A213925 -id:A213925 - cp's OEIS frontend

A223490 Smallest Fermi-Dirac factor of n.

1, 2, 3, 4, 5, 2, 7, 2, 9, 2, 11, 3, 13, 2, 3, 16, 17, 2, 19, 4, 3, 2, 23, 2, 25, 2, 3, 4, 29, 2, 31, 2, 3, 2, 5, 4, 37, 2, 3, 2, 41, 2, 43, 4, 5, 2, 47, 3, 49, 2, 3, 4, 53, 2, 5, 2, 3, 2, 59, 3, 61, 2, 7, 4, 5, 2, 67, 4, 3, 2, 71, 2, 73, 2, 3, 4, 7, 2, 79

Offset: 1

Reinhard Zumkeller, Mar 20 2013

Note that this is not equal to the smallest Fermi-Dirac prime (A050376) dividing n, which is always A020639(n). - Antti Karttunen, Apr 15 2018

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
OEIS Wiki, "Fermi-Dirac representation" of n

Cf. A223491, A050376, A028233, A000040 (subsequence).
Cf. A302023, A302024, A302786, A302792.
Cf. also A020639.

Haskell
```
a223490 = head . a213925_row
```

f[p_, e_] := p^(2^IntegerExponent[e, 2]); a[n_] := Min @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 26 2020 *)

up_to = 65537;
v050376 = vector(up_to);
A050376(n) = v050376[n];
ispow2(n) = (n && !bitand(n,n-1));
i = 0; for(n=1,oo,if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to,break));
A052331(n) = { my(s=0,e); while(n > 1, fordiv(n, d, if(((n/d)>1)&&ispow2(isprimepower(n/d)), e = vecsearch(v050376, n/d); if(!e, print("v050376 too short!"); return(1/0)); s += 2^(e-1); n = d; break))); (s); };
A001511(n) = 1+valuation(n,2);
A223490(n) = if(1==n,n,A050376(A001511(A052331(n)))); \\ Antti Karttunen, Apr 15 2018

a(n) = A213925(n,1).
A209229(A100995(a(n))) = 1; A010055(a(n)) = 1.
From Antti Karttunen, Apr 15 2018: (Start)
a(1) = 1; and for n > 1, a(n) = A050376(A302786(n)).
a(n) = n / A302792(n).
a(n) = A302023(A020639(A302024(n))).
(End)

A223491 Largest Fermi-Dirac factor of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 4, 9, 5, 11, 4, 13, 7, 5, 16, 17, 9, 19, 5, 7, 11, 23, 4, 25, 13, 9, 7, 29, 5, 31, 16, 11, 17, 7, 9, 37, 19, 13, 5, 41, 7, 43, 11, 9, 23, 47, 16, 49, 25, 17, 13, 53, 9, 11, 7, 19, 29, 59, 5, 61, 31, 9, 16, 13, 11, 67, 17, 23, 7, 71, 9

Offset: 1

Reinhard Zumkeller, Mar 20 2013

nonn

Greatest Fermi-Dirac factor of n: Largest divisor of n of the form p^(2^k), for some prime p and k >= 0, with a(1) = 1. Thus for n > 1, the largest term of A050376 that divides n. - Antti Karttunen, Apr 13 2018

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
OEIS Wiki, "Fermi-Dirac representation" of n

Cf. A223490, A050376, A034699, A000040 (subsequence), A302776, A302785, A302789 (ordinal transform).

Cf. also A006530, A034699.

Haskell
```
a223491 = last . a213925_row
```

f[p_, e_] := p^(2^Floor[Log2[e]]); a[n_] := Max @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 26 2020 *)

ispow2(n) = (n && !bitand(n,n-1));
A223491(n) = if(1==n,n,fordiv(n, d, if(ispow2(isprimepower(n/d)), return(n/d)))); \\ Antti Karttunen, Apr 13 2018

a(n) = A213925(n,A064547(n)).
A209229(A100995(a(n))) = 1; A010055(a(n)) = 1.
From Antti Karttunen, Apr 13 2018: (Start)
a(1) = 1; for n > 1, a(n) = A050376(A302785(n)).
a(n) = n/A302776(n).
(End)

A299756 Triangle read by rows in which row n is the finite increasing sequence, or set of positive integers, with FDH number n.

Original entry on oeis.org

1, 2, 3, 4, 1, 2, 5, 1, 3, 6, 1, 4, 7, 2, 3, 8, 1, 5, 2, 4, 9, 10, 1, 6, 11, 3, 4, 2, 5, 1, 7, 12, 1, 2, 3, 13, 1, 8, 2, 6, 3, 5, 14, 1, 2, 4, 15, 1, 9, 2, 7, 1, 10, 4, 5, 3, 6, 16, 1, 11, 2, 8, 1, 3, 4, 17, 1, 2, 5, 18, 3, 7, 4, 6, 1, 12, 19, 2, 9, 20, 1, 13

Offset: 1

Gus Wiseman, Feb 18 2018

Let f(n) = A050376(n) be the n-th number of the form p^(2^k) where p is prime and k >= 0. The FDH number of a set S is Product_{x in S} f(x).

