A299757 -id:A299757 - cp's OEIS frontend

A316210 Number of integer partitions of the n-th Fermi-Dirac prime into Fermi-Dirac primes.

1, 1, 2, 2, 4, 7, 11, 17, 31, 37, 54, 109, 152, 283, 380, 878, 1482, 1906, 3101, 3924, 6197, 11915, 14703, 27063, 40016, 48450, 84633, 101419, 121250, 204461, 398916, 551093, 646073, 883626, 1030952, 1397083, 2522506, 3875455, 5128718, 7741307, 8860676

Offset: 1

Gus Wiseman, Jun 26 2018

nonn

A Fermi-Dirac prime (A050376) is a number of the form p^(2^k) where p is prime and k >= 0.

			The a(6) = 7 partitions of 9 into Fermi-Dirac primes are (9), (54), (72), (333), (432), (522), (3222).

Cf. A000586, A000607, A050376, A064547, A213925, A279065, A299755, A299757, A305829, A316202, A316211, A316220.

nn=60;
FDpQ[n_]:=With[{f=FactorInteger[n]},n>1&&Length[f]==1&&MatchQ[FactorInteger[2f[[1,2]]],{{2,_}}]]
FDprimeList=Select[Range[nn],FDpQ];
ser=Product[1/(1-x^d),{d,FDprimeList}];
Table[SeriesCoefficient[ser,{x,0,FDprimeList[[n]]}],{n,Length[FDprimeList]}]

A316211 Number of strict integer partitions of n into Fermi-Dirac primes.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 2, 4, 4, 4, 6, 4, 9, 5, 10, 8, 11, 11, 12, 15, 13, 19, 16, 21, 21, 24, 26, 27, 32, 31, 37, 37, 42, 44, 47, 52, 53, 61, 61, 69, 71, 78, 82, 88, 95, 99, 108, 112, 122, 128, 137, 144, 154, 163, 172, 184, 193, 206, 216, 230, 242, 256

Offset: 0

Gus Wiseman, Jun 26 2018

nonn

A Fermi-Dirac prime (A050376) is a number of the form p^(2^k) where p is prime and k >= 0.

			The a(16) = 9 strict integer partitions of 16 into Fermi-Dirac primes:
(16),
(9,7), (11,5), (13,3),
(7,5,4), (9,4,3), (9,5,2), (11,3,2),
(7,4,3,2).

Cf. A000586, A000607, A050376, A064547, A213925, A279065, A299755, A299757, A305829, A316202, A316210, A316220.

nn=60;
FDpQ[n_]:=With[{f=FactorInteger[n]},n>1&&Length[f]==1&&MatchQ[FactorInteger[2f[[1,2]]],{{2,_}}]]
FDprimeList=Select[Range[nn],FDpQ];
ser=Product[1+x^d,{d,FDprimeList}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,0,nn}]

O.g.f.: Product_d (1 + x^d) where the product is over all Fermi-Dirac primes (A050376).

A319827 FDH numbers of relatively prime strict integer partitions.

Original entry on oeis.org

2, 6, 8, 10, 12, 14, 18, 20, 21, 22, 24, 26, 28, 30, 32, 33, 34, 35, 38, 40, 42, 44, 46, 48, 50, 52, 54, 55, 56, 57, 58, 60, 62, 63, 66, 68, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 86, 88, 90, 91, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 112

Offset: 1

Gus Wiseman, Sep 28 2018

nonn

Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. The FDH number of a strict integer partition (y_1, ..., y_k) is f(y_1) * ... * f(y_k).

			The sequence of all relatively prime strict integer partitions begins: (1), (2,1), (3,1), (4,1), (3,2), (5,1), (6,1), (4,3), (5,2), (7,1), (3,2,1), (8,1), (5,3), (4,2,1).

Cf. A050376, A056239, A064547, A213925, A289508, A289509, A290103, A299755, A299757, A319826.

nn=200;
FDfactor[n_]:=If[n==1,{},Sort[Join@@Cases[FactorInteger[n],{p_,k_}:>Power[p,Cases[Position[IntegerDigits[k,2]//Reverse,1],{m_}:>2^(m-1)]]]]];
FDprimeList=Array[FDfactor,nn,1,Union];FDrules=MapIndexed[(#1->#2[[1]])&,FDprimeList];
Select[Range[nn],GCD@@(FDfactor[#]/.FDrules)==1&]

A299758 Largest FDH number of a strict integer partition of n.

