A084400 -id:A084400 - cp's OEIS frontend

A302788 Number of times the smallest Fermi-Dirac factor of n is the smallest Fermi-Dirac factor for numbers <= n; a(1) = 1.

1, 1, 1, 1, 1, 2, 1, 3, 1, 4, 1, 2, 1, 5, 3, 1, 1, 6, 1, 2, 4, 7, 1, 8, 1, 9, 5, 3, 1, 10, 1, 11, 6, 12, 2, 4, 1, 13, 7, 14, 1, 15, 1, 5, 3, 16, 1, 8, 1, 17, 9, 6, 1, 18, 4, 19, 10, 20, 1, 11, 1, 21, 2, 7, 5, 22, 1, 8, 12, 23, 1, 24, 1, 25, 13, 9, 3, 26, 1, 6, 1, 27, 1, 14, 7, 28, 15, 29, 1, 30, 4, 10, 16, 31, 8, 32, 1, 33, 2, 11, 1, 34, 1, 35, 17

Offset: 1

Antti Karttunen, Apr 13 2018

nonn

Ordinal transform of A223490, or equally, of A302786.

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

Cf. A001511, A050376, A052331, A223490, A302786, A302789.
Cf. A084400 (gives the positions of 1's).
Cf. also A078898.

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; };
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);
A302786(n) = if(1==n, 0, A001511(A052331(n)));
v302788 = ordinal_transform(vector(up_to,n,A302786(n)));
A302788(n) = v302788[n];

A302792 a(1) = 1; for n>1, a(n) = n/(smallest Fermi-Dirac factor of n).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 4, 1, 5, 1, 4, 1, 7, 5, 1, 1, 9, 1, 5, 7, 11, 1, 12, 1, 13, 9, 7, 1, 15, 1, 16, 11, 17, 7, 9, 1, 19, 13, 20, 1, 21, 1, 11, 9, 23, 1, 16, 1, 25, 17, 13, 1, 27, 11, 28, 19, 29, 1, 20, 1, 31, 9, 16, 13, 33, 1, 17, 23, 35, 1, 36, 1, 37, 25, 19, 11, 39, 1, 16, 1, 41, 1, 28, 17, 43, 29, 44, 1, 45, 13, 23, 31, 47, 19, 48, 1

Offset: 1

Antti Karttunen, Apr 13 2018

nonn

The positive integers that are absent from this sequence are A036554, integers that have 2 as a Fermi-Dirac factor. - Peter Munn, Apr 23 2018

a(n) is the largest aliquot infinitary divisor of n, for n > 1 (cf. A077609). - Amiram Eldar, Nov 19 2022

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

Cf. A001511, A036554, A050376, A052331, A077609, A223490, A302023, A302024, A302776, A300841.
Cf. A084400 (gives the positions of 1's).
Cf. also A032742.

f[p_, e_] := p^(2^IntegerExponent[e, 2]); a[n_] := n / Min @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 27 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))));
A302792(n) = (n/A223490(n));

a(n) = {if(n==1, 1, my(f = factor(n)); for(i=1, #f~, f[i,1] = f[i,1]^(1<Amiram Eldar, Nov 19 2022

a(n) = n / A223490(n).

a(n) = A300841(A302023(A032742(A302024(n)))).

A340088 a(n) = A091732(n) / gcd(n-1, A091732(n)), where A091732 is an infinitary analog of Euler's phi function.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 3, 1, 4, 1, 6, 1, 6, 4, 1, 1, 8, 1, 12, 3, 10, 1, 6, 1, 12, 8, 2, 1, 8, 1, 15, 5, 16, 12, 24, 1, 18, 12, 4, 1, 12, 1, 30, 8, 22, 1, 30, 1, 24, 16, 12, 1, 16, 20, 18, 9, 28, 1, 24, 1, 30, 24, 5, 3, 4, 1, 48, 11, 8, 1, 24, 1, 36, 24, 18, 15, 24, 1, 60, 1, 40, 1, 36, 16, 42, 28, 10, 1, 32, 4, 66

Offset: 1

Antti Karttunen, Dec 31 2020

nonn

Conjecture: a(n) = 1 iff n = 1 or in A050376. This is an infinitary analog of Lehmer's totient conjecture from 1932.

