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.

A326836 Heinz numbers of integer partitions whose maximum part divides their sum.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 25, 27, 29, 30, 31, 32, 36, 37, 40, 41, 43, 47, 48, 49, 53, 59, 61, 63, 64, 67, 70, 71, 73, 79, 81, 83, 84, 89, 97, 101, 103, 107, 108, 109, 112, 113, 121, 125, 127, 128, 131, 135, 137, 139, 144, 149, 150, 151
Offset: 1

Views

Author

Gus Wiseman, Jul 26 2019

Keywords

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers whose maximum prime index divides their sum of prime indices.
The enumeration of these partitions by sum is given by A067538.

Examples

			The sequence of terms together with their prime indices begins:
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   12: {1,1,2}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   30: {1,2,3}
   31: {11}
   32: {1,1,1,1,1}

Crossrefs

Cf. A018818, A047993, A056239, A061395, A067538, A112798, A316413.
Cf. A326837, A326838, A326839/A326840, A326841, A326843, A326850.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[200],Divisible[Total[primeMS[#]],Max[primeMS[#]]]&]

A326837 Heinz numbers of integer partitions whose length and maximum both divide their sum.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 30, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 84, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197
Offset: 1

Views

Author

Gus Wiseman, Jul 26 2019

Keywords

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The enumeration of these partitions by sum is given by A326843.

Examples

			The sequence of terms together with their prime indices begins:
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   30: {1,2,3}
   31: {11}
   32: {1,1,1,1,1}
   37: {12}

Links

Crossrefs

The non-constant case is A326838.
The strict case is A326851.
Cf. A001222, A047993, A056239, A061395, A067538, A112798, A316413, A326836, A326843, A326847, A326848.

Programs

  • Maple
    isA326837 := proc(n)
    psigsu := A056239(n) ;
    psigma := A061395(n) ;
    psigle := numtheory[bigomega](n) ;
    if modp(psigsu,psigma) = 0 and modp(psigsu,psigle) = 0 then
        true;
    else
        false;
    end if;
end proc:
n := 1:
for i from 2 to 3000 do
    if isA326837(i) then
        printf("%d %d\n",n,i);
        n := n+1 ;
    end if;
end do: # R. J. Mathar, Aug 09 2019
  • Mathematica
    Select[Range[2,100],With[{y=Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]},Divisible[Total[y],Max[y]]&&Divisible[Total[y],Length[y]]]&]

A340830 Number of strict integer partitions of n such that every part is a multiple of the number of parts.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 3, 1, 4, 1, 4, 1, 6, 1, 5, 2, 6, 1, 8, 1, 7, 4, 7, 1, 12, 1, 8, 6, 9, 1, 16, 1, 10, 9, 11, 1, 21, 1, 12, 13, 12, 1, 28, 1, 13, 17, 16, 1, 33, 1, 19, 22, 15, 1, 45, 1, 16, 28, 25, 1, 47, 1, 28, 34, 18
Offset: 1

Views

Author

Gus Wiseman, Feb 02 2021

Keywords

Examples

			The a(n) partitions for n = 1, 6, 10, 14, 18, 20, 24, 26, 30:
  1   6     10    14     18      20     24       26      30
      4,2   6,4   8,6    10,8    12,8   16,8     18,8    22,8
            8,2   10,4   12,6    14,6   18,6     20,6    24,6
                  12,2   14,4    16,4   20,4     22,4    26,4
                         16,2    18,2   22,2     24,2    28,2
                         9,6,3          14,10    14,12   16,14
                                        12,9,3   16,10   18,12
                                        15,6,3           20,10
                                                         15,9,6
                                                         18,9,3
                                                         21,6,3
                                                         15,12,3

Crossrefs

Note: A-numbers of Heinz-number sequences are in parentheses below.
The non-strict case is A143773 (A316428).
The case where length divides sum also is A340827.
The version for factorizations is A340851.
Factorization of this type are counted by A340853.
A018818 counts partitions into divisors (A326841).
A047993 counts balanced partitions (A106529).
A067538 counts partitions whose length/max divide sum (A316413/A326836).
A072233 counts partitions by sum and length, with strict case A008289.
A102627 counts strict partitions whose length divides sum.
A326850 counts strict partitions whose maximum part divides sum.
A326851 counts strict partitions with length and maximum dividing sum.
A340828 counts strict partitions with length divisible by maximum.
A340829 counts strict partitions with Heinz number divisible by sum.
Cf. A114638, A168659, A326641, A326843 (A326837), A326849, A326852 (A326838), A330950 (A324851), A340852.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&And@@IntegerQ/@(#/Length[#])&]],{n,30}]

Formula

a(n) = Sum_{d|n} A008289(n/d, d).

A326852 Number of non-constant integer partitions of n whose length and maximum both divide n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 16, 0, 1, 7, 11, 0, 30, 0, 40, 18, 1, 0, 201, 0, 1, 38, 124, 0, 426, 0, 211, 73, 1, 48, 1391, 0, 1, 131, 1741, 0, 1774, 0, 951, 2145, 1, 0, 8345, 0, 1853, 381, 2382, 0, 6718, 2761, 10633, 623, 1, 0, 68037
Offset: 0

Views

Author

Gus Wiseman, Jul 26 2019

Keywords

Comments

The Heinz numbers of these partitions are given by A326838.

Examples

			The a(6) = 1 through a(16) = 11 partitions (empty columns not shown):
  (321)  (4211)  (52111)  (633)     (7211111)  (53322)  (8332)
                          (642)                (53331)  (8422)
                          (651)                (54222)  (8431)
                          (4332)               (54321)  (8521)
                          (4422)               (54411)  (8611)
                          (4431)               (55221)  (42222211)
                          (6222)               (55311)  (43222111)
                          (6321)                        (43321111)
                          (6411)                        (44221111)
                          (322221)                      (44311111)
                          (332211)                      (82111111)
                          (333111)
                          (422211)
                          (432111)
                          (441111)
                          (621111)

Links

Crossrefs

The possibly constant case is A326843.
The strict case is A326851.
Cf. A018818, A047993, A067538, A326837, A326838, A326842, A326849.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],!SameQ@@#&&Divisible[n,Length[#]]&&Divisible[n,Max[#]]&]],{n,0,30}]
Showing 1-4 of 4 results.

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