A046460 -id:A046460 - cp's OEIS frontend

A048288 Number of prime factors counted with multiplicity of the reverse concatenation of numbers from 1 to n.

0, 2, 2, 2, 3, 2, 2, 4, 5, 3, 2, 3, 3, 4, 6, 4, 7, 10, 4, 5, 4, 5, 4, 5, 6, 9, 9, 5, 7, 8, 3, 6, 5, 7, 9, 8, 4, 3, 6, 5, 8, 6, 3, 8, 7, 5, 7, 7, 3, 6, 3, 7, 12, 14, 3, 5, 4, 6, 3, 3, 5, 9, 6, 6, 7, 7, 4, 8, 8, 4, 9, 5, 7, 8, 10, 3, 7, 6, 4, 9, 10, 1, 3, 8, 3

Offset: 1

Paul Jasper (jasperpaul(AT)hotmail.com)

			21 = 3*7 so a(2) = 2; 321 = 3*107 so a(3) = 2; 4321 = 29*149 so a(4) = 2; etc.
a(1)=0 since 1 has no prime factors.

Sean A. Irvine, Table of n, a(n) for n = 1..106
Patrick De Geest, Reversed Smarandache Concatenated Numbers
M. Fleuren, Factors and primes of Smarandache sequences.
M. Fleuren, Smarandache Factors and Reverse factors
Carlos Rivera, Puzzle 8. Primes by Listing, The Prime Puzzles and Problems Connection.

Cf. A000422, A001222, A046460, A046461, A046468, A050679, A050680, A050681, A050682.

Join[{0},Table[PrimeOmega[FromDigits[Flatten[IntegerDigits[Range[i,1,-1]]]]],{i,2,36}]] (* Jayanta Basu, May 30 2013 *)

a(n) = A001222(A000422(n)). - Michel Marcus, Jun 14 2021

Offset and a(19) corrected and more terms from Sean A. Irvine, Jun 13 2021

Edited by N. J. A. Sloane, Sep 04 2021

A046461 Numbers k such that concatenation of numbers from 1 to k is a semiprime.

Original entry on oeis.org

3, 4, 7, 34, 97

Offset: 1

Patrick De Geest, Aug 15 1998

From Sean A. Irvine, Apr 15 2010, updated Oct 08 2015: (Start)
5053 and 9706 are definite terms of the sequence.
The next potential term is 1651.
A007908(1651) is composite, but has no known prime factor, and its least prime factor likely has at least 45 digits. (End)
If k is a multiple of 10, then k is not a term. - Chai Wah Wu, Jan 22 2020
From Jon E. Schoenfield, Oct 07 2023: (Start)
k cannot be a term if any of the following are true:
   4|k and k > 4  (2*2 would divide the concatenation)
   6|k or  6|k-2  (2*3   "     "     "        "      )
   9|k or  9|k-8  (3*3   "     "     "        "      )
  10|k            (2*5   "     "     "        "      )
  15|k or 15|k-5  (3*5   "     "     "        "      )
  25|k            (5*5   "     "     "        "      ) (End)

			A007908(691)=1304238680165623831238651513722972177904593843651*C1916, so A007908(691) is not a semiprime and 691 is not a term of this sequence.

Patrick De Geest, Normal Smarandache Concatenated Numbers, Prime factors from 1 up to n.
M. Fleuren, Factors and primes of Smarandache sequences.
M. Fleuren, Smarandache Factors and Reverse factors.
Carlos Rivera, Puzzle 8. Primes by Listing, The Prime Puzzles and Problems Connection.

Cf. A007908, A046460.

Select[Range[100], Length@FactorInteger@FromDigits@Flatten@IntegerDigits@Range@# == 2 &] (* Robert Price, Oct 11 2019 *)
Select[Range[100],PrimeOmega[FromDigits[Flatten[IntegerDigits/@Range[#]]]] == 2&] (* Harvey P. Dale, Sep 10 2022 *)

Simplified definition by Sean A. Irvine, Mar 29 2010

a(5) from Sean A. Irvine, Mar 29 2010

A046462 Numbers k such that the concatenation of numbers from 1 to k is the product of 3 primes (not necessarily distinct).