Same as A299755 with rows reversed.

			Sequence of sets begins: {}, {1}, {2}, {3}, {4}, {1,2}, {5}, {1,3}, {6}, {1,4}, {7}, {2,3}, {8}, {1,5}, {2,4}, {9}, {10}, {1,6}, {11}, {3,4}, {2,5}, {1,7}, {12}, {1,2,3}, {13}.

Row lengths are A064547.

Cf. A005117, A050376, A112798, A213925, A246867, A277098, A296150, A299090, A299755, A299757, A299759.

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)]]]]];
nn=200;FDprimeList=Array[FDfactor,nn,1,Union];
FDrules=MapIndexed[(#1->#2[[1]])&,FDprimeList];
Join@@Table[FDfactor[n]/.FDrules,{n,60}]

A299759 Triangle read by rows in which row n lists in order all FDH numbers of strict integer partitions of n.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 8, 7, 10, 12, 9, 14, 15, 24, 11, 18, 20, 21, 30, 13, 22, 27, 28, 40, 42, 16, 26, 33, 35, 36, 54, 56, 60, 17, 32, 39, 44, 45, 66, 70, 72, 84, 120, 19, 34, 48, 52, 55, 63, 78, 88, 90, 105, 108, 168, 23, 38, 51, 64, 65, 77, 96, 104, 110, 126

Offset: 1

Gus Wiseman, Feb 18 2018

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.

This sequence is a permutation of the positive integers.

			Triangle of strict partitions begins:
                  0
                 (1)
                 (2)
               (3) (21)
               (4) (31)
             (5) (41) (32)
          (6) (51) (42) (321)
        (7) (61) (43) (52) (421)
     (8) (71) (62) (53) (431) (521)
(9) (81) (72) (54) (63) (621) (531) (432).

Cf. A050376, A056239, A064547, A106400, A122111, A213925, A215366, A246867, A279065, A279614, A299755, A299757, A299758.

Row lengths give A000009.

nn=25;
FDprimeList=Select[Range[nn],MatchQ[FactorInteger[#],{{?PrimeQ,?(MatchQ[FactorInteger[2#],{{2,_}}]&)}}]&];
Table[Sort[Times@@FDprimeList[[#]]&/@Select[IntegerPartitions[n],UnsameQ@@#&]],{n,0,Length[FDprimeList]}]

A316220 Number of triangles whose weight is the n-th Fermi-Dirac prime in the multiorder of integer partitions of Fermi-Dirac primes into Fermi-Dirac primes.

Original entry on oeis.org

1, 1, 3, 3, 9, 21, 46, 95, 273, 363, 731, 3088, 6247, 24152, 46012, 319511, 1141923, 2138064, 7346404, 13530107, 45297804, 271446312

Offset: 1

Gus Wiseman, Jun 26 2018

A Fermi-Dirac prime (A050376) is a number of the form p^(2^k) where p is prime and k >= 0. An FD-partition is an integer partition of a Fermi-Dirac prime into Fermi-Dirac primes. a(n) is the number of sequences of FD-partitions whose sums are weakly decreasing and sum to the n-th Fermi-Dirac prime.

Gus Wiseman, Comcategories and Multiorders
Gus Wiseman, Illustration of the first six terms of A316220.

Cf. A050376, A063834, A064547, A213925, A269134, A281113, A299757, A305829, A316202, A316210, A316211, A316219, A316228.

nn=60;
FDpQ[n_]:=With[{f=FactorInteger[n]},n>1&&Length[f]==1&&MatchQ[FactorInteger[2f[[1,2]]],{{2,_}}]];
FDpl=Select[Range[nn],FDpQ];
fen[n_]:=fen[n]=SeriesCoefficient[Product[1/(1-x^p),{p,Select[Range[n],FDpQ]}],{x,0,n}];
Table[Sum[Times@@fen/@p,{p,Select[IntegerPartitions[FDpl[[n]]],And@@FDpQ/@#&]}],{n,Length[FDpl]}]

A319826 GCD of the strict integer partition with FDH number n; GCD of the indices (in A050376) of Fermi-Dirac prime factors of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 2, 9, 10, 1, 11, 1, 1, 1, 12, 1, 13, 1, 2, 1, 14, 1, 15, 1, 1, 1, 1, 3, 16, 1, 2, 1, 17, 1, 18, 1, 2, 1, 19, 1, 20, 1, 2, 1, 21, 1, 1, 1, 1, 1, 22, 1, 23, 1, 1, 3, 4, 1, 24, 1, 2, 1, 25, 1, 26, 1, 1, 1, 1, 1, 27, 1, 28

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).

