cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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.

Original entry on oeis.org

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

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Author

Giovanni Teofilatto, Feb 24 2006

Keywords

Examples

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

Links

Crossrefs

Cf. A165256, A167050.

Programs

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

Extensions

More terms from Stefan Steinerberger, Feb 27 2006

A342108 Smallest positive integer m with n digits and such that omega(m) = bigomega(m) = n.

Original entry on oeis.org

2, 10, 102, 1110, 10010, 101010, 1009470, 11741730, 223092870, 6469693230
Offset: 1

Views

Author

Bernard Schott, Feb 28 2021

Keywords

Comments

Equivalently: smallest n-digit squarefree number with n distinct prime factors.
Differs from A036336 where length(m) = bigomega(m) = n, when length(m) is the number of digits of m (A055642) and the n prime factors of m are counted with multiplicity (A001222).
Differs from A070842 where length(m) = omega(m) = n, when length(m) is the number of digits of m (A055642) and omega(m) is the number of distinct prime factors dividing m (A001221).
The first index for which these three sequences give three distinct terms is 4:
-> a(4) = 1110 = 2 * 3 * 5 * 37 , with length(1110) = omega(1110) = bigomega(1110) = 4.
-> A036336(4) = 1012 = 2 * 2 * 11 * 23 with length(1012) = bigomega(1012) = 4 > omega(1012) = 3.
-> A070842(4) = 1020 = 2 * 2 * 3 * 5 * 17 with length(1020) = omega(1020) = 4 < bigomega(1020) = 5.
As these terms are the smallest n-digit numbers in A167050 that is finite, this sequence is also finite with 10 terms, as for A070842.

Examples

			10010 = 2*5*7*11*13 is the smallest 5-digit number such that omega(10010) = bigomega(10010) = 5, hence a(5) = 10010.

Crossrefs

Subsequence of A167050.
Cf. A001221, A001222, A055642.
Cf. A036336, A070842, A342109.

Programs

  • Mathematica
    a={};For[n=1,n<=10,n++,For[m=10^(n-1),m<10^n,m++,If[PrimeOmega[m]==PrimeNu[m]==n,AppendTo[a,m];Break[]]]];a (* Stefano Spezia, Mar 04 2021 *)

Formula

A036336(n) <= A070842(n) <= a(n).

A342109 Largest positive integer m with n digits and such that omega(m) = bigomega(m) = n.

Original entry on oeis.org

7, 95, 994, 9982, 99858, 999570, 9998142, 99953490, 999068070, 9592993410
Offset: 1

Views

Author

Bernard Schott, Feb 28 2021

Keywords

Comments

Equivalently: largest n-digit squarefree number with n distinct prime factors (A167050).
Differs from A036337 where length(m) = bigomega(m) = n, when length(m) is the number of digits of m (A055642) and the n prime factors of m are counted with multiplicity (A001222).
Differs from A070843 where length(m) = omega(m) = n, when length(m) is the number of digits of m (A055642) and omega(m) is the number of distinct prime factors dividing m (A001221).
The first index for which these three sequences give three distinct terms is 4:
-> a(4) = 9982 = 2 * 7 * 23 * 31 with omega(9982) = bigomega(9982) = 4.
-> A036337(4) = 9999 = 3 * 3 * 11* 101 with bigomega(9999) = 4 > omega(9999) = 3.
-> A070843(4) = 9996 = 2^2 * 3 * 7^2 *17 with omega(9996) = 4 < bigomega(9996) = 6.
As these terms are the largest n-digit numbers in A167050 that is finite, this sequence is also finite with 10 terms, as for A070843.

Examples

			9592993410 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 19 * 23 * 43 and length(9592993410) = omega(9592993410) = bigomega(9592993410) = 10, so, a(10) = 9592993410 is a term; it is also the largest squarefree number with as many decimal digits as distinct prime factors (A167050).

Crossrefs

Subsequence of A167050.
Cf. A001221, A001222, A055642.
Cf. A036337, A070843, A342108.

Programs

  • Mathematica
    a={}; For[n=1,n<=10,n++,For[m=10^n-1,m>=10^(n-1),m--,If[PrimeOmega[m]==PrimeNu[m]==n,AppendTo[a, m];Break[]]]]; a (* Stefano Spezia, Mar 06 2021 *)
Showing 1-3 of 3 results.

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