A046759 -id:A046759 - cp's OEIS frontend

A055642 Number of digits in the decimal expansion of n.

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

Offset: 0

Henry Bottomley, Jun 06 2000

From Hieronymus Fischer, Jun 08 2012: (Start)
For n > 0 the first differences of A117804.
The total number of digits necessary to write down all the numbers 0, 1, 2, ..., n is A117804(n+1). (End)
Here a(0) = 1, but a different common convention is to consider that the expansion of 0 in any base b > 0 has 0 terms and digits. - M. F. Hasler, Dec 07 2018

			Examples:
999: 1 + floor(log_10(999)) = 1 + floor(2.x) = 1 + 2 = 3 or
      ceiling(log_10(999+1)) = ceiling(log_10(1000)) = ceiling(3) = 3;
1000: 1 + floor(log_10(1000)) = 1 + floor(3) = 1 + 3 = 4 or
      ceiling(log_10(1000+1)) = ceiling(log_10(1001)) = ceiling(3.x) = 4;
1001: 1 + floor(log_10(1001)) = 1 + floor(3.x) = 1 + 3 = 4 or
      ceiling(log_10(1001+1)) = ceiling(log_10(1002)) = ceiling(3.x) = 4;

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A043537, A178788, A046034, A019546, A054899, A122840, A055640, A055641, A102669-A102685, A117804, A160093, A160094, A196563, A196564, A000120, A000788, A023416, A059015 (for base 2).
Cf. A262190, A262198.
Cf. A007953: sum of digits.

a055642 :: Integer -> Int
a055642 = length . show  -- Reinhard Zumkeller, Feb 19 2012, Apr 26 2011

[ #Intseq(n): n in [0..105] ];   //  Bruno Berselli, Jun 30 2011
(Common Lisp) (defun A055642 (n) (if (zerop n) 1 (floor (log n 10)))) ; James Spahlinger, Oct 13 2012

A055642 := proc(n)
        max(1,ilog10(n)+1) ;
end proc:  # R. J. Mathar, Nov 30 2011

Join[{1}, Array[ Floor[ Log[10, 10# ]] &, 104]] (* Robert G. Wilson v, Jan 04 2006 *)
Join[{1},Table[IntegerLength[n],{n,104}]]
IntegerLength[Range[0,120]] (* Harvey P. Dale, Jul 02 2016 *)

a(n)=#Str(n) \\ M. F. Hasler, Nov 17 2008

A055642(n)=logint(n+!n,10)+1 \\ Increasingly faster than the above, for larger n. (About twice as fast for n ~ 10^7.) - M. F. Hasler, Dec 07 2018

def a(n): return len(str(n))
print([a(n) for n in range(121)]) # Michael S. Branicky, May 10 2022

def A055642(n): # Faster than len(str(n)) from ~ 50 digits on
    L = math.log10(n or 1)
    if L.is_integer() and 10**int(L)>n: return int(L or 1)
    return int(L)+1  # M. F. Hasler, Apr 08 2024

a(A046760(n)) < A050252(A046760(n)); a(A046759(n)) > A050252(A046759(n)). - Reinhard Zumkeller, Jun 21 2011
a(n) = A196563(n) + A196564(n).
a(n) = 1 + floor(log_10(n)) = 1 + A004216(n) = ceiling(log_10(n+1)) = A004218(n+1), if n >= 1. - Daniel Forgues, Mar 27 2014
a(A046758(n)) = A050252(A046758(n)). - Reinhard Zumkeller, Jun 21 2011
a(n) = A117804(n+1) - A117804(n), n > 0. - Hieronymus Fischer, Jun 08 2012
G.f.: g(x) = 1 + (1/(1-x))*Sum_{j>=0} x^(10^j). - Hieronymus Fischer, Jun 08 2012
a(n) = A262190(n) for n < 100; a(A262198(n)) != A262190(A262198(n)). - Reinhard Zumkeller, Sep 14 2015

A046758 Equidigital numbers.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 11, 13, 14, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 35, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 105, 106, 107, 109, 111, 112, 113, 115, 118, 119, 121, 122, 123, 127, 129, 131, 133, 134, 135, 137, 139

Offset: 1

N. J. A. Sloane

Write n as product of primes raised to powers, let D(n) = A050252 = total number of digits in product representation (number of digits in all the primes plus number of digits in all the exponents that are greater than 1) and l(n) = number of digits in n; sequence gives n such that D(n)=l(n).