			45 is the FDH number of (6,4), which has GCD 2, so a(45) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..16384

A left inverse of A050376.

Cf. A052331, A056239, A064547, A213925, A289508, A289509, A290103, A299755, A299757, A319825.

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];
GCD@@@Table[Reverse[FDfactor[n]/.FDrules],{n,nn}]

A319826(n) = { my(i=1,g=0,x=A052331(n)); while(x,if(x%2,g = gcd(g,i)); x>>=1; i++); (g); }; \\ (Uses the program given in A052331) - Antti Karttunen, Feb 18 2023

For all n >= 1, a(A050376(n)) = n. - Antti Karttunen, Feb 18 2023

Secondary definition added by Antti Karttunen, Feb 18 2023

A249167 a(n) = n if n <= 3, otherwise the smallest number not occurring earlier having at least one common Fermi-Dirac factor with a(n-2), but none with a(n-1).

Original entry on oeis.org

1, 2, 3, 8, 15, 4, 5, 12, 10, 21, 18, 7, 6, 28, 22, 20, 11, 24, 55, 14, 33, 26, 27, 13, 9, 39, 36, 30, 44, 32, 52, 16, 40, 48, 34, 57, 17, 19, 51, 38, 60, 46, 35, 23, 42, 92, 50, 64, 25, 56, 75, 58, 69, 29, 54, 116, 45, 68, 63, 76, 70, 100, 62, 84, 31, 66, 124, 74, 93, 37, 78, 148, 65, 72, 80

Offset: 1

Vladimir Shevelev, Dec 15 2014

nonn

Fermi-Dirac analog of A098550. Recall that every positive digit has a unique Fermi-Dirac representation as a product of distinct terms of A050376.
Conjecture: the sequence is a permutation of the positive integers.
Conjecture is true. The proof is similar to that for A098550 with minor changes. - Vladimir Shevelev, Jan 26 2015
It is interesting that while the first 10000 points (n, A098550(n)) lie on about 8 roughly straight lines, the first 10000 points (n,a(n)) here lie on only about 6 lines (cf. scatterplots of these sequences). - Vladimir Shevelev, Jan 26 2015

			a(4) is not 4, since 2 and 4 have no common Fermi-Dirac divisor; it is not 6, since a(3)=3 and 6 have the common divisor 3. So, a(4)=8, having the Fermi-Dirac representation 8=2*4.

Peter J. C. Moses, Table of n, a(n) for n = 1..10000
David L. Applegate, Hans Havermann, Bob Selcoe, Vladimir Shevelev, N. J. A. Sloane, and Reinhard Zumkeller, The Yellowstone Permutation, arXiv preprint arXiv:1501.01669, 2015 and J. Int. Seq. 18 (2015) 15.6.7..
Index entries for sequences that are permutations of the natural numbers

Cf. A098550, A050376.

Cf. A213925, A255940 (inverse).

import Data.List (delete, intersect)
a249167 n = a249167_list !! (n-1)
a249167_list = 1 : 2 : 3 : f 2 3 [4..] where
   f u v ws = g ws where
     g (x:xs) | null (intersect fdx $ a213925_row u) ||
                not (null $ intersect fdx $ a213925_row v) = g xs
              | otherwise =  x : f v x (delete x ws)
              where fdx = a213925_row x
-- Reinhard Zumkeller, Mar 11 2015

More terms from Peter J. C. Moses, Dec 15 2014

A316271 FDH numbers of strict non-knapsack partitions.

Original entry on oeis.org

24, 40, 70, 84, 120, 126, 135, 168, 198, 210, 216, 220, 231, 264, 270, 280, 286, 312, 330, 351, 360, 364, 378, 384, 408, 416, 420, 440, 456, 462, 504, 520, 528, 540, 544, 546, 552, 560, 576, 594, 600, 616, 630, 640, 646, 660, 663, 680, 696, 702, 728, 744, 748

Offset: 1

Gus Wiseman, Jun 28 2018

nonn

A strict integer partition is knapsack if every subset has a different sum.

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).

			a(1) = 24 is the FDH number of (3,2,1), which is not knapsack because 3 = 2 + 1.

Cf. A000712, A005117, A050376, A056239, A064547, A108917, A213925, A275972, A284640, A299702, A299755, A299757, A301899, A301900.

nn=1000;
sksQ[ptn_]:=And[UnsameQ@@ptn,UnsameQ@@Plus@@@Union[Subsets[ptn]]];
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],!sksQ[FDfactor[#]/.FDrules]&]

A305830 Combined weight of the n-th FDH set-system. Factor n into distinct Fermi-Dirac primes (A050376), normalize by replacing every instance of the k-th Fermi-Dirac prime with k, then add up their FD-weights (A064547).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jun 10 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. Then a(n) = w(s_1) + ... + w(s_k) where w = A064547.

