A165256 -id:A165256 - cp's OEIS frontend

A115024 Natural numbers n such that the number of prime factors of n (counted with multiplicity) is equal to the number of decimal digits of n.

2, 3, 5, 7, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 102, 105, 110, 114, 116, 117, 124, 125, 130, 138, 147, 148, 153, 154, 164, 165, 170, 171, 172, 174, 175, 182, 186, 188, 190, 195

Offset: 1

Giovanni Teofilatto, Feb 24 2006

			25 = 5*5 and 25 has two digits.
116 = 2*2*29 and 116 has three digits.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A165256, A167050.

Select[Range[2, 300], Sum[FactorInteger[ # ][[i]][[2]], {i, 1, Length[FactorInteger[ # ]]}] == Floor[Log[10, # ] + 1] &] (* Stefan Steinerberger, Feb 27 2006 *)
Select[Range[200],PrimeOmega[#]==IntegerLength[#]&] (* Harvey P. Dale, Jul 28 2020 *)

is(n)=#Str(n)==bigomega(n) \\ Charles R Greathouse IV, Feb 04 2013

More terms from Stefan Steinerberger, Feb 27 2006

A167050 Squarefree numbers with as many decimal digits as distinct prime factors.

Original entry on oeis.org

2, 3, 5, 7, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 222, 230, 231, 238, 246, 255, 258, 266, 273, 282, 285, 286, 290, 310

Offset: 1

Claudio Meller, Oct 27 2009

From Bernard Schott, Feb 02 2013: (Start)
These numbers appear in 1999 during the XV Gara Nazionale di Matematica, exercise 2, in Italia. [See Link]
Another definition (1): If p_1 < p_2 < p_3 < ... < p_r are r distinct primes, then n is in this sequence if 10^r <= n = p_1*p_2*...*p_r < 10^(r+1).
Another definition (2): If p_1 < p_2 < p_3 < ... < p_r are r distinct primes, then n = p_1*p_2*...*p_r has r digits in base ten.
These numbers are called "equilibrato" in Italian and translated "balanced" in English [see reference Crux Mathematicorum], I propose "nombres équilibrés" in French.
This sequence is finite, a proof without words:
2*3*5*7*11*13*17*19*23*29 = 6469693230 < 10^{10}
2*3*5*7*11*13*17*19*23*29*31 = 200560490130 > 10^{11}.
Two natural open questions:
--> 1) What is the last term of this sequence?
The last term is 9592993410 = 2*3*5*7*11*13*17*19*23*43.
--> 2) How many numbers in this sequence?
This sequence contains 4352 elements.
Concerning these two questions, I used the French mathematical forum les-mathematiques.net with the help of "JLT" and "Juge Ti" to confirm and solve them: see Link.
Subsequence of A115024. (End)

			138 = 2*3*23 and 138 is squarefree with three digits.

R. E. Woodrow, The Olympiad Corner, No. 226, Crux Mathematicorum, v28-n8(2002), 481.

Charles R Greathouse IV, Table of n, a(n) for n = 1..4352 (complete sequence).
XV Gara Nazionale di Matematica, Cesenatico, 7 Maggio 1999, exercise 2.
Bernard Schott, List of the 4352 terms of the sequence A167050.
Juge Ti, Bernard Schott, Jean-Louis Tu and Norbert Verdier, QDV 17: Question ouverte sans titre (French mathematical forum les-mathematiques.net).
Index to sequences related to Olympiads and other Mathematical competitions.

Intersection of A165256 and A115024.

Cf. A115024, A165256.

A001221 := proc(n) nops(numtheory[factorset](n)) ; end:
A055642 := proc(n) max(1,ilog10(n)+1) ; end:
isA167050 := proc(n) numtheory[issqrfree](n) and A055642(n) = A001221(n) end:
for n from 1 to 300 do if isA167050(n) then printf("%d,",n) ; fi; end do; # R. J. Mathar, Nov 03 2009
A Maple program is proposed by "Juge Ti" on the French mathematical forum in link for answering to the two questions (last number and cardinal of this set).

Select[Range[400],SquareFreeQ[#]&&PrimeNu[#]==IntegerLength[#]&] (* Harvey P. Dale, Jun 26 2021 *)

is(n)=issquarefree(n)&&#Str(n)==omega(n) \\ Charles R Greathouse IV, Feb 04 2013

Intersection of A005117 and A165256.

Definition rephrased and formula added by R. J. Mathar, Nov 05 2009

A327786 Numbers whose number of distinct prime factors is greater than the sum of their digits.

Original entry on oeis.org

10, 100, 110, 210, 1000, 1001, 1010, 1020, 1100, 1110, 2010, 2100, 10000, 10010, 10020, 10100, 10101, 10110, 10200, 11000, 11010, 11100, 20010, 20020, 20100, 21000, 100000, 100002, 100010, 100011, 100020, 100100, 100110, 100200, 101000, 101010, 101100, 102000

Offset: 1

Metin Sariyar, Sep 25 2019

The sequence is infinite since every number of the form 10^k for k >= 1 is in the sequence. It can be proved that 210 is the largest term with distinct digits.

			For a(4) = 210, 2 + 1 + 0 = 3, 210 = 2*3*5*7 with 4 distinct factors, 4 > 3 so 210 is a term.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 291 terms from Metin Sariyar)

Cf. A001221, A165256.

[k:k in [2..110000]| #PrimeDivisors(k) gt &+Intseq(k)]; // Marius A. Burtea, Oct 07 2019

Select[Range[10^6], Total[IntegerDigits[#]]Total[IntegerDigits[#]]&] (* Harvey P. Dale, Jul 07 2020 *)

isok(n) = omega(n) > sumdigits(n); \\ Michel Marcus, Sep 25 2019

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A115024 Natural numbers n such that the number of prime factors of n (counted with multiplicity) is equal to the number of decimal digits of n.

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A167050 Squarefree numbers with as many decimal digits as distinct prime factors.

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Maple

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A327786 Numbers whose number of distinct prime factors is greater than the sum of their digits.

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Magma

Mathematica

PARI