The term "equidigital number" was coined by Recamán (1995). - Amiram Eldar, Mar 10 2024

			For n = 125 = 5^3, l(n) = 3 but D(n) = 2. So 125 is not a member of this sequence.

Bernardo Recamán Santos, Equidigital representation: problem 2204, J. Rec. Math., Vol. 27, No. 1 (1995), pp. 58-59.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J. P. Delahaye, "Primes Hunters", Economical and Prodigal Numbers (Text in French). [Wayback Machine link]
R. G. E. Pinch, Economical numbers, arXiv:math/9802046 [math.NT], 1998.
Eric Weisstein's World of Mathematics, Equidigital Number..
Wikipedia, Equidigital number.

Cf. A046759, A046760, A050252, A055642, A073048.

a046758 n = a046758_list !! (n-1)
a046758_list = filter (\n -> a050252 n == a055642 n) [1..]
-- Reinhard Zumkeller, Jun 21 2011

edQ[n_] := Total[IntegerLength[DeleteCases[Flatten[FactorInteger[n]], 1]]] == IntegerLength[n]; Join[{1}, Select[Range[140], edQ]] (* Jayanta Basu, Jun 28 2013 *)

for(n=1, 100, s=""; F=factor(n); for(i=1, #F[, 1], s=concat(s, Str(F[i, 1])); s=concat(s, Str(F[i, 2]))); c=0; for(j=1, #F[, 2], if(F[j, 2]==1, c++)); if(#digits(n)==#s-c, print1(n, ", "))) \\ Derek Orr, Jan 30 2015

from itertools import count, islice
from sympy import factorint
def A046758_gen(): # generator of terms
    return (n for n in count(1) if n == 1 or len(str(n)) == sum(len(str(p))+(len(str(e)) if e > 1 else 0) for p, e in factorint(n).items()))
A046758_list = list(islice(A046758_gen(),20)) # Chai Wah Wu, Feb 18 2022

A050252(a(n)) = A055642(a(n)). - Reinhard Zumkeller, Jun 21 2011

More terms from Eric W. Weisstein

A046760 Wasteful numbers.

Original entry on oeis.org

4, 6, 8, 9, 12, 18, 20, 22, 24, 26, 28, 30, 33, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 108, 110, 114

Offset: 1

N. J. A. Sloane

Write n as product of primes raised to powers, let D(n) = number of digits in product, l(n) = number of digits in n; sequence gives n such that D(n)>l(n).

A050252(a(n)) > A055642(a(n)). - Reinhard Zumkeller, Jun 21 2011

			For n = 125 = 5^3, l(n) = 3 but D(n) = 2. So 125 is not a term of this sequence. [clarified by _Derek Orr_, Jan 30 2015]

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J.-P. Delahaye, Les chasseurs de nombres premiers.
R. G. E. Pinch, Economical numbers., arXiv:math/9802046 [math.NT], 1998.
Eric Weisstein's World of Mathematics, Wasteful Number.
Wikipedia, Extravagant number.

Cf. A046758, A046759.

a046760 n = a046760_list !! (n-1)
a046760_list = filter (\n -> a050252 n > a055642 n) [1..]
-- Reinhard Zumkeller, Aug 02 2013

Cases[Range[115], n_ /; Length[Flatten[IntegerDigits[FactorInteger[n] /. 1 -> Sequence[]]]] > Length[IntegerDigits[n]]] (* Jean-François Alcover, Mar 21 2011 *)

for(n=1,100,s="";F=factor(n);for(i=1,#F[,1],s=concat(s,Str(F[i,1]));s=concat(s,Str(F[i,2])));c=0;for(j=1,#F[,2],if(F[j,2]==1,c++));if(#digits(n)<#s-c,print1(n,", "))) \\ Derek Orr, Jan 30 2015

from itertools import count, islice
from sympy import factorint
def A046760_gen(): # generator of terms
    return (n for n in count(1) if len(str(n)) < sum(len(str(p))+(len(str(e)) if e > 1 else 0) for p, e in factorint(n).items()))
A046760_list = list(islice(A046760_gen(),20)) # Chai Wah Wu, Feb 18 2022

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]

A254318 Hyper equidigital numbers.

