A223491 -id:A223491 - cp's OEIS frontend

A302785 Index of the largest Fermi-Dirac factor of n, a(1) = 0 by convention: a(n) = A302778(A223491(n)).

0, 1, 2, 3, 4, 2, 5, 3, 6, 4, 7, 3, 8, 5, 4, 9, 10, 6, 11, 4, 5, 7, 12, 3, 13, 8, 6, 5, 14, 4, 15, 9, 7, 10, 5, 6, 16, 11, 8, 4, 17, 5, 18, 7, 6, 12, 19, 9, 20, 13, 10, 8, 21, 6, 7, 5, 11, 14, 22, 4, 23, 15, 6, 9, 8, 7, 24, 10, 12, 5, 25, 6, 26, 16, 13, 11, 7, 8, 27, 9, 28, 17, 29, 5, 10, 18, 14, 7, 30, 6, 8, 12, 15, 19, 11, 9, 31, 20, 7, 13, 32, 10, 33, 8, 5

Offset: 1

Antti Karttunen, Apr 13 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

A left inverse of A050376.
Cf. A052331, A223491, A240535, A302778, A302786, A302788, A302789 (ordinal transform).
Cf. also A061395.

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));
A302785(n) = if(1==n,0, my(e); fordiv(n, d, if(ispow2(isprimepower(n/d)), e = vecsearch(v050376, n/d); if(e, return(e), print("v050376 too short!"); return(1/0)))));

a(n) = A302778(A223491(n)).
For n > 1, A050376(a(n)) = A223491(n).
For n >= 1, a(A050376(n)) = n.

A064547 Sum of binary digits (or count of 1-bits) in the exponents of the prime factorization of n.

Original entry on oeis.org

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

Offset: 1

Wouter Meeussen, Oct 09 2001

This sequence is different from A058061 for n containing 6th, 8th, ..., k-th powers in its prime decomposition, where k runs through the integers missing from A064548.
For n > 1, n is a product of a(n) distinct members of A050376. - Matthew Vandermast, Jul 13 2004
For n > 1: a(n) = length of n-th row in A213925. - Reinhard Zumkeller, Mar 20 2013
Number of Fermi-Dirac factors of n. - Peter Munn, Dec 27 2019

			For n = 54, n = 2^1 * 3^3 with exponents (1) and (11) in binary, so a(54) = A000120(1) + A000120(3) = 1 + 2 = 3.

Antti Karttunen, Table of n, a(n) for n = 1..32768 (terms 1..2000 from Harry J. Smith)
Index entries for sequences computed from exponents in factorization of n.
Index entries for sequences related to binary expansion of n.

Cf. A000028 (positions of odd terms), A000379 (of even terms).
Cf. A050376 (positions of ones), A268388 (terms larger than ones).
Row lengths of A213925.
A000120, A007814, A028234, A037445, A052331, A064989, A067029, A156552, A223491, A286574 are used in formulas defining this sequence.
Cf. A005117, A058061 (to which A064548 relates), A138302.
Cf. other sequences counting factors of n: A001221, A001222.
Cf. other sequences where a(n) depends only on the prime signature of n: A181819, A267116, A268387.
A003961, A007913, A008833, A059895, A059896, A059897, A225546 are used to express relationship between terms of this sequence.
Cf. A077761, A176699, A037992, A382294.

a064547 1 = 0
a064547 n = length $ a213925_row n  -- Reinhard Zumkeller, Mar 20 2013

expts:=proc(n) local t1,t2,t3,t4,i; if n=1 then RETURN([0]); fi; if isprime(n) then RETURN([1]); fi; t1:=ifactor(n); if nops(factorset(n))=1 then RETURN([op(2,t1)]); fi; t2:=nops(t1); t3:=[]; for i from 1 to t2 do t4:=op(i,t1); if nops(t4) = 1 then t3:=[op(t3),1]; else t3:=[op(t3),op(2,t4)]; fi; od; RETURN(t3); end;
A000120 := proc(n) local w,m,i; w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end:
LamMos:= proc(n) local t1,t2,t3,i; t1:=expts(n); add( A000120(t1[i]),i=1..nops(t1)); end; # N. J. A. Sloane, Dec 20 2007
# alternative Maple program:
A064547:= proc(n) local F;
F:= ifactors(n)[2];
add(convert(convert(f[2],base,2),`+`),f=F)
end proc:
map(A064547,[$1..100]); # Robert Israel, May 17 2016

