A050376 -id:A050376 - cp's OEIS frontend

A334872 Number of steps needed to reach either 1 or one of the "Fermi-Dirac primes" (A050376) when starting from n and iterating with A334870.

0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 2, 0, 0, 1, 0, 4, 2, 1, 0, 3, 0, 1, 2, 8, 0, 3, 0, 1, 2, 1, 4, 2, 0, 1, 2, 5, 0, 3, 0, 16, 4, 1, 0, 2, 0, 1, 2, 32, 0, 3, 4, 9, 2, 1, 0, 6, 0, 1, 8, 2, 4, 3, 0, 64, 2, 5, 0, 3, 0, 1, 2, 128, 8, 3, 0, 4, 0, 1, 0, 10, 4, 1, 2, 17, 0, 5, 8, 256, 2, 1, 4, 3, 0, 1, 16, 2, 0, 3, 0, 33, 6

Offset: 1

Antti Karttunen, Jun 08 2020

nonn

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

Cf. A050376 (positions of zeros after 1), A302777, A334859, A334865, A334870, A334871.

A209229(n) = (n && !bitand(n,n-1));
A302777(n) = A209229(isprimepower(n));
A334870(n) = if(issquare(n),sqrtint(n),my(c=core(n), m=n); forprime(p=2, , if(!(c % p), m/=p; break, m*=p)); (m));
A334872(n) = { my(s=0); while(n>1 && !A302777(n), s++; n = A334870(n)); (s); };

\\ Much faster, A302777 like in above:
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A052126(n) = if(1==n,n,(n/vecmax(factor(n)[, 1])));
A334872(n) = { my(s=0); while(n>1 && !A302777(n), if(issquarefree(n), return(s+A048675(A052126(n)))); if(issquare(n), s++; n = sqrtint(n), s += A048675(core(n)); n /= core(n))); (s); };

If n = 1 or A302777(n) = 1, a(n) = 0, otherwise a(n) = 1 + a(A334870(n)).

For all n >= 1, a(n) <= A334871(n).

A304536 Binary weight of terms of A304533; Number of terms of A050376 in "Fermi-Dirac factorization" of A304531(1+n).

Original entry on oeis.org

0, 1, 2, 1, 2, 1, 2, 1, 2, 3, 1, 2, 3, 2, 3, 2, 3, 2, 4, 2, 3, 4, 3, 5, 2, 3, 4, 3, 4, 3, 4, 1, 2, 3, 2, 3, 2, 3, 2, 3, 4, 1, 2, 3, 2, 3, 2, 3, 2, 3, 4, 2, 3, 4, 3, 4, 3, 4, 3, 5, 3, 4, 5, 4, 6, 3, 4, 5, 4, 5, 4, 5, 2, 3, 4, 3, 4, 3, 4, 3, 5, 3, 4, 3, 4, 5, 4, 6, 4, 5, 3, 4, 5, 4, 5, 1, 2, 3, 2, 3, 2, 3, 2, 3, 4, 2

Offset: 0

Antti Karttunen, May 14 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 0..74431
Index entries for sequences related to binary expansion of n

Cf. A000120, A050376, A064547, A304531, A304533, A304535.

PARI
```
A304536(n) = hammingweight(A304533(n));
```

a(n) = A000120(A304533(n)).

a(n) = A064547(A304531(1+n)).

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}]

A322822 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(2) = -1, f(n) = 0 if n is a Fermi-Dirac prime (A050376) > 2, and f(n) = A300840(n) for all other numbers.

Original entry on oeis.org

1, 2, 3, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 9, 3, 3, 10, 3, 11, 12, 13, 3, 7, 3, 14, 15, 16, 3, 9, 3, 17, 18, 19, 20, 21, 3, 22, 23, 11, 3, 12, 3, 24, 25, 26, 3, 27, 3, 28, 29, 30, 3, 15, 31, 16, 32, 33, 3, 34, 3, 35, 36, 37, 38, 18, 3, 39, 40, 20, 3, 21, 3, 41, 42, 43, 44, 23, 3, 45, 3, 46, 3, 47, 48, 49, 50, 24, 3, 25, 51, 52, 53, 54, 55, 27, 3, 56, 57, 58, 3, 29, 3, 30

Offset: 1

Antti Karttunen, Dec 29 2018

nonn

For all i, j: a(i) = a(j) => A322823(i) = A322823(j).