Original entry on oeis.org

2, 3, 4, 5, 7, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 28, 29, 31, 32, 35, 36, 37, 39, 41, 43, 46, 47, 49, 50, 53, 54, 58, 59, 61, 64, 67, 69, 71, 72, 73, 79, 81, 83, 89, 92, 93, 97, 98, 100, 101, 103, 104, 105, 106, 107, 109, 113, 116, 119

Offset: 1

Michel Lagneau, Jan 28 2015

The distinction between the equidigital numbers (A046758) is that only the distinct digits are counted instead of all digits. Hence the definition:

Write n as product of primes raised to powers, let D(n) = total number of distinct digits in product representation (number of distinct digits in all the primes and number of distinct digits in all the exponents that are greater than 1) and nbd(n) = A043537(n) = number of distinct digits in n; sequence gives n such that D(n) = nbd(n).

			116 is in the sequence because 116 = 2^2*29 => D(116)= A043537(116)=2.

Michel Lagneau, Table of n, a(n) for n = 1..10000

Cf. A043537, A046760, A046758, A046759, A254315, A254317, A254319, A254321.

Cases[Range[400], n_ /; Length[Union[Flatten[IntegerDigits[FactorInteger[n] /. 1 -> Sequence[]]]]]==Length[Union[Flatten[IntegerDigits[n]]]]]

for(n=1,100,s=[];F=factor(n);for(i=1,#F[,1],s=concat(s,digits(F[i,1]));if(F[i,2]>1,s=concat(s,digits(F[i,2]))));if(#vecsort(digits(n),,8)==#vecsort(s,,8),print1(n,", "))) \\ Derek Orr, Jan 30 2015

A379373 Numbers k such that A050252(k) <= A055642(k).

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 11, 13, 14, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 35, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 105, 106, 107, 109, 111, 112, 113, 115, 118, 119, 121, 122, 123, 125, 127, 128, 129, 131, 133, 134, 135

Offset: 1

Paolo Xausa, Dec 21 2024

Some authors call these numbers "economical numbers" (see links) and refer to A046759 as "frugal numbers", while other authors define "economical numbers" to be A046759.

First differs from A046758 at n = 54, where a(54) = 125 is missing from A046758.

			112 is a term because 112 = (2^4)*7; the total number of digits of (2, 4, 7) = 1 + 1 + 1 <= the number of digits of 112 (3).
125 is a term because 125 = 5^3; the total number of digits of (5, 3) = 1 + 1 <= the number of digits of 125 (3).

Paolo Xausa, Table of n, a(n) for n = 1..10000
Richard G. E. Pinch, Economical numbers, arXiv:math/9802046 [math.NT], 1998.
Giovanni Resta, Economical numbers, Numbers Aplenty, 2013.

Union of A046758 and A046759.
Complement of A046760.
Cf. A050252, A055642.

A379373Q[k_] := Total[IntegerLength[Select[Flatten[FactorInteger[k]], # > 1 &]]] <= IntegerLength[k];
Select[Range[200], A379373Q]

from sympy import factorint
def ok(n): return n > 0 and sum(len(str(p))+(len(str(e)) if e>1 else 0) for p, e in factorint(n).items()) <= len(str(n))
print([k for k in range(136) if ok(k)]) # Michael S. Branicky, Dec 22 2024

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

Original entry on oeis.org

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]

A379538 Square array read by ascending antidiagonals: T(n,k) is the k-th frugal number in base n.

Original entry on oeis.org

1, 1, 27, 1, 32, 32, 1, 27, 49, 49, 1, 27, 64, 64, 64, 1, 81, 81, 81, 81, 81, 1, 64, 125, 125, 121, 98, 121, 1, 64, 81, 243, 128, 125, 121, 125, 1, 81, 81, 125, 250, 162, 128, 125, 128, 1, 125, 125, 125, 243, 256, 169, 169, 128, 135, 1, 125, 128, 128, 128, 343, 289, 243, 243, 169, 147

Offset: 2

Paolo Xausa, Dec 25 2024

A frugal number in base n is a number with more digits (in its base n representation) than the total number of digits (in base n representation) 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).