Table[Plus@@(DigitCount[Last/@FactorInteger[k], 2, 1]), {k, 105}]

a(n) = {my(f = factor(n)[,2]); sum(k=1, #f, hammingweight(f[k]));} \\ Michel Marcus, Feb 10 2016

from sympy import factorint
def wt(n): return bin(n).count("1")
def a(n):
    f=factorint(n)
    return sum([wt(f[i]) for i in f]) # Indranil Ghosh, May 30 2017

;; uses memoizing-macro definec
(definec (A064547 n) (cond ((= 1 n) 0) (else (+ (A000120 (A067029 n)) (A064547 (A028234 n))))))
;; Antti Karttunen, Feb 09 2016

;; uses memoizing-macro definec
(definec (A064547 n) (if (= 1 n) 0 (+ (A000120 (A007814 n)) (A064547 (A064989 n)))))
;; Antti Karttunen, Feb 09 2016

a(m*n) <= a(m)*a(n). - Reinhard Zumkeller, Mar 20 2013
From Antti Karttunen, Feb 09 2016: (Start)
a(1) = 0, and for n > 1, a(n) = A000120(A067029(n)) + a(A028234(n)).
a(1) = 0, and for n > 1, a(n) = A000120(A007814(n)) + a(A064989(n)).
(End)
a(n) = log_2(A037445(n)). - Vladimir Shevelev, May 13 2016
a(n) = A286574(A156552(n)). - Antti Karttunen, May 28 2017
Additive with a(p^e) = A000120(e). - Jianing Song, Jul 28 2018
a(n) = A000120(A052331(n)). - Peter Munn, Aug 26 2019
From Peter Munn, Dec 18 2019: (Start)
a(A000379(n)) mod 2 = 0.
a(A000028(n)) mod 2 = 1.
A001221(n) <= a(n) <= A001222(n).
A001221(n) < a(n) => a(n) < A001222(n).
a(n) = A001222(n) if and only if n is in A005117.
a(n) = A001221(n) if and only if n is in A138302.
a(n^2) = a(n).
a(A003961(n)) = a(n).
a(A225546(n)) = a(n).
a(n) = a(A007913(n)) + a(A008833(n)).
a(A050376(n)) = 1.
a(A059897(n,k)) + 2 * a(A059895(n,k)) = a(n) + a(k).
a(A059896(n,k)) + a(A059895(n,k)) = a(n) + a(k).
Alternative definition: a(1) = 0; a(n * m) = a(n) + 1 for m = A050376(k) > A223491(n).
(End)
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{p prime} f(1/p) = 0.13605447049622836522... (A382294), where f(x) = -x + Sum_{k>=0} x^(2^k)/(1+x^(2^k)). - Amiram Eldar, Sep 28 2023
a(n) << log n/log log n. - Charles R Greathouse IV, Nov 29 2024

A213925 Triangle read by rows: n-th row contains Fermi-Dirac representation of n.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Mar 20 2013

Unique factorization of n into distinct prime powers of form p^(2^k), cf. A050376.