Original entry on oeis.org

2, 5, 10, 13, 14, 15, 31, 51, 61, 67, 73

Offset: 1

Patrick De Geest, Aug 15 1998

Patrick De Geest, Normal Smarandache Concatenated Numbers, Prime factors from 1 up to n
M. Fleuren, Factors and primes of Smarandache sequences.
Carlos Rivera, Puzzle 8. Primes by Listing, The Prime Puzzles & Problems Connection.

Cf. A001222, A046460.

q:= n-> is(numtheory[bigomega](parse(cat($1..n)))=3):
select(q, [$1..35])[];  # Alois P. Heinz, Apr 10 2021

Select[Range[100],
PrimeOmega@FromDigits@Flatten@IntegerDigits@Range@# == 3 &] (* Robert Price, Oct 11 2019 *)

A046460(a(n)) = 3.

a(10)-a(11) from Sean A. Irvine, Apr 10 2021

A046463 Numbers k such that the concatenation of numbers from 1 to k is the product of 4 primes (not necessarily distinct).

Original entry on oeis.org

9, 16, 23, 29, 37, 38, 43, 58, 59

Offset: 1

Patrick De Geest, Aug 15 1998

Patrick De Geest, Normal Smarandache Concatenated Numbers, Prime factors from 1 up to n
M. Fleuren, Factors and primes of Smarandache sequences.
Carlos Rivera, Primes by Listing, The Prime Puzzles & Problems Connection.

Cf. A046460.

Select[Range[60],PrimeOmega[FromDigits[Flatten[IntegerDigits/@ Range[#]]]] == 4&] (* Harvey P. Dale, Jan 15 2013 *)

A046464 Numbers k such that the concatenation of numbers from 1 to k is the product of 5 primes (not necessarily distinct).

Original entry on oeis.org

8, 17, 25, 41, 47, 55, 56, 63, 77, 94, 101, 103, 107

Offset: 1

Patrick De Geest, Aug 15 1998

Patrick De Geest, Normal Smarandache Concatenated Numbers, Prime factors from 1 up to n
M. Fleuren, Factors and primes of Smarandache sequences.
M. Fleuren, Smarandache Factors and Reverse factors
Carlos Rivera, Puzzle 8. Primes by Listing, The Prime Puzzles & Problems Connection.

Cf. A007908, A046460, A050682.

Select[Range[100],PrimeOmega@FromDigits@Flatten@ IntegerDigits@ Range@# == 5 &] (* Robert Price, Oct 11 2019 *)

a(9)-a(13) from Sean A. Irvine, Apr 10 2021

A046468 Numbers k such that the concatenation of numbers from 1 to k is the product of 9 primes (not necessarily distinct).

Original entry on oeis.org

26, 36, 40, 52, 74, 90, 102, 112, 114, 115

Offset: 1

Patrick De Geest, Aug 15 1998

Patrick De Geest, Normal Smarandache Concatenated Numbers, Prime factors from 1 up to n
M. Fleuren, Factors and primes of Smarandache sequences.
M. Fleuren, Smarandache Factors and Reverse factors
Carlos Rivera, Puzzle 8. Primes by Listing, The Prime Puzzles & Problems Connection.

Cf. A046460.

Select[Range[52],PrimeOmega[FromDigits[Flatten[IntegerDigits/@ Range[ #]]]] ==9&] (* Harvey P. Dale, Nov 12 2017 *)

a(5)-a(10) from Sean A. Irvine, Apr 10 2021

A046465 Numbers k such that the concatenation of numbers from 1 to k is the product of 6 primes (not necessarily distinct).

Original entry on oeis.org

11, 12, 18, 19, 22, 24, 32, 33, 39, 49, 57, 65, 70, 71, 75, 76, 78, 79, 81, 83, 85, 87, 88, 91, 95, 96, 99, 105, 110

Offset: 1

Patrick De Geest, Aug 15 1998

Patrick De Geest, Normal Smarandache Concatenated Numbers, Prime factors from 1 up to n
M. Fleuren, Smarandache Factors and Reverse factors
Carlos Rivera, Puzzle 8. Primes by Listing, The Prime Puzzles & Problems Connection.

