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 1-8 of 8 results.

A078135 Numbers which cannot be written as a sum of squares > 1.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 19, 23
Offset: 1

Views

Author

Reinhard Zumkeller, Nov 19 2002

Keywords

Comments

Numbers such that A078134(n)=0.
"Numbers which cannot be written as sum of squares > 1" is equivalent to "Numbers which cannot be written as sum of squares of primes." Equivalently, numbers which can be written as the sum of nonzero squares can also be written as sum of the squares of primes." cf. A090677 = number of ways to partition n into sums of squares of primes. - Jonathan Vos Post, Sep 20 2006
The sequence is finite with a(12)=23 as last member. Proof: When k=a^2+b^2+..., k+4 = 2^2+a^2+b^2+... If k can be written as sum of the squares of primes, k+4 also has this property. As 24,25,26,27 have the property, by induction, all numbers > 23 can be written as sum of squares>1. - Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Apr 07 2007
Also, numbers which cannot be written as sum of squares of 2 and 3 (see A078137 for the proof). Explicit representation as sum of squares of primes, or rather of squares of 2 and 3, for numbers m>23: we have m=c*2^2+d*3^2, where c:=(floor(m/4) - 2*(m mod 4))>=0, d:=m mod 4. For that, the finiteness of the sequence is proved constructively. - Hieronymus Fischer, Nov 11 2007
Also numbers n such that every integer partition of n contains a squarefree number. For example, 21 does not belong to the sequence because there are integer partitions of 21 containing no squarefree numbers, namely: (12,9), (9,8,4), (9,4,4,4). - Gus Wiseman, Dec 14 2018

Links

Crossrefs

Cf. A000290, A001422, A078137, A078139, A078136, A078129.
Cf. A090677.
Cf. A078134, A078139, A090677, A078137, A134754, A134755.
Cf. A005117, A101417, A320322, A322526, A322546, A322547.

Programs

  • Mathematica
    nn=100;
ser=Product[If[SquareFreeQ[n],1,1/(1-x^n)],{n,nn}];
Join@@Position[CoefficientList[Series[ser,{x,0,nn}],x],0]-1 (* Gus Wiseman, Dec 14 2018 *)

Formula

A090677(a(n)) = 0. - Jonathan Vos Post, Sep 20 2006 [corrected by Joerg Arndt, Dec 16 2018]
A033183(a(n)) = 0. - Reinhard Zumkeller, Nov 07 2009

A322527 Number of integer partitions of n whose product of parts is a power of a squarefree number (A072774).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 13, 18, 21, 31, 34, 45, 51, 63, 72, 88, 97, 120, 128, 158, 174, 201, 222, 264, 287, 333, 359, 416, 441, 518, 557, 631, 684, 770, 833, 954, 1017, 1141, 1222, 1378, 1475, 1643, 1755, 1939, 2097, 2327, 2471, 2758, 2928, 3233, 3470, 3813, 4085
Offset: 0

Views

Author

Gus Wiseman, Dec 14 2018

Keywords

Examples

			The a(1) = 1 through a(8) = 18 integer partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (52)       (44)
             (111)  (31)    (41)     (42)      (61)       (53)
                    (211)   (221)    (51)      (331)      (71)
                    (1111)  (311)    (222)     (421)      (422)
                            (2111)   (321)     (511)      (521)
                            (11111)  (411)     (2221)     (611)
                                     (2211)    (3211)     (2222)
                                     (3111)    (4111)     (3311)
                                     (21111)   (22111)    (4211)
                                     (111111)  (31111)    (5111)
                                               (211111)   (22211)
                                               (1111111)  (32111)
                                                          (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
Missing from the list for n = 7 through 9:
  (43)   (62)    (54)
  (322)  (332)   (63)
         (431)   (432)
         (3221)  (522)
                 (621)
                 (3222)
                 (3321)
                 (4311)
                 (32211)

Crossrefs

Cf. A003963, A005117, A038041, A062503, A064573, A072774, A295193, A302505, A306021, A319169, A320322, A322526, A322528, A322530.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],SameQ@@Last/@FactorInteger[Times@@#]&]],{n,30}]

A322530 Number of integer partitions of n with no 1's whose product of parts is a squarefree number.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 1, 2, 1, 1, 3, 2, 2, 4, 3, 3, 3, 6, 5, 5, 5, 6, 8, 8, 9, 8, 11, 8, 12, 13, 16, 14, 13, 16, 21, 18, 21, 25, 22, 24, 27, 35, 33, 33, 32, 37, 42, 47, 48, 48, 52, 51, 59, 70, 68, 65, 69, 80, 87, 90, 103, 100, 96, 103, 123, 128, 135, 136, 132, 153
Offset: 0

Views

Author

Gus Wiseman, Dec 14 2018

Keywords

Comments

Such a partition must be strict and its parts must also be squarefree.