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

Cf. A050376, A052330, A052331, A302777, A300840, A322823.

Cf. also A322805, A322807.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
ispow2(n) = (n && !bitand(n,n-1));
A302777(n) = ispow2(isprimepower(n));
A050376list(up_to) = { my(v=vector(up_to), i=0); for(n=1,oo,if(A302777(n), i++; v[i] = n); if(i == up_to,return(v))); };
v050376 = A050376list(up_to);
A050376(n) = v050376[n];
A052330(n) = { my(p=1,i=1); while(n>0, if(n%2, p *= A050376(i)); i++; n >>= 1); (p); };
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); };
A300840(n) = A052330(A052331(n)>>1);
A322822aux(n) = if((2==n),-1,if(A302777(n),0,A300840(n)));
v322822 = rgs_transform(vector(up_to,n,A322822aux(n)));
A322822(n) = v322822[n];

A366074 The number of "Fermi-Dirac primes" (A050376) that are unitary divisors of n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Sep 28 2023

First differs from A293439 at n = 128.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Unitary Divisor.

Cf. A050376, A064547, A077610, A077761, A209229, A366073.

f[p_, e_] := If[e == 2^IntegerExponent[e, 2], 1, 0]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecsum(apply(x -> (x == 1 << valuation(x, 2)), factor(n)[, 2]));

Additive with a(p^e) = A209229(e).

Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = -P(2) + Sum_{k>=1} (P(2^k) - P(2^k+1)) = -0.13145993422430119364..., where P(s) is the prime zeta function.

A366246 The number of infinitary divisors of n that are "Fermi-Dirac primes" (A050376) and terms of A366242.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Oct 05 2023

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

Cf. A050376, A064547, A077761, A139351, A366242, A366243, A366247.

s[0] = 0; s[n_] := s[n] = s[Floor[n/4]] + If[OddQ[Mod[n, 4]], 1, 0]; f[p_, e_] := s[e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

s(e) = if(e>3, s(e\4)) + e%2 \\ after Charles R Greathouse IV at A139351
a(n) = vecsum(apply(s, factor(n)[, 2]));

Additive with a(p^e) = A139351(e).
a(n) = A064547(n) - A366247(n).
a(n) = A064547(A366244(n)).
a(n) >= 0, with equality if and only if n is in A366243.
a(n) <= A064547(n), with equality if and only if n is in A366242.
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) = -0.25705126777012995187..., where f(x) = - x + Sum_{k>=0} (x^(4^k)/(1+x^(4^k))).

A050378 Number of factorizations into members of A050376 by prime signature. A050377(A025487).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 4, 2, 1, 4, 4, 4, 2, 6, 4, 4, 2, 6, 8, 4, 6, 1, 4, 4, 10, 8, 4, 6, 2, 8, 4, 10, 12, 8, 10, 2, 8, 8, 6, 14, 4, 12, 4, 16, 8, 10, 4, 12, 8, 6, 14, 8, 20, 1, 4, 16, 12, 14, 4, 12, 16, 10, 20, 8, 20, 2, 8, 24, 8, 12, 14, 8, 16, 6, 20, 16, 4, 10

Offset: 1

Christian G. Bower, Nov 15 1999

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A025487, A050376, A050377.

More terms from Amiram Eldar, Jan 29 2019

A050380 Number of ordered factorizations into terms of A050376 by prime signature.

