A377369 -id:A377369 - cp's OEIS frontend

A379537 Frugal numbers in base 2: numbers k such that A377369(k) < A070939(k).

1, 27, 32, 49, 64, 81, 121, 125, 128, 135, 147, 162, 169, 189, 192, 243, 250, 256, 289, 297, 320, 338, 343, 351, 361, 363, 375, 384, 405, 448, 486, 507, 512, 513, 529, 539, 567, 576, 578, 605, 621, 625, 637, 640, 648, 675, 686, 704, 722, 729, 750, 768, 783, 832

Offset: 1

Paolo Xausa, Dec 25 2024

A frugal number in base 2 is a number with more bits than the total number of bits of its prime factorization (including exponents > 1).
Following the definition by Pinch (1998), 1 is considered a frugal number.
Some authors call these numbers "economical numbers", as in A046759 which, according to the definition provided here, lists frugal numbers in base 10 (additionally, A046759 does not include 1).

			32 is a term because 32 = 2^5 = 10_2^101_2; the total number of bits of (10_2, 101_2) = 5 < the number of bits of 32 = 100000_2 (6).
135 is a term because 135 = 3^3*5 = 11_2^11_2*101_2; the total number of bits of (11_2, 11_2, 101_2) = 7 < the number of bits of 135 = 10000111_2 (8).

Paolo Xausa, Table of n, a(n) for n = 1..10000
Richard G. E. Pinch, Economical numbers, arXiv:math/9802046 [math.NT], 1998.
Wikipedia, Frugal number.

Row n = 2 of A379538.

Cf. A046759, A070939, A377369, A379373.

A379537Q[k_] := Total[BitLength[Select[Flatten[FactorInteger[k]], # > 1 &]]] < BitLength[k];
Select[Range[1000], A379537Q]

A050252 Number of digits in the prime factorization of n (counting terms of the form p^1 as p).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 2, 3, 3, 2, 4, 2, 3, 3, 3, 2, 3, 2, 4, 3, 3, 2, 3, 2, 3, 3, 4, 2, 3, 3, 3, 3, 3, 2, 4, 2, 3, 3, 2, 3, 4, 2, 4, 3, 3, 2, 4, 2, 3, 3, 4, 3, 4, 2, 3, 2, 3, 2, 4, 3, 3, 3, 4, 2, 4, 3, 4, 3, 3, 3, 3, 2, 3, 4, 4, 3, 4

Offset: 1

Eric W. Weisstein

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Factorization.

Cf. A046758, A073048, A055642, A020639, A027748, A124010, A110475.

Cf. also A377369 (analog for base 2).

a050252 1 = 1
a050252 n = sum $ map a055642 $
            (a027748_row n) ++ (filter (> 1) $ a124010_row n)
-- Reinhard Zumkeller, Aug 03 2013, Jun 21 2011

nd[n_]:=Total@IntegerLength@Select[Flatten@FactorInteger[n],#>1&];Table[If[n==1,1,nd[n]],{n,102}] (* Vladimir Joseph Stephan Orlovsky, Jan 30 2012 *)

from sympy import factorint
def a(n): return 1 if n == 1 else sum(len(str(p))+(len(str(e)) if e>1 else 0) for p, e in factorint(n).items())
print([a(n) for n in range(1, 103)]) # Michael S. Branicky, Dec 27 2024

a(A192010(n)) = n and a(m) != n for m < A192010(n);

a(A046759(n)) < A055642(A046759(n)); a(A046758(n)) = A055642(A046758(n)); a(A046760(n)) > A055642(A046760(n)). [Reinhard Zumkeller, Jun 21 2011]

cp's OEIS Frontend

A379537 Frugal numbers in base 2: numbers k such that A377369(k) < A070939(k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica