A086263 -id:A086263 - cp's OEIS frontend

A343819 Numbers k such that k and k+1 have the same number of Fermi-Dirac factors (A064547).

2, 3, 4, 14, 16, 20, 21, 26, 27, 32, 33, 34, 35, 38, 44, 45, 50, 51, 57, 62, 63, 64, 68, 74, 75, 76, 85, 86, 91, 92, 93, 94, 98, 99, 104, 111, 115, 116, 117, 118, 122, 123, 124, 133, 135, 141, 142, 143, 144, 145, 146, 147, 158, 161, 171, 175, 176, 177, 187, 189

Offset: 1

Amiram Eldar, Apr 30 2021

nonn

Since the number of infinitary divisors of k is A037445(k) = 2^A064547(k), this is also the sequence of numbers k such that k and k+1 have the same number of infinitary divisors.

			2 is a term since A064547(2) = A064547(3) = 1.

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

Cf. A064547, A306985, A343818.
Similar sequences: A005237, A006049.
Subsequence of A086263.

fd[1] = 0; fd[n_] := Plus @@ DigitCount[FactorInteger[n][[;;,2]], 2, 1]; Select[Range[200], fd[#] == fd[# + 1] &]

A263990 Nonsquare numbers k such that k and k+1 are semiprimes.

Original entry on oeis.org

14, 21, 33, 34, 38, 57, 85, 86, 93, 94, 118, 122, 133, 141, 142, 145, 158, 177, 201, 202, 205, 213, 214, 217, 218, 253, 298, 301, 302, 326, 334, 381, 393, 394, 445, 446, 453, 481, 501, 514, 526, 537, 542, 553, 565, 622, 633, 634, 694, 697, 698, 706, 717, 745, 766, 778, 793, 802, 817, 842, 865, 878

Offset: 1

Zak Seidov, Oct 31 2015

nonn

If k and k+1 are semiprimes then k+1 is always nonsquare while k can be a square (see A263951). The sequence gives the nonsquare terms of A070552. Each of the numbers k and k+1 is a product of two distinct primes.
Numbers that are terms in A070552 but not in A263951.
The subsequence of triples of consecutive squarefree semiprimes is A039833. - R. J. Mathar, Aug 13 2019

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Subsequence of A070552, A086263.

Cf. A006881, A039833, A109288, A263951.

Select[Range[1000], ! IntegerQ[Sqrt[#]] && 2 == PrimeOmega[#] == PrimeOmega[# + 1] &]

is(n)=if(n%2, isprime((n+1)/2) && bigomega(n)==2 && !isprimepower(n), isprime(n/2) && bigomega(n+1)==2) \\ Charles R Greathouse IV, Apr 25 2016

a(n) = A109288(n) - 1. - Amiram Eldar, Aug 08 2025

A075039 Smallest of three consecutive squarefree numbers having equal numbers of prime factors.

Original entry on oeis.org

33, 85, 93, 141, 201, 213, 217, 301, 393, 445, 633, 697, 921, 1041, 1137, 1261, 1309, 1345, 1401, 1641, 1761, 1837, 1885, 1893, 1941, 1981, 2013, 2101, 2181, 2217, 2305, 2361, 2433, 2461, 2517, 2641, 2665, 2721, 2733, 3097, 3385, 3601, 3693, 3729, 3865

Offset: 1

Amarnath Murthy, Sep 03 2002

nonn

			33 is a member as 33, 34 and 35 are of the form p*q where p and q are primes.

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

Cf. A001221, A001222, A005117, A052214, A086263, A039833.

Select[Range[4000], AllTrue[# + Range[0, 2], SquareFreeQ] && Equal @@ PrimeNu[# + Range[0, 2]] &] (* Amiram Eldar, Feb 24 2021 *)

A001221(a(n)) = A001222(a(n)) = A001221(a(n)+1) = A001222(a(n)+1).

More terms from Matthew Conroy, Sep 08 2002
Edited by Reinhard Zumkeller, Jul 14 2003
Offset corrected by Amiram Eldar, Feb 24 2021

A124300 Greatest x such that mu(n+k) = mu(n) for 0<=k<=x, where mu=A008683.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Oct 25 2006

nonn

a(n+1) = a(n) - 1 if a(n) > 0;
a(A124301(n)) = n and a(m) < n for m < A124301(n);
a(A068781(n)) > 0; a(A086263(n)) > 0.

R. Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Moebius Function

cp's OEIS Frontend

A343819 Numbers k such that k and k+1 have the same number of Fermi-Dirac factors (A064547).

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A263990 Nonsquare numbers k such that k and k+1 are semiprimes.

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A075039 Smallest of three consecutive squarefree numbers having equal numbers of prime factors.

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A124300 Greatest x such that mu(n+k) = mu(n) for 0<=k<=x, where mu=A008683.

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