A068069 -id:A068069 - cp's OEIS frontend

A067665 The start of a record-setting run of consecutive integers i with distinct A001222(i).

1, 6, 15, 60, 726, 6318, 189375, 755968, 683441871, 33714015615

Offset: 1

G. L. Honaker, Jr., Feb 03 2002

The list of indices of record terms in A068796;
n is in the sequence if A068796(n) is larger than A068796(m) when n is larger than m. For the known terms, f(a(n)) = n+1. Is that true for all n? In other words, is the monotonic subsequence of A068796 identical to A020725?
a(11) > 10^13. - Giovanni Resta, Jan 08 2014

			The values of f(n) for n=1 to 15 are 2,1,2,2,2,3,3,2,1,3,2,3,2,1,4. Records occur at f(1)=2, f(6)=3 and f(15)=4.

J.-M. De Koninck, J. B. Friedlander, and F. Luca, On strings of consecutive integers with a distinct number of prime factors, Proc. Amer. Math. Soc., 137 (2009), 1585-1592.

Cf. A001222, A067650, A068069, A068796, A068797.

bigomega[n_] := Plus@@Last/@FactorInteger[n]; f[n_] := For[k=1; s={bigomega[n]}, True, k++, If[MemberQ[s, z=bigomega[n+k]], Return[k], AppendTo[s, z]]]; For[n=1; max=0, True, n++, If[f[n]>max, Print[n, " ", max=f[n]]]]

a(n,lim=1e12,startAt=1)={
  forstep(i=startAt-1,lim,10^6-n,
    my(v=vectorsmall(min(10^6,lim\1-i),j,bigomega(j+i)));
    for(j=n,#v,if(#vecsort(v[j-n+1..j],,8)==n,return(j+i-n+1)))
  )
}; \\ Charles R Greathouse IV, Jul 03 2013

More terms from Shyam Sunder Gupta, Feb 08 2002
Edited by Robert G. Wilson v, Feb 20 2002
Edited by Dean Hickerson, Mar 05 2002
a(10) from Donovan Johnson, Oct 15 2008

A072664 End of smallest run of n consecutive integers with n, n-1, ..., 1 distinct prime factors in that order.

Original entry on oeis.org

2, 7, 107, 2187, 578309, 12239309, 45640671719, 29591195200067

Offset: 1

Rick L. Shepherd, Jun 30 2002

nonn

Using a musical analogy, each run is a "crescendo" of primality where each subsequent member of the run is gradually "more prime" in the sense of having one fewer distinct prime factors (see A001221). These a(n) are the peaks of crescendos of increasing length. a(7) is greater than 60000000.

This sequence was inspired by A068069, where the members of the runs have n different numbers of distinct prime factors, 1 through n, but where the order is not specified.

			a(1)=2 because 2 is prime and therefore the smallest integer with exactly one distinct prime factor. a(2)=7 because 6=2*3 and 7 (prime) is the smallest run of consecutive integers with exactly 2 and 1 distinct prime factors in that order. a(3)=107 because 105=3*5*7, 106=2*53 and 107 (prime) is the smallest run with exactly 3, 2 and 1 distinct prime factors in that order. Note that a(1), a(2), a(3), a(5) and a(6) are prime but that a(4)=2187=3^7 is not.

Cf. A086560 (smallest start with run pattern 1, 2, ..., n), A072665 (center with run pattern n+1, n, ..., 2, 1, 2, ..., n, n+1), A068069 (run order not specified), A001221 (omega(n)).

a(7) from Donovan Johnson, Jan 24 2009

a(8) from Donovan Johnson, Jul 19 2011

A084296 Triangle: number of distinct prime factors in n-th primorial numbers when n prime factors first appears and in n-1 subsequent integers after.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 4, 1, 2, 2, 5, 1, 2, 2, 3, 6, 2, 2, 3, 2, 2, 7, 3, 2, 3, 3, 2, 4, 8, 2, 3, 2, 4, 2, 3, 2, 9, 2, 3, 3, 3, 2, 4, 3, 4, 10, 3, 3, 2, 2, 2, 4, 3, 3, 2, 11, 1, 4, 3, 2, 4, 5, 4, 3, 3, 4, 12, 3, 3, 4, 2, 3, 6, 2, 3, 5, 4, 3, 13, 3, 4, 2, 3, 3, 3, 3, 3, 3, 6, 2, 4, 14, 2, 3, 2, 4, 5, 4, 5, 3, 3, 6, 4

Offset: 1

Labos Elemer, May 27 2003

Omega-values(=A001221) in the subsequent neighborhood of radius n, for primorial numbers are usually neither all distinct or all equal items as it is required in A068069, A045983 sequences.

			n-th row of table consists of n numbers A001221[A002110(n+j)], j=0...n-1:
1,
2,1,
3,1,1,
4,1,2,2,
5,1,2,2,3,
6,2,2,3,2,2,
7,3,2,3,3,2,4,
Rows starts with n at indices which are central polygonal numbers:a[A000124(n)]=n; rows ends at a[A000217(n)] terms, at triangular number indices.

Cf. A001221, A002110, A068069, A045983, A000217, A000124.

lf[x_] := Length[FactorInteger[x]] q[x_] := Apply[Times, Table[Prime[w], {w, 1, x}]] Flatten[Table[Table[lf[q[n]+j], {j, 0, n-1}], {n, 1, 20}], 1]

cp's OEIS Frontend

A067665 The start of a record-setting run of consecutive integers i with distinct A001222(i).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A072664 End of smallest run of n consecutive integers with n, n-1, ..., 1 distinct prime factors in that order.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A084296 Triangle: number of distinct prime factors in n-th primorial numbers when n prime factors first appears and in n-1 subsequent integers after.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica