User: Michael Engling

Michael Engling has authored 3 sequences.

A193280 Triangle read by rows: row n contains, in increasing order, all the distinct sums of distinct proper divisors of n.

0, 1, 1, 1, 2, 3, 1, 1, 2, 3, 4, 5, 6, 1, 1, 2, 3, 4, 5, 6, 7, 1, 3, 4, 1, 2, 3, 5, 6, 7, 8, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 1, 2, 3, 7, 8, 9, 10, 1, 3, 4, 5, 6, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15

Offset: 1

Michael Engling, Jul 20 2011

Row n > 1 contains A193279(n) terms. In row n the first term is 1 and the last term is sigma(n) - n (= A000203(n) - n). Row 1 contains 0 because 1 has no proper divisors.

			Row 10 is 1,2,3,5,6,7,8 the possible sums obtained from the proper divisors 1, 2, and 5 of 10.
Triangle starts:
  0;
  1;
  1;
  1,2,3;
  1;
  1,2,3,4,5,6;
  1;
  1,2,3,4,5,6,7;
  1,3,4;
  1,2,3,5,6,7,8;
  1;
  1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16;

Nathaniel Johnston, Rows 1..150, flattened

Cf. A119347, A119348, A193279.

with(linalg): print(0); for n from 2 to 12 do dl:=convert(numtheory[divisors](n) minus {n}, list): t:=nops(dl): print(op({seq(innerprod(dl, convert(2^t+i, base, 2)[1..t]), i=1..2^t-1)})): od: # Nathaniel Johnston, Jul 23 2011

A193279 Number of distinct sums of distinct proper divisors of n.

Original entry on oeis.org

0, 1, 1, 3, 1, 6, 1, 7, 3, 7, 1, 16, 1, 7, 7, 15, 1, 21, 1, 22, 7, 7, 1, 36, 3, 7, 7, 28, 1, 42, 1, 31, 7, 7, 7, 55, 1, 7, 7, 50, 1, 54, 1, 31, 27, 7, 1, 76, 3, 31, 7, 31, 1, 66, 7, 64, 7, 7, 1, 108, 1, 7, 29, 63, 7, 78, 1, 31, 7, 72, 1, 123, 1, 7, 31, 31

Offset: 1

Michael Engling, Jul 20 2011

nonn

a(n)=1 if and only if n is prime.
a(n)=n-1 if n is a power of 2.
a(n)=n if n is an even perfect number (is the converse true?)
Note: the count excludes an empty subset of proper divisors that would give 0 as a sum. - Antti Karttunen, Mar 07 2018

Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 10000 terms from Amiram Eldar)

Cf. A193280.

Cf. A119347 (allows also n to be included in the sums), A378447 (differences).

with(linalg): a:=proc(n) local dl,t: dl:=convert(numtheory[divisors](n) minus {n}, list): t:=nops(dl): return nops({seq(innerprod(dl, convert(2^t+i, base, 2)[1..t]), i=1..2^t-1)}): end: seq(a(n), n=1..76); # Nathaniel Johnston, Jul 23 2011

a[n_] := Module[{d = Most @ Divisors[n], x}, Count[CoefficientList[Product[1 + x^i, {i, d}], x], ?(# > 0 &)] - 1]; Array[a, 100] (* _Amiram Eldar, Jun 13 2020 *)

\\ Slow and naive:
A193279(n) = if(1==n,0,my(pds = (divisors(n)[1..(numdiv(n)-1)]), maxsum = vecsum(pds), sums = vector(maxsum), psetsiz = (2^length(pds))-1, k = 0, s); for(i=1,psetsiz,s = vecsum(choosebybits(pds,i)); if(!sums[s],k++;sums[s]++)); (k)); \\ Antti Karttunen, Mar 07 2018

A193279(n) = { my(p=1); fordiv(n, d, if(dAntti Karttunen, Nov 29 2024

A193279(n) = { my(c=[0]); fordiv(n,d, if(dA119347) - Antti Karttunen, Nov 29 2024

A166540 Number of ways to place 2 nonattacking kings on an n X n X n raumschach board.

Original entry on oeis.org

0, 0, 0, 193, 1548, 6714, 21280, 55395, 125748, 257908, 489024, 870885, 1473340, 2388078, 3732768, 5655559, 8339940, 12009960, 16935808, 23439753, 31902444, 42769570, 56558880, 73867563, 95379988, 121875804, 154238400, 193463725, 240669468, 297104598, 364159264

Offset: 0

Michael Engling, Oct 16 2009

We consider that kings "attack" any square that differs by at most one in any combination of the indices from its current space. A logical extension of sequence A061995.

Andrew Woods, Table of n, a(n) for n = 0..1000
Wikipedia, Three-dimensional chess
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A061995, A194605.

[0] cat [(n^6 -(3*n-2)^3)/2: n in [1..35]]; // G. C. Greubel, Apr 03 2019

LinearRecurrence[{7,-21,35,-35,21,-7,1}, {0, 0, 0, 193, 1548, 6714, 21280, 55395}, 35] (* G. C. Greubel, May 16 2016; a(7) appended by Georg Fischer, Apr 03 2019 *)

{a(n) = if(n==0,0, (n^6 -(3*n-2)^3)/2)}; \\ G. C. Greubel, Apr 03 2019

# Two non-attacking kings in n x n x n cubic "board" .
m=int(input('What\'s the biggest board to investigate? '))
for n in range (0, m+1):
    sum=0
    for x1 in range (1, n+1):
      for y1 in range (1, n+1):
        for z1 in range (1, n+1):
          for x2 in range (1, n+1):
            for y2 in range (1, n+1):
              for z2 in range (1, n+1):
                if abs(x1-x2)>1 or abs(y1-y2)>1 or abs(z1-z2)>1:
                  sum=sum+1
    sum=sum//2
    print(n, sum)

[0]+[(n^6 -(3*n-2)^3)/2 for n in (1..35)] # G. C. Greubel, Apr 03 2019

a(n) = (n^6 - (3*n-2)^3) / 2 = (n^6)/2 - (27*n^3)/2 + 27*n^2 - 18*n + 4 for n>0. - Andrew Woods, Aug 30 2011
G.f.: x^3*(193 +197*x -69*x^2 +35*x^3 +4*x^4)/(1-x)^7. - Colin Barker, Jan 09 2013
E.g.f.: (8 -8*x +4*x^2 +63*x^3 +65*x^4 +15*x^5 +x^6)*exp(x)/2 -4. - G. C. Greubel, Apr 03 2019

cp's OEIS Frontend

User: Michael Engling

A193280 Triangle read by rows: row n contains, in increasing order, all the distinct sums of distinct proper divisors of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

A193279 Number of distinct sums of distinct proper divisors of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

A166540 Number of ways to place 2 nonattacking kings on an n X n X n raumschach board.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

Sage

Formula