Original entry on oeis.org

1, 1, 2, 2, 3, 5, 6, 10, 6, 10, 14, 22, 18, 18, 32, 44, 44, 31, 76, 60, 89, 24, 84, 108, 56, 167, 164, 174, 84, 217, 246, 98, 364, 435, 342, 240, 522, 222, 550, 174, 500, 768, 324, 598, 1074, 660, 654, 1224, 674, 1188, 306, 1434, 1608, 120

Offset: 1

Christian G. Bower, Nov 15 1999

nonn

Amiram Eldar, Table of n, a(n) for n = 1..400 (terms 1..124 from R. J. Mathar)

Cf. A050376, A050379, A025487.

a(n) = A050379(A025487(n)).

A176699 Fermi-Dirac composite numbers that are not a sum of two Fermi-Dirac primes (A050376).

Original entry on oeis.org

145, 187, 205, 217, 219, 221, 247, 301, 325, 343, 415, 427, 475, 517, 535, 553, 555, 583, 637, 667, 671, 697, 715, 781, 783, 793, 795, 805, 807, 817, 835, 847, 851, 871, 895, 901, 905, 925, 959, 1003, 1005, 1027, 1045, 1057, 1059, 1075, 1081, 1135, 1141, 1147

Offset: 1

Vladimir Shevelev, Apr 24 2010, Apr 26 2010

nonn

We define a Fermi-Dirac composite number as a positive integer with at least two factors in its factorization over distinct terms of A050376.

They are those c for which A064547(c) >= 2, namely c= 6, 8, 10, 12,..., 62, 63, 64, 65, ..., or the complement of A050376 with respect to the natural numbers > 1.

			291 = 3*97 is a Fermi-Dirac composite number, equal to 289+2, the sum of two Fermi-Dirac primes. Therefore 291 is not in the sequence.

Vladimir S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature Sciences 4 (1996), 28-43.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Simon Litsyn and Vladimir Shevelev, On factorization of integers with restrictions on the exponents, INTEGERS: El. J. of Combin. Number Theory, 7 (2007), Article #A33, 1-35.

Cf. A050376, A025583, A164376.

A064547 := proc(n) f := ifactors(n)[2] ; a := 0 ; for p in f do a := a+wt(op(2, p)) ; end do: a ; end proc:
A050376 := proc(n) local a; if n = 1 then 2; else for a from procname(n-1)+1 do if A064547(a) = 1 then return a; end if; end do: end if; end proc:
isA176699 := proc(n) local pi,q ; if A064547(n) < 2 then return false; end if; for pi from 1 do if A050376(pi) > n then return true; else q := n-A050376(pi) ; if A064547(q) = 1 then return false; end if; end if; end do; end proc:
for n from 2 to 1000 do if isA176699(n) then printf("%d,\n",n) ; end if; end do: # R. J. Mathar, Jun 16 2010

pow2Q[n_] := n == 2^IntegerExponent[n, 2]; fdpQ[n_] := PrimePowerQ[n] && pow2Q[FactorInteger[n][[1, 2]]]; With[{m = 1200}, p = Select[Range[m], fdpQ]; Complement[Range[m], Join[{1}, p, Plus @@@ Subsets[p, {2}]]]] (* Amiram Eldar, Oct 05 2023 *)

Edited and extended by R. J. Mathar, Jun 16 2010

More terms from Amiram Eldar, Oct 05 2023

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

A334872 Number of steps needed to reach either 1 or one of the "Fermi-Dirac primes" (A050376) when starting from n and iterating with A334870.

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A304536 Binary weight of terms of A304533; Number of terms of A050376 in "Fermi-Dirac factorization" of A304531(1+n).

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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).

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A322822 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(2) = -1, f(n) = 0 if n is a Fermi-Dirac prime (A050376) > 2, and f(n) = A300840(n) for all other numbers.

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A366074 The number of "Fermi-Dirac primes" (A050376) that are unitary divisors of n.

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A366246 The number of infinitary divisors of n that are "Fermi-Dirac primes" (A050376) and terms of A366242.

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A050378 Number of factorizations into members of A050376 by prime signature. A050377(A025487).

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A050380 Number of ordered factorizations into terms of A050376 by prime signature.

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A176699 Fermi-Dirac composite numbers that are not a sum of two Fermi-Dirac primes (A050376).

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