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.

A326152 Number of integer partitions of n whose product of parts is 2 * n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 3, 2, 2, 0, 5, 0, 2, 3, 5, 0, 7, 0, 5, 3, 2, 0, 10, 2, 2, 5, 5, 0, 9, 0, 9, 3, 2, 3, 14, 0, 2, 3, 10, 0, 9, 0, 5, 9, 2, 0, 17, 2, 7, 3, 5, 0, 14, 3, 10, 3, 2, 0, 19, 0, 2, 9, 13, 3, 9, 0, 5, 3, 9, 0, 27, 0, 2, 9, 5, 3, 9, 0, 17, 10, 2, 0
Offset: 0

Views

Author

Gus Wiseman, Jun 09 2019

Keywords

Comments

Also the number of orderless factorizations of 2 * n into factors > 1 with sum at most n.
The Heinz numbers of these partitions are given by A326151.

Examples

			The a(8) = 3 through a(16) = 5 partitions (empty columns not shown) (A = 10):
  (44)    (63)    (541)   (831)      (74111)   (A311)      (841111)
  (422)   (3321)  (5221)  (6411)     (722111)  (651111)    (8221111)
  (2222)                  (62211)              (53211111)  (442111111)
                          (432111)                         (4222111111)
                          (3222111)                        (22222111111)

Crossrefs

Cf. A003963, A057567, A057568, A096276, A114324, A301987, A319005, A326149, A326151, A326156, A326157.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Times@@#==2*n&]],{n,0,30}]

A326151 Numbers whose product of prime indices is twice their sum of prime indices.

Original entry on oeis.org

49, 63, 65, 81, 150, 154, 190, 198, 364, 468, 580, 840, 952, 1080, 1224, 1480, 2128, 2288, 2736, 3440, 5152, 5280, 6624, 8480, 9408, 10816, 12096, 12992, 15552, 16704, 19520, 24960, 26752, 27776, 35712, 44800, 45440, 56576, 57600, 66304, 85248, 101120, 118272
Offset: 1

Views

Author

Gus Wiseman, Jun 09 2019

Keywords

Comments

The only squarefree terms are 65, 154, and 190. See A326157 for a proof.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose product of parts is twice their sum of parts. The enumeration of these partitions by sum is given by A326152.

Examples

			The sequence of terms together with their prime indices begins:
     49: {4,4}
     63: {2,2,4}
     65: {3,6}
     81: {2,2,2,2}
    150: {1,2,3,3}
    154: {1,4,5}
    190: {1,3,8}
    198: {1,2,2,5}
    364: {1,1,4,6}
    468: {1,1,2,2,6}
    580: {1,1,3,10}
    840: {1,1,1,2,3,4}
    952: {1,1,1,4,7}
   1080: {1,1,1,2,2,2,3}
   1224: {1,1,1,2,2,7}
   1480: {1,1,1,3,12}
   2128: {1,1,1,1,4,8}
   2288: {1,1,1,1,5,6}
   2736: {1,1,1,1,2,2,8}
   3440: {1,1,1,1,3,14}

Links

Crossrefs

Satisfies A003963(a(n)) = 2 * A056239(a(n)).
Cf. A112798, A301987, A325037, A325038, A325044, A326149, A326152, A326153/A326154, A326156, A326157.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Times@@primeMS[#]==2*Plus@@primeMS[#]&]
  • PARI
    is(k) = {my(f=factor(k)); for(i=1, #f~, f[i, 1]=primepi(f[i, 1])); factorback(f)==2*sum(i=1, #f~, f[i, 2]*f[i, 1]); } \\ Jinyuan Wang, Jun 27 2020

A379844 Squarefree numbers x such that the product of prime indices of x is a multiple of the sum of prime indices of x.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 43, 47, 53, 59, 61, 65, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 154, 157, 163, 165, 167, 173, 179, 181, 190, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241
Offset: 1

Views

Author

Gus Wiseman, Jan 19 2025

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The sum and product of prime indices are A056239 and A003963 respectively.

Crossrefs

Squarefree case of A326149.
For nonprime instead of squarefree we have A326150.
The non-prime case is A326158.
Partitions of this type are counted by A379733, see A379735.
The even case is A379845, counted by A380221.
A003963 multiplies together prime indices.
A005117 lists the squarefree numbers.
A056239 adds up prime indices.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- divisible: A057567, ranks A326155
- divisor: A057568, ranks A326149
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029, A379720
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A000720, A001222, A036844, A112798, A324850, A324851, A324925, A326151, A326153/A326154, A326156, A326157.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],SquareFreeQ[#]&&Divisible[Times@@prix[#],Plus@@prix[#]]&]

Formula

Satisfies A056239(a(n))|A003963(a(n)).

A379845 Even squarefree numbers x such that the product of prime indices of x is a multiple of the sum of prime indices of x.

Original entry on oeis.org

2, 30, 154, 190, 390, 442, 506, 658, 714, 874, 1110, 1118, 1254, 1330, 1430, 1786, 1794, 1798, 1958, 2310, 2414, 2442, 2470, 2730, 2958, 3034, 3066, 3266, 3390, 3534, 3710, 3770, 3874, 3914, 4042, 4466, 4526, 4758, 4930, 5106, 5434, 5474, 5642, 6090, 6106
Offset: 1

Views

Author

Gus Wiseman, Jan 20 2025

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The sum and product of prime indices are A056239 and A003963 respectively.

Examples

			The terms together with their prime indices begin:
     2: {1}
    30: {1,2,3}
   154: {1,4,5}
   190: {1,3,8}
   390: {1,2,3,6}
   442: {1,6,7}
   506: {1,5,9}
   658: {1,4,15}
   714: {1,2,4,7}
   874: {1,8,9}
  1110: {1,2,3,12}

Crossrefs

Even squarefree case of A326149.
For nonprime instead of even we have A326158.
Squarefree case of A379319.
Even case of A379844.
Partitions of this type are counted by A380221, see A379733, A379735.
A003963 multiplies together prime indices.
A005117 lists the squarefree numbers.
A056239 adds up prime indices.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- multiple: A057567, ranks A326155
- divisor: A057568, ranks A326149
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029, A379720
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A000720, A001222, A112798, A175508, A324850, A324851, A326150, A326151, A326153/A326154, A326156, A326157.

Programs

  • Mathematica
    Select[Range[2,1000],EvenQ[#]&&SquareFreeQ[#]&&Divisible[Times@@prix[#],Plus@@prix[#]]&]
Showing 1-4 of 4 results.

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