A048107 - OEIS (original) (raw)

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86

COMMENTS

Numbers with at most one 2 and no 3s or higher in their prime exponents. - Charles R Greathouse IV, Aug 25 2016

A disjoint union of A005117 and A060687. The asymptotic density of this sequence is (6/Pi^2) * (1 + Sum_{p prime} 1/(p*(p+1))) = A059956 * (1 + A179119) = A059956 + A271971 = 0.8086828238... - Amiram Eldar, Nov 07 2020

Numbers k such that 2*A325973(k) = A034448(k)+A048250(k) > A000203(k), or equally, numbers k for which A325973(k) > A325974(k). Proof: for squarefree numbers the left hand side = 2*sigma(k). For k in A060687, i.e., when k = p^2 * r, with r squarefree, A034448(k)+A048250(k) = (p^2 + 1)*sigma(r) + (p + 1)*sigma(r) = (p^2 + p + 2)*sigma(r) = sigma(k) + sigma(r) > sigma(k). But for p^e * r, e >= 3, A034448(k)+A048250(k) = (p^e + 1)*sigma(r) + (p + 1)*sigma(r) = (p^e + p + 2)*sigma(r) < sigma(p^e)*sigma(r) = sigma(k). Likewise if the powerful part of k contains at least two distinct prime factors, then also A034448(k)+A048250(k) < A000203(k). - Antti Karttunen, Oct 05 2025

FORMULA

Numbers for which 2^(r(n)+1) > d(n), where r = A001221, d = A000005.

EXAMPLE

n = 420 = 2*2*3*5*7, 4 distinct prime factors, 24 divisors of which 16 are unitary and 8 are not; ud(n) > nud(n) and 2^(4+1) = 32 is larger than d, the number of divisors.

MATHEMATICA

Select[Range[500], 2^(1 + PrimeNu[#]) > DivisorSigma[0, #] &] (* G. C. Greubel, May 05 2017 *)

PROG

(PARI) is(n)=my(f=factor(n)[, 2], t); for(i=1, #f, if(f[i]>1, if(t||f[i]>2, return(0), t=1))); 1 \\ Charles R Greathouse IV, Sep 17 2015

CROSSREFS

Cf. A000005, A001221, A005117, A034444, A034448, A048105, A048250, A059956, A060687, A179119, A271971, A325973, A325974.