A160649 -id:A160649 - cp's OEIS frontend

A160650 a(n) = A001222(A160649(n)) = A160649(n+1) - A160649(n); where A001222(m) is the sum of prime-factorization exponents of m (or, A001222(m) is the number of primes dividing m, counted with multiplicity).

1, 1, 2, 2, 3, 1, 3, 2, 1, 3, 2, 1, 4, 3, 1, 5, 1, 2, 4, 3, 1, 5, 1, 4, 2, 4, 6, 3, 1, 2, 3, 1, 5, 2, 2, 1, 4, 2, 6, 3, 3, 5, 1, 3, 3, 5, 3, 7, 4, 1, 4, 6, 4, 3, 1, 2, 6, 2, 5, 1, 3, 2, 1, 5, 2, 2, 4, 1, 2, 4, 5, 2, 3, 4, 2, 6, 3, 4, 1, 3, 1, 4, 3, 1, 3, 3, 4, 5, 1, 3, 3, 5, 1, 5, 3, 2, 5, 3, 7, 2, 4, 2, 2, 2, 1

Offset: 1

Leroy Quet, May 21 2009

nonn

Cf. A001222, A160649.

Extended by Ray Chandler, Jun 16 2009

A375508 Begin A160649 with n instead of 2; a(n) is the position in the new sequence at which it generates the same numbers as A160649 or a(n)=0 if it doesn't.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 5, 4, 1, 3, 1, 1, 3, 2, 1, 2, 1, 1, 6, 2, 5, 1, 5, 4, 1, 1, 3, 3, 2, 2, 1, 1, 5, 1, 4, 3, 2, 1, 2, 2, 1, 1, 3, 2, 2, 5, 1, 1, 4, 2, 3, 1, 2, 1, 3, 2, 7, 1, 7, 6, 6, 5, 5, 1, 4, 3, 1, 1, 4, 1, 2, 3, 1, 1, 2, 9, 9, 8, 1, 8, 1, 7

Offset: 1

James C. McMahon, Aug 18 2024

nonn

The indices of the matching entries of A160649 and this sequence do not necessarily have to be the same (see Examples).

			Using () to indicate the point at which the new sequence generates the same numbers as A160649:
A160649: 2, 3, 4, 6, 8, 11, 12...  a(1)=1
Start=3: (3), 4, 6, 8, 11, 12...   a(2)=1
Start=4: (4), 6, 8, 11, 12, 15...  a(3)=1
Start=5: 5, (6), 8, 11, 12, 15...  a(4)=2

James C. McMahon, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Histogram of the frequency of a(n) = k, n = 1..2^20, k = 1..14. Maximum in the dataset is k = 41.
Michael De Vlieger, Histogram of the frequency of a(n) = k, n = 1..2^24, k = 1..14. Maximum in the dataset is k = 57.
Wikipedia, Kruskal count

Cf. A160649.

Lim=88;pseq1=NestList[#+PrimeOmega[#]&,2,Lim] (* pseq1 is base sequence A160649 *); pseq={}; Do[ i=1; s=n; While[!MemberQ[pseq1, s], s=s+PrimeOmega[s]; i++]; AppendTo[pseq, i], {n,2, Lim}];pseq (* pseq is A375508 *)

A094222 a(n+1) = a(n) + (number of distinct prime factors of a(n)) for n>1; a(1)=1, a(2)=2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 16, 17, 18, 20, 22, 24, 26, 28, 30, 33, 35, 37, 38, 40, 42, 45, 47, 48, 50, 52, 54, 56, 58, 60, 63, 65, 67, 68, 70, 73, 74, 76, 78, 81, 82, 84, 87, 89, 90, 93, 95, 97, 98, 100, 102, 105, 108, 110, 113, 114, 117, 119, 121, 122, 124

Offset: 1

Reinhard Zumkeller, May 28 2004

nonn

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A001221, A160649.

A094222 := proc(n)
    option remember;
    if n <= 2 then
        n;
    else
        procname(n-1)+A001221(procname(n-1)) ;
    end if;
end proc:
seq(A094222(n),n=1..30) ; # R. J. Mathar, Jul 09 2016

a[1] = 1; a[2] = 2; a[n_] := a[n] = a[n-1] + PrimeNu[a[n-1]]; Array[a, 66] (* Jean-François Alcover, Sep 13 2016 *)

A337455 Numbers of the form m + bigomega(m) with m a positive integer.

Original entry on oeis.org

1, 3, 4, 6, 8, 11, 12, 14, 15, 16, 17, 18, 20, 21, 23, 24, 27, 28, 30, 31, 32, 33, 35, 36, 37, 38, 40, 41, 42, 44, 45, 47, 48, 51, 53, 54, 55, 57, 58, 59, 60, 62, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 84, 85, 87, 88, 89, 90, 92, 93

Offset: 1

Nathan J. McDougall, Aug 27 2020

nonn

If a(n) = m + A001222(m) then (a(n) - m) <= log(a(n))/log(2).

It appears that a(n)/n may converge to a constant around ~ 1.49.

			a(7) = 10 + A001222(10) = 10 + 2 = 12

Petr Kucheriaviy, On numbers not representable as n + ω(n), arXiv preprint (2022). arXiv:2203.12006 [math.NT]

Cf. A001222 (bigomega), A064800, A358973.
Numbers of the form k^n+n where k is prime are subsequences: A006127 (k=2), A104743 (k=3), A104745 (k=5), A226199 (k=7), A226737 (k=11).
Subsequences include A008864, A101340, and A160649 (excluding the first term).

m = 100; Select[Union @ Table[n + PrimeOmega[n], {n, 1, m}], # <= m &] (* Amiram Eldar, Aug 28 2020 *)

upto(limit)=Set(select(t->t<=limit, apply(m->m+bigomega(m), [1..limit]))) \\ Andrew Howroyd, Aug 27 2020

list(lim)=my(v=List()); forfactored(n=1, lim\1-1, my(t=n[1]+bigomega(n)); if(t<=lim, listput(v, t))); Set(v) \\ Charles R Greathouse IV, Dec 07 2022

Kucheriaviy proves that a(n) << n log log n and conjectures that a(n) ≍ n, that is, these numbers have positive lower density. - Charles R Greathouse IV, Dec 07 2022

cp's OEIS Frontend

A160650 a(n) = A001222(A160649(n)) = A160649(n+1) - A160649(n); where A001222(m) is the sum of prime-factorization exponents of m (or, A001222(m) is the number of primes dividing m, counted with multiplicity).

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Extensions

A375508 Begin A160649 with n instead of 2; a(n) is the position in the new sequence at which it generates the same numbers as A160649 or a(n)=0 if it doesn't.

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A094222 a(n+1) = a(n) + (number of distinct prime factors of a(n)) for n>1; a(1)=1, a(2)=2.

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Mathematica

A337455 Numbers of the form m + bigomega(m) with m a positive integer.

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Mathematica

PARI

PARI

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