A213980 -id:A213980 - cp's OEIS frontend

A222312 a(n) = n + A001222(n) - 1.

0, 2, 3, 5, 5, 7, 7, 10, 10, 11, 11, 14, 13, 15, 16, 19, 17, 20, 19, 22, 22, 23, 23, 27, 26, 27, 29, 30, 29, 32, 31, 36, 34, 35, 36, 39, 37, 39, 40, 43, 41, 44, 43, 46, 47, 47, 47, 52, 50, 52, 52, 54, 53, 57, 56, 59, 58, 59, 59, 63, 61, 63, 65, 69, 66, 68, 67, 70, 70, 72, 71, 76, 73, 75, 77, 78, 78, 80, 79, 84, 84, 83

Offset: 1

N. J. A. Sloane, Feb 16 2013

nonn

a(n) = n + (number of prime factors counted with multiplicity) - 1.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A001222, A213980.

with(numtheory); [seq(n+bigomega(n)-1,n=1..100)];

Table[n+PrimeOmega[n]-1,{n,90}] (* Harvey P. Dale, Feb 01 2014 *)

A214625 Let n=r_1r_2...*r_k is Fermi-Dirac factorization of n (see comment). Set g(n) = n + k - 1 and g_i, i>=0 (g_0(n) = n, g_1=g), is i-th iteration of g. a(n) is the minimal i such that g_i(n) is in A050376.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 1, 0, 0, 1, 0, 3, 2, 1, 0, 4, 0, 3, 2, 1, 0, 6, 0, 5, 4, 3, 2, 1, 0, 7, 6, 5, 0, 4, 0, 3, 2, 1, 0, 1, 0, 3, 2, 1, 0, 3, 3, 2, 2, 1, 0, 16, 0, 15, 14, 13, 12, 11, 0, 10, 9, 8, 0, 7, 0, 6, 5, 4, 3, 2, 0, 1, 0, 1, 0, 13, 13

Offset: 2

Vladimir Shevelev, Feb 16 2013

nonn

Recall that every n>=2 has a unique factorization over distinct numbers from A050376 which is called Fermi-Dirac factorization of n. The sequence is a dual to A213980.

Conjecture: a(n) exists for every n >= 2.

			Since 24 = 2*3*4, then g_1(24) = 24 + 3 - 1 = 26; analogously, g_1(26) = 26 +2 -1 = 27, g_1(27) = 27 + 2 - 1 = 28, g_1(28) = 28 + 2 - 1 = 29 is in A050376. We used 4 iterations, therefore, a(24) = 4.

Amiram Eldar, Table of n, a(n) for n = 2..10000

Cf. A050376, A213980.

f[1]=0; f[n_] := Plus @@ (DigitCount[Last /@ FactorInteger[n], 2, 1]); g[n_] := n + f[n] - 1; a[n_] := Length @ FixedPointList[g, n]; Array[a, 30] (* Amiram Eldar, Sep 17 2019 *)

a(63) corrected by Amiram Eldar, Sep 17 2019

cp's OEIS Frontend

A222312 a(n) = n + A001222(n) - 1.

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A214625 Let n=r_1r_2...*r_k is Fermi-Dirac factorization of n (see comment). Set g(n) = n + k - 1 and g_i, i>=0 (g_0(n) = n, g_1=g), is i-th iteration of g. a(n) is the minimal i such that g_i(n) is in A050376.

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cp's OEIS Frontend

A222312 a(n) = n + A001222(n) - 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A214625 Let n=r_1*r_2*...*r_k is Fermi-Dirac factorization of n (see comment). Set g(n) = n + k - 1 and g_i, i>=0 (g_0(n) = n, g_1=g), is i-th iteration of g. a(n) is the minimal i such that g_i(n) is in A050376.

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Author

Keywords

Comments

Examples

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Programs

Mathematica

Extensions

A214625 Let n=r_1r_2...*r_k is Fermi-Dirac factorization of n (see comment). Set g(n) = n + k - 1 and g_i, i>=0 (g_0(n) = n, g_1=g), is i-th iteration of g. a(n) is the minimal i such that g_i(n) is in A050376.