Original entry on oeis.org

1, 2, 3, 6, 8, 12, 24, 30, 42, 60, 120, 168, 216, 280, 420, 840, 1080, 1512, 1890, 2520, 3780, 7560, 9240, 11880, 16632, 20790, 27720, 41580, 83160, 98280, 120960, 154440, 216216, 270270, 360360, 540540, 1081080, 1330560, 1572480, 1921920, 2471040, 3459456, 4324320

Offset: 1

Gus Wiseman, Feb 18 2018

nonn

Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. Every positive integer n has a unique factorization of the form n = f(s_1)*...*f(s_k) where the s_i are strictly increasing positive integers. This determines a unique strict integer partition (s_k...s_1) whose FDH number is then defined to be n.

			Sequence of strict integer partitions realizing each maximum begins: () (1) (2) (21) (31) (32) (321) (421) (521) (432) (4321) (5321) (6321) (5431) (5432) (54321) (64321) (65321) (65421) (65431) (65432).

Cf. A005518, A050376, A064547, A213925, A279065, A299755, A299757, A299759.

nn=150;
FDprimeList=Select[Range[nn],MatchQ[FactorInteger[#],{{?PrimeQ,?(MatchQ[FactorInteger[2#],{{2,_}}]&)}}]&];
Table[Max[Times@@FDprimeList[[#]]&/@Select[IntegerPartitions[n],UnsameQ@@#&]],{n,0,Length[FDprimeList]}]

A300351 Triangle whose n-th row lists in order all Heinz numbers of integer partitions of n into odd parts.

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 16, 11, 20, 32, 22, 25, 40, 64, 17, 44, 50, 80, 128, 34, 55, 88, 100, 160, 256, 23, 68, 110, 125, 176, 200, 320, 512, 46, 85, 121, 136, 220, 250, 352, 400, 640, 1024, 31, 92, 170, 242, 272, 275, 440, 500, 704, 800, 1280, 2048, 62, 115, 184

Offset: 1

Gus Wiseman, Mar 03 2018

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			Triangle of partitions into odd parts begins:
                   0
                  (1)
                  (11)
                (3) (111)
               (31) (1111)
            (5) (311) (11111)
        (51) (33) (3111) (111111)
    (7) (511) (331) (31111) (1111111)
(71) (53) (5111) (3311) (311111) (11111111)

Cf. A000009, A031368, A056239, A066208, A078408, A215366, A246867, A299759, A299757, A300063, A300272.

Table[Sort[Times@@Prime/@#&/@Select[IntegerPartitions[n],And@@OddQ/@#&]],{n,0,12}]

A316228 Numbers whose Fermi-Dirac prime factorization sums to a Fermi-Dirac prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 22, 23, 24, 25, 28, 29, 31, 34, 36, 37, 39, 40, 41, 43, 46, 47, 48, 49, 52, 53, 55, 56, 58, 59, 61, 63, 66, 67, 71, 73, 76, 79, 81, 82, 83, 88, 89, 90, 94, 97, 100, 101, 103, 104, 107, 108, 109, 112

Offset: 1

Gus Wiseman, Jun 27 2018

nonn

A Fermi-Dirac prime (A050376) is a number of the form p^(2^k) where p is prime and k >= 0. Every positive integer has a unique factorization into distinct Fermi-Dirac primes.

			Sequence of multiarrows in the form "number: sum <= factors" begins:
   2:  2 <= {2}
   3:  3 <= {3}
   4:  4 <= {4}
   5:  5 <= {5}
   6:  5 <= {2,3}
   7:  7 <= {7}
   9:  9 <= {9}
  10:  7 <= {2,5}
  11: 11 <= {11}
  12:  7 <= {3,4}
  13: 13 <= {13}
  14:  9 <= {2,7}
  16: 16 <= {16}
  17: 17 <= {17}
  18: 11 <= {2,9}
  19: 19 <= {19}
  20:  9 <= {4,5}
  22: 13 <= {2,11}
  23: 23 <= {23}
  24:  9 <= {2,3,4}

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A050376, A064547, A100118, A213925, A299757, A305829, A316202, A316210, A316211, A316220.