Examples

			The a(26) = 11 integer partitions:
  (26),
  (15,11), (19,7), (21,5), (23,3),
  (13,7,6), (13,10,3), (13,11,2), (17,7,2), (19,5,2),
  (11,7,5,3).

Crossrefs

First differences of A322526.
Cf. A002865, A003963, A005117, A064573, A072774, A073576, A302505, A319005, A319057, A319169, A320322, A322527, A322528, A322529.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],!MemberQ[#,1]&&SquareFreeQ[Times@@#]&]],{n,30}]

A322528 Number of integer partitions of n whose parts all have the same number of prime factors (counted with multiplicity) and whose product of parts is a power of a squarefree number (A072774).

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 3, 5, 4, 7, 2, 7, 4, 7, 7, 9, 3, 10, 5, 12, 9, 8, 6, 14, 10, 12, 10, 14, 11, 20, 13, 18, 13, 16, 16, 25, 16, 19, 20, 26, 18, 30, 19, 27, 26, 27, 22, 38, 30, 37, 28, 38, 32, 43, 37, 46, 40, 47, 40, 66, 49, 58, 56, 64, 56, 73, 58, 76, 70, 85
Offset: 0

Views

Author

Gus Wiseman, Dec 14 2018

Keywords

Examples

			The a(1) = 1 through a(8) = 5 integer partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (32)     (33)      (52)       (44)
                    (1111)  (11111)  (222)     (1111111)  (53)
                                     (111111)             (2222)
                                                          (11111111)

Crossrefs

Cf. A002865, A003963, A005117, A064573 , A072774, A295193, A302505, A306017, A319169, A320322, A322526, A322527, A322529, A322530, A322531.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],And[SameQ@@PrimeOmega/@#,SameQ@@Last/@FactorInteger[Times@@#]]&]],{n,30}]

Extensions

More terms from Alois P. Heinz, Dec 14 2018

A322529 Number of integer partitions of n whose parts all have the same number of prime factors (counted with or without multiplicity) and whose product of parts is a squarefree number.

Original entry on oeis.org

1, 1, 2, 2, 1, 3, 2, 3, 2, 2, 4, 2, 3, 3, 4, 4, 4, 3, 5, 4, 5, 6, 6, 6, 6, 6, 8, 6, 7, 9, 8, 11, 8, 11, 11, 11, 12, 13, 13, 15, 13, 17, 17, 18, 18, 17, 20, 22, 21, 24, 24, 24, 26, 29, 28, 33, 30, 35, 34, 38, 38, 45, 42, 43, 45, 48, 52, 54, 55, 59, 59, 65, 65, 72, 73
Offset: 0

Views

Author

Gus Wiseman, Dec 14 2018

Keywords

Comments

Such a partition must be strict (unless it is all 1's) and its parts must also be squarefree.

Examples

			The a(30) = 8 integer partitions:
  (30),
  (17,13),(19,11),(23,7),
  (17,11,2),(23,5,2),
  (13,7,5,3,2),
  (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1).

Links

Crossrefs

Cf. A002865, A003963, A005117, A038041, A064573, A073576, A302505, A319056, A320322, A321717, A321718, A322526, A322527, A322528, A322530, A322531.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],And[SameQ@@PrimeOmega/@#,SquareFreeQ[Times@@#]]&]],{n,30}]

Extensions

a(51)-a(69) from Jinyuan Wang, Jun 27 2020
a(70) onwards from Lucas A. Brown, Aug 17 2024

A322531 Heinz numbers of integer partitions whose parts all have the same number of prime factors (counted with or without multiplicity) and whose product of parts is a squarefree number.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 11, 13, 15, 16, 17, 29, 31, 32, 33, 41, 43, 47, 51, 55, 59, 64, 67, 73, 79, 83, 85, 93, 101, 109, 113, 123, 127, 128, 137, 139, 149, 155, 157, 163, 165, 167, 177, 179, 181, 187, 191, 199, 201, 205, 211, 233, 241, 249, 255, 256, 257, 269, 271
Offset: 1

Views

Author

Gus Wiseman, Dec 14 2018

Keywords

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
All entries are themselves squarefree numbers (except the powers of 2).
The first odd term not in this sequence but in A302521 is 141, which is the MM-number (see A302242) of {{1},{2,3}}.

Examples

			The sequence of all integer partitions whose parts all have the same number of prime factors and whose product of parts is a squarefree number begins: (), (1), (2), (1,1), (3), (1,1,1), (5), (6), (3,2), (1,1,1,1), (7), (10), (11), (1,1,1,1,1), (5,2), (13), (14), (15), (7,2), (5,3), (17), (1,1,1,1,1,1).