For all i, j > 1: a(i) = a(j) => A302777(i) = A302777(j), if the above conjecture holds.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Wikipedia, Lehmer's totient problem

Cf. A050376, A084400, A091732, A302777, A340087, A340089.

Cf. also A160595.

ispow2(n) = (n && !bitand(n,n-1));
A302777(n) = ispow2(isprimepower(n));
A091732(n) = { my(m=1); while(n > 1, fordiv(n, d, if((dA302777(n/d), m *= ((n/d)-1); n = d; break))); (m); };
A340088(n) = { my(x=A091732(n)); (x/gcd(n-1, x)); };

a(n) = A091732(n) / A340087(n) = A091732(n) / gcd(n-1, A091732(n)).

For all n >= 1, a(A084400(n)) = 1.

A219964 a(n) = product(i >= 0, (P(n, i)/P(n-1, i))^(2^i)) where P(n, i) = product(p prime, n/2^(i+1) < p <= n/2^i).

Original entry on oeis.org

1, 1, 2, 3, 2, 5, 3, 7, 4, 1, 5, 11, 9, 13, 7, 1, 16, 17, 1, 19, 25, 1, 11, 23, 81, 1, 13, 1, 49, 29, 1, 31, 256, 1, 17, 1, 1, 37, 19, 1, 625, 41, 1, 43, 121, 1, 23, 47, 6561, 1, 1, 1, 169, 53, 1, 1, 2401, 1, 29, 59, 1, 61, 31, 1, 65536, 1, 1, 67, 289, 1, 1, 71

Offset: 0

Peter Luschny and Arie Groeneveld, Mar 30 2013

nonn

a(n) is 1 or a prime or an even power of a prime (A084400, A050376).

If n > 0 then a(n) = 1 if and only if n is an element of A110473.

			a(20) = (7/(5*7))^2*((3*5)/3)^4 = 25.
a(22) = ((13*17*19)/(11*13*17*19))*((7*11)/7)^2 = 11.

Peter Luschny, Table of n, a(n) for n = 0..1000

Cf. A220027, the partial products of a(n).

genSeq=: 3 :0
p=. x: i.&.(_1&p:) y1=.y+1
i=.(#~y1>])&.> <:@((i.@>.&.(2&^.)y1)*])&.> p
y{.(;p(^2x^0,i.@<:@#)&.>i) (;i) } y1$1
)

A219964 := proc(n) local l, m, z;
if isprime(n) then RETURN(n) fi;
z := 1; l := n - 1; m := n;
do l := iquo(l, 2); m := iquo(m, 2);
   if l = 0 then break fi;
   if l < m then if isprime(l+1) then RETURN((l+1)^z) fi fi;
   z := z + z;
od; 1 end:  seq(A219964(k), k=0..71);

a[n_] := Module[{l, m, z}, If[PrimeQ[n] , Return[n] ]; z = 1; l = Max[0, n - 1]; m = n; While[True, l = Quotient[l, 2]; m = Quotient[m, 2]; If[l == 0 , Break[]]; If[l < m , If[ PrimeQ[l+1], Return[(l+1)^z]]]; z = z+z]; 1]; Table[a[k], {k, 0, 71}] (* Jean-François Alcover, Jan 15 2014, after Maple *)

def A219964(n):
    if is_prime(n): return n
    z = 1; l = max(0,n-1); m = n
    while true:
        l = l // 2
        m = m // 2
        if l == 0: break
        if l < m:
            if is_prime(l+1): return (l+1)^z
        z = z + z
    return 1
[A219964(n) for n in (0..71)]

a(n) = A220027(n) / A220027(n-1).

cp's OEIS Frontend

A302788 Number of times the smallest Fermi-Dirac factor of n is the smallest Fermi-Dirac factor for numbers <= n; a(1) = 1.

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A302792 a(1) = 1; for n>1, a(n) = n/(smallest Fermi-Dirac factor of n).

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A340088 a(n) = A091732(n) / gcd(n-1, A091732(n)), where A091732 is an infinitary analog of Euler's phi function.

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A219964 a(n) = product(i >= 0, (P(n, i)/P(n-1, i))^(2^i)) where P(n, i) = product(p prime, n/2^(i+1) < p <= n/2^i).

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