FDfactor[n_]:=If[n==1,{},Sort[Join@@Cases[FactorInteger[n],{p_,k_}:>Power[p,Cases[Position[IntegerDigits[k,2]//Reverse,1],{m_}->2^(m-1)]]]]];
Select[Range[2,200],Length[FDfactor[Total[FDfactor[#]]]]==1&]

A316265 FDH numbers of strict integer partitions with prime parts.

Original entry on oeis.org

1, 3, 4, 7, 11, 12, 19, 21, 25, 28, 33, 41, 44, 47, 57, 61, 75, 76, 77, 83, 84, 97, 100, 121, 123, 132, 133, 139, 141, 151, 164, 169, 175, 183, 188, 197, 209, 228, 231, 233, 241, 244, 249, 271, 275, 287, 289, 291, 300, 307, 308, 329, 332, 347, 361, 363, 388

Offset: 1

Gus Wiseman, Jun 28 2018

nonn

Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. The FDH number of a strict integer partition (y_1,...,y_k) is f(y_1)*...*f(y_k).

			Sequence of strict integer partitions with prime parts, preceded by their FDH numbers, begins:
   1: ()
   3: (2)
   4: (3)
   7: (5)
  11: (7)
  12: (3,2)
  19: (11)
  21: (5,2)
  25: (13)
  28: (5,3)
  33: (7,2)
  41: (17)
  44: (7,3)
  47: (19)
  57: (11,2)
  61: (23)
  75: (13,2)
  76: (11,3)
  77: (7,5)
  83: (29)
  84: (5,3,2)

Cf. A000586, A045450, A050376, A064547, A213925, A299755, A299757, A316185, A316266, A316267.

nn=100;
FDfactor[n_]:=If[n==1,{},Sort[Join@@Cases[FactorInteger[n],{p_,k_}:>Power[p,Cases[Position[IntegerDigits[k,2]//Reverse,1],{m_}->2^(m-1)]]]]];
FDprimeList=Array[FDfactor,nn,1,Union];FDrules=MapIndexed[(#1->#2[[1]])&,FDprimeList];
Select[Range[nn],And@@PrimeQ/@(FDfactor[#]/.FDrules)&]

A316266 FDH numbers of strict integer partitions with prime parts and prime length.

Original entry on oeis.org

12, 21, 28, 33, 44, 57, 75, 76, 77, 84, 100, 123, 132, 133, 141, 164, 175, 183, 188, 209, 228, 231, 244, 249, 275, 287, 291, 300, 308, 329, 332, 363, 388, 399, 417, 427, 451, 453, 475, 484, 492, 507, 517, 525, 532, 556, 564, 581, 591, 604, 627, 671, 676, 679

Offset: 1

Gus Wiseman, Jun 28 2018

nonn

Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. The FDH number of a strict integer partition (y_1,...,y_k) is f(y_1)*...*f(y_k).

			Sequence of strict integer partitions with prime parts and prime length, preceded by their FDH numbers, begins:
  12: (3,2)
  21: (5,2)
  28: (5,3)
  33: (7,2)
  44: (7,3)
  57: (11,2)
  75: (13,2)
  76: (11,3)
  77: (7,5)
  84: (5,3,2)

Cf. A000586, A045450, A050376, A064547, A213925, A299755, A299757, A316185, A316265, A316267.

nn=1000;
FDfactor[n_]:=If[n==1,{},Sort[Join@@Cases[FactorInteger[n],{p_,k_}:>Power[p,Cases[Position[IntegerDigits[k,2]//Reverse,1],{m_}->2^(m-1)]]]]];
FDprimeList=Array[FDfactor,nn,1,Union];FDrules=MapIndexed[(#1->#2[[1]])&,FDprimeList];
Select[Range[nn],And[PrimeQ[Length[FDfactor[#]]],And@@PrimeQ/@(FDfactor[#]/.FDrules)]&]

A316267 FDH numbers of strict integer partitions of prime numbers with a prime number of prime parts.