			First rows:
.     1:    1
.     2:    2
.     3:    3
.     4:    4
.     5:    5
.     6:    2  3
.     7:    7
.     8:    2  4                   8 = 2^2^0 * 2^2^1
.     9:    9
.    10:    2  5
.......
.   990:    2   5  9  11
.   991:  991
.   992:    2  16 31             992 = 2^2^0 * 2^2^2 * 31^2^0
.   993:    3 331
.   994:    2   7 71
.   995:    5 199
.   996:    3   4 83
.   997:  997
.   998:    2 499
.   999:    3   9 37             999 = 3^2^0 * 3^2^1 * 37^2^0
.  1000:    2   4  5  25        1000 = 2^2^0 * 2^2^1 * 5^2^0 * 5^2^1 .

Alois P. Heinz, Rows n = 1..8000, flattened (first 1000 rows from Reinhard Zumkeller)
OEIS Wiki, "Fermi-Dirac representation" of n.

Cf. A050376.

For n > 1: A064547 (row lengths), A181894 (row sums), A223490, A223491.

a213925 n k = a213925_row n !! (k-1)
a213925_row 1 = [1]
a213925_row n = reverse $ fd n (reverse $ takeWhile (<= n) a050376_list)
   where fd 1 _      = []
         fd x (q:qs) = if m == 0 then q : fd x' qs else fd x qs
                       where (x',m) = divMod x q
a213925_tabf = map a213925_row [1..]

T:= n-> `if`(n=1, [1], sort([seq((l-> seq(`if`(l[j]=1, i[1]^(2^(j-1)), [][]),
             j=1..nops(l)))(convert(i[2], base, 2)), i=ifactors(n)[2])]))[]:
seq(T(n), n=1..60);  # Alois P. Heinz, Feb 20 2018

nmax = 50; FDPrimes = Reap[k = 1; While[lim = nmax^(1/k); lim > 2, Sow[Prime[Range[PrimePi[lim]]]^k]; k = 2 k]][[2, 1]] // Flatten // Union;
f[1] = 1; f[n_] := Reap[m = n; Do[If[m == 1, Break[], If[Divisible[m, p], m = m/p; Sow[p]]], {p, Reverse[FDPrimes]}]][[2, 1]] // Reverse;
Array[f, nmax] // Flatten (* Jean-François Alcover, Feb 05 2019 *)

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

Product_{k=1..A064547(n)} T(n,k) = n.

Example corrected (row 992) by Reinhard Zumkeller, Mar 11 2015

A223490 Smallest Fermi-Dirac factor of n.

Original entry on oeis.org

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)

A302789 Number of times the largest Fermi-Dirac factor of n is the largest Fermi-Dirac factor for numbers <= n; a(1) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 3, 1, 1, 2, 1, 4, 3, 2, 1, 4, 1, 2, 3, 4, 1, 5, 1, 2, 3, 2, 5, 4, 1, 2, 3, 6, 1, 6, 1, 4, 5, 2, 1, 3, 1, 2, 3, 4, 1, 6, 5, 7, 3, 2, 1, 7, 1, 2, 7, 4, 5, 6, 1, 4, 3, 8, 1, 8, 1, 2, 3, 4, 7, 6, 1, 5, 1, 2, 1, 9, 5, 2, 3, 8, 1, 9, 7, 4, 3, 2, 5, 6, 1, 2, 9, 4, 1, 6, 1, 8, 10

Offset: 1

Antti Karttunen, Apr 13 2018

nonn

Ordinal transform of A223491, or equally, of A302785.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A050376, A052331, A223491, A302785, A302788.
Cf. A084400 (gives the positions of 1's).
Cf. also A078899.

f[n_] := Max@Table[{p, e} = pe; p^(2^(Length[IntegerDigits[e, 2]]-1)), {pe, FactorInteger[n]}];
b[_] = 1;
a[n_] := a[n] = With[{t = f[n]}, b[t]++];
Array[a, 105] (* Jean-François Alcover, Dec 18 2021 *)

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
ispow2(n) = (n && !bitand(n, n-1));
A223491(n) = if(1==n,n,fordiv(n, d, if(ispow2(isprimepower(n/d)), return(n/d))));
v302789 = ordinal_transform(vector(up_to,n,A223491(n)));
A302789(n) = v302789[n];

