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.

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A322553 Odd numbers whose product of prime indices is a prime power.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 17, 19, 21, 23, 25, 27, 31, 41, 49, 53, 57, 59, 63, 67, 81, 83, 97, 103, 109, 115, 121, 125, 127, 131, 133, 147, 157, 159, 171, 179, 189, 191, 211, 227, 241, 243, 277, 283, 289, 311, 331, 343, 353, 361, 367, 371, 393, 399, 401, 419, 431, 441
Offset: 1

Views

Author

Gus Wiseman, Dec 15 2018

Keywords

Comments

Differs from A322400 in having 1 and lacking 377, the MM-number of {{1,2},{1,3}}.
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.

Examples

			The multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}. The sequence of multiset partitions whose MM-numbers belong to this sequence begins:
   1: {}
   3: {{1}}
   5: {{2}}
   7: {{1,1}}
   9: {{1},{1}}
  11: {{3}}
  17: {{4}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  23: {{2,2}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  31: {{5}}
  41: {{6}}
  49: {{1,1},{1,1}}
  53: {{1,1,1,1}}
  57: {{1},{1,1,1}}
  59: {{7}}
  63: {{1},{1},{1,1}}
  67: {{8}}
  81: {{1},{1},{1},{1}}
  83: {{9}}
  97: {{3,3}}

Crossrefs

Cf. A003963, A056239, A112798, A290103, A302242, A320325, A320698.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1,100,2],#==1||PrimePowerQ[Times@@primeMS[#]]&]
  • PARI
    isok(n) = {if (n % 2, my(f = factor(n), pk = prod(k=1, #f~, primepi(f[k,1]))); (pk == 1) || isprimepower(pk););} \\ Michel Marcus, Dec 16 2018

A330106 Number of integer partitions of n whose product is a powerful number.

Original entry on oeis.org

0, 0, 0, 0, 2, 2, 5, 5, 9, 11, 18, 19, 30, 36, 51, 62, 87, 104, 141, 171, 225, 271, 349, 419, 534, 643, 804, 965, 1197, 1431, 1766, 2106, 2571, 3063, 3719, 4410, 5325, 6305, 7567, 8939, 10678, 12572, 14961, 17567, 20804, 24389, 28775, 33626, 39551, 46106
Offset: 0

Views

Author

Gus Wiseman, Dec 05 2019

Keywords

Examples

			The a(4) = 2 through a(10) = 18 partitions:
  (4)   (41)   (33)    (331)    (8)       (9)        (55)
  (22)  (221)  (42)    (421)    (44)      (81)       (82)
               (222)   (2221)   (422)     (333)      (91)
               (411)   (4111)   (2222)    (441)      (433)
               (2211)  (22111)  (3311)    (4221)     (442)
                                (4211)    (22221)    (811)
                                (22211)   (33111)    (3322)
                                (41111)   (42111)    (3331)
                                (221111)  (222111)   (4222)
                                          (411111)   (4411)
                                          (2211111)  (22222)
                                                     (42211)
                                                     (222211)
                                                     (331111)
                                                     (421111)
                                                     (2221111)
                                                     (4111111)
                                                     (22111111)

Crossrefs

The strict version is A330216.
Powerful numbers are A001694.
Partitions whose product is a perfect power are A320322.
Cf. A000041, A001055, A001597, A003963, A064573, A320325.

Programs

  • Mathematica
    powQ[n_]:=Min@@Last/@FactorInteger[n]>1;
Table[Length[Select[IntegerPartitions[n],powQ[Times@@#]&]],{n,0,30}]

A330216 Number of strict integer partitions of n whose product is a powerful number.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 3, 4, 3, 4, 5, 7, 8, 8, 10, 12, 12, 15, 18, 19, 20, 24, 25, 28, 38, 41, 43, 50, 55, 63, 79, 85, 88, 104, 116, 124, 143, 157, 173, 197, 214, 235, 274, 294, 319, 363, 393, 430, 487, 529, 577, 647, 692, 752, 856, 925, 992, 1099
Offset: 0

Views

Author

Gus Wiseman, Dec 05 2019

Keywords

Examples

			The a(n) partitions for n = 4, 9, 12, 13, 16, 17, 18:
  (4)  (9)    (8,4)      (9,4)    (16)         (9,8)      (12,6)
       (8,1)  (9,3)      (6,4,3)  (9,4,3)      (16,1)     (16,2)
              (6,3,2,1)  (8,4,1)  (12,3,1)     (8,6,3)    (9,8,1)
                         (9,3,1)  (9,4,2,1)    (9,6,2)    (8,6,3,1)
                                  (6,4,3,2,1)  (10,5,2)   (9,4,3,2)
                                               (12,3,2)   (9,6,2,1)
                                               (9,4,3,1)  (10,5,2,1)
                                                          (12,3,2,1)

Crossrefs

The non-strict version is A330106.
Powerful numbers are A001694.
Partitions whose product is a perfect power are A320322.
Cf. A000009, A045778, A001597, A003963, A064573, A320325.

Programs

  • Mathematica
    powQ[n_]:=Min@@Last/@FactorInteger[n]>1;
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&powQ[Times@@#]&]],{n,0,30}]

A353699 Heinz numbers of integer partitions whose product equals their length.

Original entry on oeis.org

2, 6, 20, 36, 56, 176, 240, 416, 864, 1088, 1344, 2432, 3200, 5888, 8448, 14848, 23040, 31744, 35840, 39936, 75776, 167936, 208896, 331776, 352256, 450560, 516096, 770048, 802816, 933888, 1736704, 2457600, 3866624, 4259840, 4521984, 7995392, 12976128, 17563648
Offset: 1

Views

Author

Gus Wiseman, May 19 2022

Keywords

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

Examples

			The terms together with their prime indices begin:
      2: {1}
      6: {1,2}
     20: {1,1,3}
     36: {1,1,2,2}
     56: {1,1,1,4}
    176: {1,1,1,1,5}
    240: {1,1,1,1,2,3}
    416: {1,1,1,1,1,6}
    864: {1,1,1,1,1,2,2,2}
   1088: {1,1,1,1,1,1,7}
   1344: {1,1,1,1,1,1,2,4}
   2432: {1,1,1,1,1,1,1,8}
   3200: {1,1,1,1,1,1,1,3,3}
   5888: {1,1,1,1,1,1,1,1,9}
   8448: {1,1,1,1,1,1,1,1,2,5}
  14848: {1,1,1,1,1,1,1,1,1,10}
  23040: {1,1,1,1,1,1,1,1,1,2,2,3}
  31744: {1,1,1,1,1,1,1,1,1,1,11}
  35840: {1,1,1,1,1,1,1,1,1,1,3,4}
  39936: {1,1,1,1,1,1,1,1,1,1,2,6}
  75776: {1,1,1,1,1,1,1,1,1,1,1,12}

Crossrefs

Length is A001222, counted by A008284, distinct A001221.
Product is A003963, counted by A339095, firsts A318871.
A similar sequence is A353503, counted by A353506.
These partitions are counted by A353698.
A005361 gives product of signature, firsts A353500 (sorted A085629).
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A353394 gives product of shadows of prime indices, firsts A353397.
Cf. A000720, A003586, A114640, A182850, A320325, A325131, A325755, A353399, A353507.

Programs

  • Mathematica
    Select[Range[1000],Times@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]^k]==PrimeOmega[#]&]
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