Cf. A046460.

Select[Range[100], PrimeOmega@FromDigits@Flatten@IntegerDigits@Range@# == 6 &] (* Robert Price, Oct 11 2019 *)

a(12)-a(29) from Sean A. Irvine, Apr 10 2021

A046466 Numbers k such that the concatenation of numbers from 1 to k is the product of 7 primes (not necessarily distinct).

Original entry on oeis.org

28, 30, 42, 44, 46, 69, 72, 82, 84, 86, 93, 109, 116

Offset: 1

Patrick De Geest, Aug 15 1998

Patrick De Geest, Normal Smarandache Concatenated Numbers, Prime factors from 1 up to n
M. Fleuren, Smarandache Factors and Reverse factors
Carlos Rivera, Puzzle 8. Primes by Listing, The Prime Puzzles & Problems Connection.

Cf. A046460.

Select[Range[100], PrimeOmega@FromDigits@Flatten@IntegerDigits@Range@# == 7 &] (* Robert Price, Oct 11 2019 *)

a(6)-a(13) from Sean A. Irvine, Apr 10 2021

A046467 Numbers k such that the concatenation of numbers from 1 to k is the product of 8 primes (not necessarily distinct).

Original entry on oeis.org

6, 21, 27, 35, 48, 53, 60, 62, 66, 92, 106, 108, 117

Offset: 1

Patrick De Geest, Aug 15 1998

Patrick De Geest, Normal Smarandache Concatenated Numbers, Prime factors from 1 up to n
M. Fleuren, Smarandache Factors and Reverse factors
Carlos Rivera, Puzzle 8. Primes by Listing, The Prime Puzzles & Problems Connection.

Cf. A046460.

Select[Range[100], PrimeOmega@FromDigits@Flatten@IntegerDigits@Range@# == 8 &] (* Robert Price, Oct 11 2019 *)

a(9)-a(13) from Sean A. Irvine, Apr 10 2021

A050676 Let b(n) = number of prime factors (with multiplicity) of concatenation of numbers from 1 to n; sequence gives smallest number m with b(m) = n.

Original entry on oeis.org

1, 3, 2, 9, 8, 11, 28, 6, 26, 20

Offset: 1

Patrick De Geest, Aug 15 1999

a(11) <= 148, a(12) = 104, a(13) = 100, a(14) = 54, a(15) <= 184. - Michael S. Branicky, Dec 14 2022 using factorizations at the De Geest link

Patrick De Geest, Normal Smarandache Concatenated Numbers, Prime factors from 1 up to n
Micha Fleuren, Factors and primes of Smarandache sequences.
Micha Fleuren, Smarandache Factors and Reverse factors
Carlos Rivera, Puzzle 8. Primes by Listing, The Prime Puzzles & Problems connection.

Cf. A007908, A050678, A046460.

Join[{1},Table[i=1; While[PrimeOmega[FromDigits[Flatten[IntegerDigits[Range[i]]]]]!=n,i++]; i,{n,2,10}]] (* Jayanta Basu, May 30 2013 *)

cp's OEIS Frontend

A048288 Number of prime factors counted with multiplicity of the reverse concatenation of numbers from 1 to n.

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A046461 Numbers k such that concatenation of numbers from 1 to k is a semiprime.

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A046462 Numbers k such that the concatenation of numbers from 1 to k is the product of 3 primes (not necessarily distinct).

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A046463 Numbers k such that the concatenation of numbers from 1 to k is the product of 4 primes (not necessarily distinct).

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A046464 Numbers k such that the concatenation of numbers from 1 to k is the product of 5 primes (not necessarily distinct).

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A046468 Numbers k such that the concatenation of numbers from 1 to k is the product of 9 primes (not necessarily distinct).

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A046465 Numbers k such that the concatenation of numbers from 1 to k is the product of 6 primes (not necessarily distinct).

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A046466 Numbers k such that the concatenation of numbers from 1 to k is the product of 7 primes (not necessarily distinct).

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A046467 Numbers k such that the concatenation of numbers from 1 to k is the product of 8 primes (not necessarily distinct).

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