A302776 a(1) = 1; for n>1, a(n) = n/(largest Fermi-Dirac factor of n).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 3, 1, 1, 2, 1, 4, 3, 2, 1, 6, 1, 2, 3, 4, 1, 6, 1, 2, 3, 2, 5, 4, 1, 2, 3, 8, 1, 6, 1, 4, 5, 2, 1, 3, 1, 2, 3, 4, 1, 6, 5, 8, 3, 2, 1, 12, 1, 2, 7, 4, 5, 6, 1, 4, 3, 10, 1, 8, 1, 2, 3, 4, 7, 6, 1, 5, 1, 2, 1, 12, 5, 2, 3, 8, 1, 10, 7, 4, 3, 2, 5, 6, 1, 2, 9, 4, 1, 6, 1, 8, 15

Offset: 1

Antti Karttunen, Apr 13 2018

nonn

For n > 1, a(n) = the smallest positive number d such that n/d is a "Fermi-Dirac prime", a term of A050376.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A050376, A052331, A223491, A302023, A302024, A302777, A302792.
Cf. A084400 (gives the positions of 1's).
Cf. also A052126, A284600.

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

A209229(n) = (n && !bitand(n,n-1));
A302777(n) = A209229(isprimepower(n));
A302776(n) = if(1==n,n,fordiv(n, d, if(A302777(n/d), return(d))));

a(n) = n / A223491(n).

a(n) = A302023(A052126(A302024(n))).

A346088 Smallest divisor d of n for which A002034(d) = A002034(n), where A002034(n) is the smallest positive integer k such that k! is a multiple 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, 27, 7, 29, 5, 31, 32, 11, 17, 7, 9, 37, 19, 13, 5, 41, 7, 43, 11, 9, 23, 47, 16, 49, 25, 17, 13, 53, 27, 11, 7, 19, 29, 59, 5, 61, 31, 7, 32, 13, 11, 67, 17, 23, 7, 71, 9, 73, 37, 25, 19, 11, 13, 79, 16, 27, 41, 83, 7, 17, 43, 29, 11, 89

Offset: 1

Antti Karttunen, Jul 05 2021

nonn

			36 has 9 divisors: 1, 2, 3, 4, 6, 9, 12, 18, 36. When A002034 is applied to them, one obtains values [1, 2, 3, 4, 3, 6, 4, 6, 6], thus there are three divisors that obtain the maximal value 6 obtained at 36 itself, of which divisor 9 is the smallest, and therefore a(36) = 9.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A002034, A345935, A345936, A346089.
Cf. also A344758.
Differs from A223491 for the first time at n=27, where a(27) = 27, while A223491(27) = 9.

A002034(n) = if(1==n,n,my(s=factor(n)[, 1], k=s[#s], f=Mod(k!, n)); while(f, f*=k++); (k)); \\ After code in A002034.
A346088(n) = { my(x=A002034(n)); fordiv(n,d,if(A002034(d)==x, return(d))); };

a(n) = n / A346089(n).

cp's OEIS Frontend

A302785 Index of the largest Fermi-Dirac factor of n, a(1) = 0 by convention: a(n) = A302778(A223491(n)).

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A064547 Sum of binary digits (or count of 1-bits) in the exponents of the prime factorization of n.

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A213925 Triangle read by rows: n-th row contains Fermi-Dirac representation of n.

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A223490 Smallest Fermi-Dirac factor of n.

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A302789 Number of times the largest Fermi-Dirac factor of n is the largest Fermi-Dirac factor for numbers <= n; a(1) = 1.

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A302776 a(1) = 1; for n>1, a(n) = n/(largest Fermi-Dirac factor of n).

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A346088 Smallest divisor d of n for which A002034(d) = A002034(n), where A002034(n) is the smallest positive integer k such that k! is a multiple of n.

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