			Sequence of FDH set-systems (a list containing all finite sets of finite sets of positive integers) begins:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{2}}
   5: {{3}}
   6: {{},{1}}
   7: {{4}}
   8: {{},{2}}
   9: {{1,2}}
  10: {{},{3}}
  11: {{5}}
  12: {{1},{2}}
  13: {{1,3}}
  14: {{},{4}}
  15: {{1},{3}}
  16: {{6}}
  17: {{1,4}}
  18: {{},{1,2}}
  19: {{7}}
  20: {{2},{3}}
  21: {{1},{4}}
  22: {{},{5}}
  23: {{2,3}}
  24: {{},{1},{2}}
  25: {{8}}
  26: {{},{1,3}}
  27: {{1},{1,2}}

Cf. A003963, A050376, A056239, A061775, A064547, A213925, A279065, A279614, A299755, A299756, A299757, A302242, A302243, A305829, A305832.

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];
Table[Total[Length/@(FDfactor/@(FDfactor[n]/.FDrules))],{n,nn}]

A181894 Sum of factors from A050376 in Fermi-Dirac representation of n.

Original entry on oeis.org

0, 2, 3, 4, 5, 5, 7, 6, 9, 7, 11, 7, 13, 9, 8, 16, 17, 11, 19, 9, 10, 13, 23, 9, 25, 15, 12, 11, 29, 10, 31, 18, 14, 19, 12, 13, 37, 21, 16, 11, 41, 12, 43, 15, 14, 25, 47, 19, 49, 27, 20, 17, 53, 14, 16, 13, 22, 31, 59, 12, 61, 33, 16, 20, 18, 16, 67, 21, 26

Offset: 1

Vladimir Shevelev, Mar 31 2012

nonn

Fermi-Dirac analog of A008472. Also, since a(q) = q iff q is in A050376, then for n = Product_{q is in A050376} q, we have a(n) = Sum_{q is in A050376} a(q). Therefore, it is natural to call a(n) the Fermi-Dirac integer logarithm of n (Cf. A001414).

			For n = 54, the Fermi-Dirac representation is 54 = 2*3*9, then a(54) = 2+3+9 = 14.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001414, A008472, A050376, A064547, A213925.

a181894 1 = 0
a181894 n = sum $ a213925_row n  -- Reinhard Zumkeller, Mar 20 2013

FermiDiracSum[n_] := Module[{e, ex, p, s}, If[n <= 1, 0, {p, e} = Transpose[FactorInteger[n]]; s = 0; Do[d = IntegerDigits[e[[i]], 2]; ex = DeleteCases[Reverse[2^Range[0, Length[d] - 1]] d, 0]; s = s + Total[p[[i]]^ex], {i, Length[e]}]; s]]; Table[FermiDiracSum[n], {n, 100}] (* T. D. Noe, Apr 05 2012 *)

a(n) = if(n == 1, 0, my(f = factor(n), p = f[, 1], e = f[, 2], s = 0, b); for(i = 1, #p, b = binary(e[i]); for(j = 0, #b-1, if(b[#b-j], s += p[i]^(2^j)))); s); \\ Amiram Eldar, May 02 2025

a(n) = A008472(n) iff n is squarefree; if n is squarefree, then also a(n) = A001414(n), but here conversely, generally speaking, is not true. For example, a(24) = A001414(24). More generally, if n is duplicate or quadruplicate squarefree number, then also a(n) = A001414(n).

For n > 1: a(n) = Sum_{k=1..A064547(n)} A213925(n,k). - Reinhard Zumkeller, Mar 20 2013

cp's OEIS Frontend

A223490 Smallest Fermi-Dirac factor of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A223491 Largest Fermi-Dirac factor of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A299756 Triangle read by rows in which row n is the finite increasing sequence, or set of positive integers, with FDH number n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A299759 Triangle read by rows in which row n lists in order all FDH numbers of strict integer partitions of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A316220 Number of triangles whose weight is the n-th Fermi-Dirac prime in the multiorder of integer partitions of Fermi-Dirac primes into Fermi-Dirac primes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A319826 GCD of the strict integer partition with FDH number n; GCD of the indices (in A050376) of Fermi-Dirac prime factors of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A249167 a(n) = n if n <= 3, otherwise the smallest number not occurring earlier having at least one common Fermi-Dirac factor with a(n-2), but none with a(n-1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Extensions

A316271 FDH numbers of strict non-knapsack partitions.

Views

Author

Keywords