A335428 -id:A335428 - cp's OEIS frontend

A368781 The maximal exponent in the unique factorization of n in terms of distinct "Fermi-Dirac primes".

0, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Jan 05 2024

First differs from A335428 at n = 36. Differs from A050377, A344417 and A347437 at n = 1 and then at n = 36.

In the unique factorization of n in terms of distinct "Fermi-Dirac primes", n is represented as a product of prime powers (A246655) whose exponents are powers of 2 (A000079). a(n) is the maximal exponent of these prime powers (and not the maximal exponent of the exponents that are powers of 2). Thus, a(n) is a power of 2 for n >= 2.

			For n = 972 = 2^2 * 3^5, the unique factorization of 972 in terms of distinct "Fermi-Dirac primes" is 2^(2^1) * 3^(2^0) * 3^(2^2). Therefore, a(972) = 2^2 = 4.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wikipedia, Fermi-Dirac prime.

Cf. A000079, A050376, A051903, A053644, A182979, A246655, A299090, A353897.

Cf. A050377, A335428, A344417, A347437.

a[n_] := 2^Floor[Log2[Max[FactorInteger[n][[;; , 2]]]]]; a[1] = 0; Array[a, 100]

a(n) = if(n > 1, 2^exponent(vecmax(factor(n)[, 2])), 0);

from sympy import factorint
def A368781(n): return 1<1 else 0 # Chai Wah Wu, Apr 11 2025

a(n) = A053644(A051903(n)).
a(n) = 2^(A299090(n)-1) for n >= 2.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + Sum_{k>=1} (2^(k-1) * (1 - 1/zeta(2^k))) = 1.56056154773294953123... .
a(n) = A051903(A353897(n)). - Amiram Eldar, May 07 2024

A335426 a(1) = 0; thereafter a(2^(2^k)) = 0 for k > 0, and for other even numbers n, a(n) = 1+a(n/2), and for odd numbers n, a(n) = 2*a(A064989(n)).

Original entry on oeis.org

0, 1, 2, 0, 4, 3, 8, 1, 0, 5, 16, 4, 32, 9, 6, 0, 64, 1, 128, 6, 10, 17, 256, 5, 0, 33, 2, 10, 512, 7, 1024, 1, 18, 65, 12, 2, 2048, 129, 34, 7, 4096, 11, 8192, 18, 8, 257, 16384, 6, 0, 1, 66, 34, 32768, 3, 20, 11, 130, 513, 65536, 8, 131072, 1025, 12, 2, 36, 19, 262144, 66, 258, 13, 524288, 3, 1048576, 2049, 2, 130, 24, 35, 2097152, 8, 0, 4097

Offset: 1

Antti Karttunen, Jun 15 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences computed from indices in prime factorization

Cf. A001146, A003961, A048675, A064989, A209229, A225546, A248663, A335427, A335428.

Cf. A082522 (gives indices of zeros after a(1)=0).

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A209229(n) = (n && !bitand(n,n-1));
A335426(n) = if(n<=2,n-1, if(!(n%2),if(A209229(n) && A209229(valuation(n,2)),0,1+A335426(n/2)), 2*A335426(A064989(n))));
\\ Alternatively:
A335426(n) = if(1==n,0, my(e=isprimepower(n)); if(e>1 && A209229(e), 0, if(!(n%2),1+A335426(n/2), 2*A335426(A064989(n)))));

a(1) = 0, and then after, a(2^(2^k)) = 0 for k > 0, and for other even numbers n, a(n) = 1+a(n/2), and for odd numbers n, a(n) = 2*a(A064989(n)).
a(n) = A335427(A225546(n)).
a(A003961(n)) = 2 * a(n).

cp's OEIS Frontend

A368781 The maximal exponent in the unique factorization of n in terms of distinct "Fermi-Dirac primes".

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