Crossrefs

Cf. A003963, A005117, A038041, A056239, A073576, A112798, A302242, A302505, A306017, A319056, A319169, A320324, A321717, A321718, A322526, A322528.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],And[SameQ@@PrimeOmega/@primeMS[#],SquareFreeQ[Times@@primeMS[#]]]&]

A319877 Numbers whose product of prime indices (A003963) is a square of a squarefree number (A062503).

Original entry on oeis.org

1, 7, 9, 14, 18, 23, 25, 28, 36, 46, 50, 56, 72, 92, 97, 100, 112, 121, 144, 151, 161, 169, 175, 183, 184, 185, 194, 195, 200, 207, 224, 225, 227, 242, 288, 289, 302, 322, 338, 350, 366, 368, 370, 388, 390, 400, 414, 448, 450, 454, 484, 541, 576, 578, 604, 644
Offset: 1

Views

Author

Gus Wiseman, Dec 17 2018

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 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}}. This sequence lists all MM-numbers of 2-regular multiset multisystems (meaning all vertex-degrees are 2).

Examples

			The sequence of multiset multisystems whose MM-numbers belong to the sequence begins:
    1: {}
    7: {{1,1}}
    9: {{1},{1}}
   14: {{},{1,1}}
   18: {{},{1},{1}}
   23: {{2,2}}
   25: {{2},{2}}
   28: {{},{},{1,1}}
   36: {{},{},{1},{1}}
   46: {{},{2,2}}
   50: {{},{2},{2}}
   56: {{},{},{},{1,1}}
   72: {{},{},{},{1},{1}}
   92: {{},{},{2,2}}
   97: {{3,3}}
  100: {{},{},{2},{2}}
  112: {{},{},{},{},{1,1}}
  121: {{3},{3}}
  144: {{},{},{},{},{1},{1}}
  151: {{1,1,2,2}}
  161: {{1,1},{2,2}}
  169: {{1,2},{1,2}}
  175: {{2},{2},{1,1}}
  183: {{1},{1,2,2}}
  184: {{},{},{},{2,2}}
  185: {{2},{1,1,2}}
  194: {{},{3,3}}
  195: {{1},{2},{1,2}}
  200: {{},{},{},{2},{2}}

Crossrefs

Cf. A003963, A005117, A005176, A062503, A064573, A072774, A295193, A302505, A319878, A319899, A320325, A322526, A322527, A322530.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Or[#==1,SameQ[##,2]&@@Last/@FactorInteger[Times@@primeMS[#]]]&]

A319878 Odd numbers whose product of prime indices (A003963) is a square of a squarefree number (A062503).

Original entry on oeis.org

1, 7, 9, 23, 25, 97, 121, 151, 161, 169, 175, 183, 185, 195, 207, 225, 227, 289, 541, 661, 679, 687, 781, 841, 847, 873, 957, 961, 1009, 1089, 1193, 1427, 1563, 1589, 1681, 1819, 1849, 1879, 1895, 2023, 2043, 2167, 2193, 2209, 2231, 2425, 2437, 2585, 2601
Offset: 1

Views

Author

Gus Wiseman, Dec 17 2018

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 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}}. This sequence lists all MM-numbers of 2-regular (all vertex-degrees are 2) multiset partitions (no empty parts).

Examples

			The sequence of multiset partitions whose MM-numbers belong to the sequence begins:
    1: {}
    7: {{1,1}}
    9: {{1},{1}}
   23: {{2,2}}
   25: {{2},{2}}
   97: {{3,3}}
  121: {{3},{3}}
  151: {{1,1,2,2}}
  161: {{1,1},{2,2}}
  169: {{1,2},{1,2}}
  175: {{2},{2},{1,1}}
  183: {{1},{1,2,2}}
  185: {{2},{1,1,2}}
  195: {{1},{2},{1,2}}
  207: {{1},{1},{2,2}}
  225: {{1},{1},{2},{2}}
  227: {{4,4}}
  289: {{4},{4}}
  541: {{1,1,3,3}}
  661: {{5,5}}
  679: {{1,1},{3,3}}
  687: {{1},{1,3,3}}
  781: {{3},{1,1,3}}
  841: {{1,3},{1,3}}
  847: {{1,1},{3},{3}}
  873: {{1},{1},{3,3}}
  957: {{1},{3},{1,3}}
  961: {{5},{5}}

Crossrefs

Cf. A003963, A005117, A005176, A062503, A064573, A072774, A295193, A302505, A319877, A319899, A320325, A322526, A322527, A322530.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1,100,2],Or[#==1,SameQ[##,2]&@@Last/@FactorInteger[Times@@primeMS[#]]]&]
Showing 1-8 of 8 results.

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