			Array begins:
  n\k| 1    2    3    4    5    6    7    8    9    10  ...
  ---------------------------------------------------------
   2 | 1,  27,  32,  49,  64,  81, 121, 125, 128,  135, ... = A379537
   3 | 1,  32,  49,  64,  81,  98, 121, 125, 128,  169, ...
   4 | 1,  27,  64,  81, 121, 125, 128, 169, 243,  256, ...
   5 | 1,  27,  81, 125, 128, 162, 169, 243, 256,  289, ...
   6 | 1,  81, 125, 243, 250, 256, 289, 343, 361,  375, ...
   7 | 1,  64,  81, 125, 243, 343, 361, 375, 405,  486, ...
   8 | 1,  64,  81, 125, 128, 243, 343, 512, 529,  567, ...
   9 | 1,  81, 125, 128, 243, 256, 343, 625, 729,  768, ...
  10 | 1, 125, 128, 243, 256, 343, 512, 625, 729, 1024, ... = A046759 (without the initial 1)
  ...       |                                         \______ A379539 (main diagonal)
         A377478
T(2,10) = 135 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); and 135 is the tenth number with this property.

Richard G. E. Pinch, Economical numbers, arXiv:math/9802046 [math.NT], 1998.
Giovanni Resta, Frugal numbers, Numbers Aplenty, 2013.
Wikipedia, Frugal number.

Cf. A377478 (column k = 2), A379537 (row n = 2), A046759 (row n = 10), A379539 (main diagonal).

Cf. A379373.

Module[{dmax = 15, a, m}, a = Table[m = 0; Table[While[Total[IntegerLength[Select[Flatten[FactorInteger[++m]], # > 1 &], n]] >= IntegerLength[m, n]]; m, dmax-n+2], {n, dmax+1, 2, -1}]; Array[Diagonal[a, # - dmax] &, dmax]]

A254319 Hyper economical numbers.

Original entry on oeis.org

27, 108, 125, 128, 129, 135, 138, 143, 159, 160, 184, 187, 189, 196, 207, 209, 216, 219, 243, 249, 256, 259, 265, 276, 295, 297, 329, 341, 351, 375, 403, 429, 451, 458, 469, 497, 512, 529, 621, 625, 671, 679, 729, 781, 795, 837, 841, 892, 896, 908, 916, 932

Offset: 1

Michel Lagneau, Jan 28 2015

The distinction between the economical numbers (A046759) is that the distinct digits are counted only instead all digits. Hence the definition:

Write n as product of primes raised to powers, let D(n) = total number of distinct digits in product representation (number of distinct digits in all the primes and number of distinct digits in all the exponents that are greater than 1) and nbd(n) = A043537(n) number of distinct digits in n; sequence gives n such that D(n) < nbd(n).

			27 is in the sequence because 27 = 3 ^ 3 => D(27)= 1 < nbd(27)=2.

Cf. A043537, A046760, A046758, A046759, A254318, A254321.

Cases[Range[400], n_ /; Length[Union[Flatten[IntegerDigits[FactorInteger[n] /. 1 -> Sequence[]]]]]< Length[Union[Flatten[IntegerDigits[n]]]]]

for(n=1,10^3,s=[];F=factor(n);for(i=1,#F[,1],s=concat(s,digits(F[i,1]));if(F[i,2]>1,s=concat(s,digits(F[i,2]))));if(#vecsort(digits(n),,8)>#vecsort(s,,8),print1(n,", "))) \\ Derek Orr, Jan 30 2015

cp's OEIS Frontend

A172460 Partial sums of economical numbers A046759.

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A055642 Number of digits in the decimal expansion of n.

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A046758 Equidigital numbers.

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A046760 Wasteful numbers.

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

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A254318 Hyper equidigital numbers.

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A379373 Numbers k such that A050252(k) <= A055642(k).

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