Original entry on oeis.org

12, 21, 57, 123, 249, 417, 532, 699, 867, 1100, 1389, 1463, 1509, 1708, 2049, 2068, 2307, 2324, 2913, 3116, 3147, 3157, 3273, 3325, 3619, 3903, 4227, 4268, 4636, 4821, 5079, 5225, 5324, 5516, 5739, 6308, 6391, 6524, 6621, 6644, 7469, 8092, 8193, 8225, 8457

Offset: 1

Gus Wiseman, Jun 28 2018

nonn

Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. The FDH number of a strict integer partition (y_1,...,y_k) is f(y_1)*...*f(y_k).

			Sequence of strict integer partitions of prime numbers with a prime number of prime parts, preceded by their FDH numbers, begins:
    12: (3,2)
    21: (5,2)
    57: (11,2)
   123: (17,2)
   249: (29,2)
   417: (41,2)
   532: (11,5,3)
   699: (59,2)
   867: (71,2)
  1100: (13,7,3)
  1389: (101,2)
  1463: (11,7,5)
  1509: (107,2)
  1708: (23,5,3)

Cf. A000586, A045450, A050376, A064547, A213925, A299755, A299757, A316185, A316265, A316266.

nn=1000;
FDfactor[n_]:=If[n==1,{},Sort[Join@@Cases[FactorInteger[n],{p_,k_}:>Power[p,Cases[Position[IntegerDigits[k,2]//Reverse,1],{m_}->2^(m-1)]]]]];
FDprimeList=Array[FDfactor,nn,1,Union];FDrules=MapIndexed[(#1->#2[[1]])&,FDprimeList];
Select[Range[nn],And[PrimeQ[Total[FDfactor[#]/.FDrules]],PrimeQ[Length[FDfactor[#]]],And@@PrimeQ/@(FDfactor[#]/.FDrules)]&]

A319829 FDH numbers of strict integer partitions of odd numbers.

Original entry on oeis.org

2, 4, 6, 7, 10, 11, 12, 16, 18, 19, 20, 21, 25, 26, 30, 31, 33, 34, 35, 36, 41, 46, 47, 48, 52, 53, 54, 55, 56, 57, 58, 60, 61, 63, 68, 71, 74, 75, 78, 79, 80, 83, 86, 88, 90, 91, 92, 93, 95, 97, 98, 99, 102, 103, 105, 108, 109, 116, 118, 119, 121, 123, 125

Offset: 1

Gus Wiseman, Sep 28 2018

nonn

Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. The FDH number of a strict integer partition (y_1, ..., y_k) is f(y_1) * ... * f(y_k).

			The sequence of all strict integer partitions of odd numbers begins: (1), (3), (2,1), (5), (4,1), (7), (3,2), (9), (6,1), (11), (4,3), (5,2), (13), (8,1), (4,2,1), (15), (7,2), (10,1), (5,4), (6,3), (17), (12,1), (19), (9,2), (8,3), (21), (6,2,1), (7,4), (5,3,1), (11,2), (14,1), (4,3,2).

Complement of A319828.

Cf. A050376, A064547, A213925, A299755, A299757, A300061, A300063, A319241, A319242, A319827.

nn=200;
FDfactor[n_]:=If[n==1,{},Sort[Join@@Cases[FactorInteger[n],{p_,k_}:>Power[p,Cases[Position[IntegerDigits[k,2]//Reverse,1],{m_}:>2^(m-1)]]]]];
FDprimeList=Array[FDfactor,nn,1,Union];FDrules=MapIndexed[(#1->#2[[1]])&,FDprimeList];
Select[Range[nn],OddQ[Total[FDfactor[#]/.FDrules]]&]

cp's OEIS Frontend

A316210 Number of integer partitions of the n-th Fermi-Dirac prime into Fermi-Dirac primes.

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A316211 Number of strict integer partitions of n into Fermi-Dirac primes.

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A319827 FDH numbers of relatively prime strict integer partitions.

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A299758 Largest FDH number of a strict integer partition of n.

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A300351 Triangle whose n-th row lists in order all Heinz numbers of integer partitions of n into odd parts.

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A316228 Numbers whose Fermi-Dirac prime factorization sums to a Fermi-Dirac prime.

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A316265 FDH numbers of strict integer partitions with prime parts.

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A316266 FDH numbers of strict integer partitions with prime parts and prime length.

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A316267 FDH numbers of strict integer partitions of prime numbers with a prime number of prime parts.

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