A115024 -id:A115024 - cp's OEIS frontend

A165256 Numbers whose number of distinct prime factors equals the number of digits in the number.

2, 3, 4, 5, 7, 8, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 33, 34, 35, 36, 38, 39, 40, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 65, 68, 69, 72, 74, 75, 76, 77, 80, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 98, 99, 102, 105, 110, 114, 120, 126, 130, 132

Offset: 1

Parthasarathy Nambi, Sep 11 2009

Is this sequence finite? If the answer is yes, is there any estimate for the number of terms of this sequence? - Parthasarathy Nambi, Nov 16 2009
This sequence is finite since there are only finitely many primes less than 10. - Charles R Greathouse IV, Feb 04 2013
Specifically, all terms have <= 10 digits since primorial(k) = A002110(k) has > k digits for k > 10. - Michael S. Branicky, Apr 13 2023

			The number of distinct prime factors of 4 is 1, which is the same as the number of digits in 4, so 4 is in the sequence.
The number of distinct prime factors of 21 is 2, which is the same as the number of digits in 21, so 21 is in the sequence.
However, 25 is NOT in the sequence because the number of distinct prime factors of 25 is 1, which does not match the number of digits in 25.

Charles R Greathouse IV, Table of n, a(n) for n = 1..7812 (complete sequence)
Michael S. Branicky, Python program

Cf. A001221, A055642, A115024, A002110.

omega := proc(n) nops(numtheory[factorset](n)) ; end: A055642 := proc(n) max(1, ilog10(n)+1) ; end: A165256 := proc(n) option remember; local a; if n = 1 then 2; else for a from procname(n-1)+1 do if A055642(a) = omega(a) then RETURN(a) ; fi; od: fi; end: seq(A165256(n),n=1..120) ; # R. J. Mathar, Sep 17 2009

Select[Range[200], IntegerLength[#] == Length[FactorInteger[#]] &]  (* Harvey P. Dale, Mar 20 2011 *)

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

# see link for alternate version producing full sequence instantly
from sympy import primefactors
def ok(n): return len(primefactors(n)) == len(str(n))
print([k for k in range(10**5) if ok(k)]) # Michael S. Branicky, Apr 13 2023

Extended by R. J. Mathar, Sep 17 2009

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

A359491 Numbers k with the property that the set of decimal digits of k matches the set of first digits of the prime factors of k.

Original entry on oeis.org

2, 3, 5, 7, 333, 23532, 33165, 77322, 175175, 232152, 321372, 373212, 515375, 712236, 2249232, 2321232, 2971332, 3372138, 3611322, 4313331, 5773131, 12322332, 23147124, 42323112, 72325232, 113338575, 123221232, 132232224, 172232112, 212322912, 221437272, 273233331

Offset: 1

John R Phelan, Jan 02 2023

Analogous to an acrostic, in which the first digit of each prime factor also forms the number itself.

There could also be a sequence based on the set of decimal digits of k matching the set of last digits of the prime factors of k; 373212 = 2*2*3*3*7*1481 and 73222329312 = 2*2*2*2*2*3*3*11*79*307*953 are examples of numbers in both sequences.

			a(5)=333 has prime factors 3*3*37, the first digits of which are 3, 3 and 3, matching the set of digits in 333.
a(10)=232152 has prime factors 2*3*2*17*569*2, the first digits of which are 2, 3, 2, 1, 5 and 2, matching the set of digits in 232152.

John R Phelan, Java program

Cf. A115024.

Java
```
See Links
```

is(n) = { my (d=digits(n), f=factor(n)); #d==bigomega(f) && vecsort(d) == vecsort(concat(vector(#f~, k, vector(f[k,2], z, digits(f[k,1])[1])))) } \\ Rémy Sigrist, Jan 28 2023

from sympy import factorint
def ok(n): return sorted(str(n)) == sorted(s[0] for s in map(str, factorint(n, multiple=True)))
print([k for k in range(1, 10**5) if ok(k)]) # Michael S. Branicky, Jan 08 2023

cp's OEIS Frontend

A165256 Numbers whose number of distinct prime factors equals the number of digits in the number.

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

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Author

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Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A359491 Numbers k with the property that the set of decimal digits of k matches the set of first digits of the prime factors of k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Java

PARI

Python