A279065 -id:A279065 - 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).

A322028 Number of distinct orders of primeness among the prime factors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 3, 1, 2, 2, 1, 2, 3, 1, 2, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2

Offset: 1

Gus Wiseman, Nov 24 2018

nonn

The order of primeness (A078442) of a prime number p is the number of times one must apply A000720 to obtain a nonprime number.

			a(105) = 3 because the prime factors of 105 = 3*5*7 have 3 different orders of primeness, namely 2, 3, and 1 respectively.

Alois P. Heinz, Table of n, a(n) for n = 1..65536
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included at A006450 with permission of the author]

Cf. A000720, A006450, A007097, A007821, A049076, A076610, A078442, A109082, A114537, A184155, A276625, A279065, A322027, A322030.

with(numtheory):
p:= proc(n) option remember;
      `if`(isprime(n), 1+p(pi(n)), 0)
    end:
a:= n-> nops(map(p, factorset(n))):
seq(a(n), n=1..120);  # Alois P. Heinz, Nov 24 2018

Table[If[n==1,0,Length[Union[Length[NestWhileList[PrimePi,PrimePi[#],PrimeQ]]&/@FactorInteger[n][[All,1]]]]],{n,100}]

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]}]

A316094 FDH numbers of strict integer partitions with odd parts.

Original entry on oeis.org

1, 2, 4, 7, 8, 11, 14, 16, 19, 22, 25, 28, 31, 32, 38, 41, 44, 47, 50, 53, 56, 61, 62, 64, 71, 76, 77, 79, 82, 83, 88, 94, 97, 100, 101, 103, 106, 107, 109, 112, 113, 121, 122, 124, 127, 128, 131, 133, 137, 139, 142, 149, 151, 152, 154, 157, 158, 163, 164, 166

Offset: 1

Gus Wiseman, Jun 24 2018

nonn

Also numbers n such that A305829(n) is odd.

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 all integer partitions with distinct odd parts begins (), (1), (3), (5), (3,1), (7), (5,1), (9), (11), (7,1), (13), (5,3), (15), (9,1), (11,1), (17), (7,3), (19), (13,1), (21), (5,3,1), (23), (15,1), (9,3), (25), (11,3), (7,5), (27), (17,1), (29), (7,3,1), (19,1), (31).

Cf. A000700, A050376, A064547, A213925, A258116, A279065, A299755, A299757, A305829.

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],OddQ[Times@@(FDfactor[#]/.FDrules)]&]

A316264 FDH numbers of strict integer partitions with odd length and all odd parts.

Original entry on oeis.org

2, 4, 7, 11, 16, 19, 25, 31, 41, 47, 53, 56, 61, 71, 79, 83, 88, 97, 101, 103, 107, 109, 113, 121, 127, 128, 131, 137, 139, 149, 151, 152, 154, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 200, 211, 223, 224, 227, 229, 233, 239, 241, 248, 251, 257

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 all strict odd integer partitions begins (1), (3), (5), (7), (9), (11), (13), (15), (17), (19), (21), (1,3,5), (23), (25), (27), (29), (1,3,7), (31).

Cf. A000700, A050376, A064547, A213925, A258116, A279065, A299755, A299757, A305829, A316094.

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[OddQ[Length[FDfactor[#]]],OddQ[Times@@(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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A322028 Number of distinct orders of primeness among the prime factors of n.

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

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A316094 FDH numbers of strict integer partitions with odd parts.

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A316264 FDH numbers of strict integer partitions with odd length and all odd parts.

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