A064920 -id:A064920 - cp's OEIS frontend

A064921 Iterate A064920 until a prime is reached.

2, 3, 3, 5, 3, 7, 5, 5, 3, 11, 3, 13, 5, 7, 5, 17, 5, 19, 5, 5, 3, 23, 3, 5, 5, 11, 3, 29, 3, 31, 17, 13, 5, 11, 5, 37, 5, 7, 3, 41, 3, 43, 5, 13, 3, 47, 5, 13, 5, 19, 5, 53, 5, 7, 5, 5, 3, 59, 5, 61, 17, 7, 13, 17, 5, 67, 5, 5, 5, 71, 5, 73, 5, 19, 3, 17, 5, 79, 5, 29, 3, 83, 5, 5, 5, 31, 5

Offset: 2

Reinhard Zumkeller, Oct 14 2001

nonn

Well defined since A064920(n) < n for nonprimes.

a(p) = p for all primes p.

			a(18) = 5 as A064920(18) = 8 and A064920(8) = 5.

Harry J. Smith, Table of n, a(n) for n = 2..1000

Cf. A064920, A064922, A064917.

gpf(n)= { local(f); f=factor(n)~; return(f[1, length(f)]) } { for (n=2, 1000, m=n; while (!isprime(m), g=gpf(m); m=m / g + g - 1); write("b064921.txt", n, " ", m) ) } \\ Harry J. Smith, Sep 29 2009

A064922 Number of iterations in A064920 to reach a prime.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 1, 1, 3, 0, 3, 0, 2, 1, 2, 0, 2, 0, 2, 2, 4, 0, 4, 2, 3, 1, 4, 0, 4, 0, 1, 1, 3, 1, 3, 0, 3, 2, 4, 0, 4, 0, 3, 1, 5, 0, 3, 1, 3, 1, 3, 0, 3, 2, 3, 3, 5, 0, 3, 0, 2, 2, 2, 1, 3, 0, 3, 3, 3, 0, 4, 0, 4, 1, 5, 1, 3, 0, 3, 1, 5, 0, 3, 3, 4, 1, 3, 0, 5, 1, 4, 2, 4, 1, 4, 0, 3, 1, 5, 0, 5, 0, 3, 3, 4

Offset: 2

Reinhard Zumkeller, Oct 14 2001

nonn

a(p) = 0 for all primes p.

			a(18) = 2 as A064920(A064920(18)) = A064920(8) = 5 = A064921(18).

Harry J. Smith, Table of n, a(n) for n = 2..1000

Cf. A064920, A064921, A064918.

gpf(n)= { local(f); f=factor(n)~; return(f[1, length(f)]) } { for (n=2, 1000, m=n; a=0; while (!isprime(m), g=gpf(m); m=m / g + g - 1; a++); write("b064922.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 29 2009

A064923 a(n) = Min { k | A064920(k) = n }.

Original entry on oeis.org

2, 3, 6, 5, 10, 7, 14, 16, 24, 11, 22, 13, 26, 39, 52, 17, 34, 19, 38, 57, 76, 23, 46, 69, 72, 115, 120, 29, 58, 31, 62, 64, 96, 155, 160, 37, 74, 111, 148, 41, 82, 43, 86, 129, 172, 47, 94, 141, 144, 235, 240, 53, 106, 159, 162, 265, 270, 59, 118, 61, 122, 183, 244

Offset: 2

Reinhard Zumkeller, Oct 14 2001

nonn

A064920(a(n)) = n. a(n) = n iff n is prime.

Harry J. Smith, Table of n, a(n) for n = 2..1000

Cf. A064920, A064924, A064919.

gpf(n)= { local(f); f=factor(n)~; return(f[1, length(f)]) } { for (n=2, 1000, k=1; until (m==n, k++; g=gpf(k); m=k / g + g - 1); write("b064923.txt", n, " ", k) ) } \\ Harry J. Smith, Sep 29 2009

A064916 a(n) = n/lpf(n) + lpf(n) - 1, where lpf = A020639 = least prime factor.

Original entry on oeis.org

2, 3, 3, 5, 4, 7, 5, 5, 6, 11, 7, 13, 8, 7, 9, 17, 10, 19, 11, 9, 12, 23, 13, 9, 14, 11, 15, 29, 16, 31, 17, 13, 18, 11, 19, 37, 20, 15, 21, 41, 22, 43, 23, 17, 24, 47, 25, 13, 26, 19, 27, 53, 28, 15, 29, 21, 30, 59, 31, 61, 32, 23, 33, 17, 34, 67, 35, 25, 36, 71, 37, 73, 38, 27

Offset: 2

Reinhard Zumkeller, Oct 14 2001

nonn

a(n) = A032742(n) + A020639(n) - 1; a(n) <= n and for n > 1 a(n) = n iff n is prime.

			a(18) = 18/2 + 2 - 1 = 10;
a(19) = 19/19 + 19 - 1 = 19.

Harvey P. Dale, Table of n, a(n) for n = 2..1001

Cf. A064920, A064917, A064918, A032742, A020639, A064919.

lpf[n_]:=Module[{lp=FactorInteger[n][[1,1]]},n/lp+lp-1]; Array[lpf, 80, 2] (* Harvey P. Dale, Sep 25 2011 *)

lpf(n)= { local(f); f=factor(n); return(f[1, 1]) } { for (n=2, 1000, L=lpf(n); a=n / L + L - 1; write("b064916.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 28 2009

a(n) = my(p = vecmin(factor(n)[,1])); n/p + p - 1; \\ Michel Marcus, Jun 19 2018

A064924 If n is prime then a(n) = n; for the subsequent nonprime positions a(n + k) = (k+1)*n; then at the next prime position a new subsequence begins.

Original entry on oeis.org

2, 3, 6, 5, 10, 7, 14, 21, 28, 11, 22, 13, 26, 39, 52, 17, 34, 19, 38, 57, 76, 23, 46, 69, 92, 115, 138, 29, 58, 31, 62, 93, 124, 155, 186, 37, 74, 111, 148, 41, 82, 43, 86, 129, 172, 47, 94, 141, 188, 235, 282, 53, 106, 159, 212, 265, 318, 59, 118, 61, 122, 183, 244

Offset: 2

Reinhard Zumkeller, Oct 14 2001

A064920(a(n)) = n.

			a(7) = A007917(7) * (A064722(7) + 1) = 7 * (0 + 1) = 7; a(8) = A007917(8) * (A064722(8) + 1) = 7 * (1 + 1) = 14; a(9) = A007917(9) * (A064722(9) + 1) = 7 * (2 + 1) = 21; a(10) = A007917(10) * (A064722(10) + 1) = 7 * (3 + 1) = 28; a(11) = 11.

R. Zumkeller, Table of n, a(n) for n = 2..10000

Cf. A064920, A064923, A007917, A064722, A064919.

import Data.List (genericTake)
a064924 n = a064924_list !! (n-1)
a064924_list = concat $ zipWith (\p g -> genericTake g [p, 2 * p ..])
   a000040_list $ zipWith (-) (tail a000040_list) a000040_list
-- Reinhard Zumkeller, Jul 05 2013

a[n_?PrimeQ] := n; a[n_] := NextPrime[n, -1]*(n - NextPrime[n, -1] + 1); Table[a[n], {n, 2, 64}] (* Jean-François Alcover, Sep 19 2011 *)
Flatten[First[#]Range[Last[#]-First[#]]&/@Partition[Prime[Range[20]],2,1]] (* Harvey P. Dale, May 03 2012 *)

{ for (n=2, 10000, if (isprime(n), a=m=n; k=2, a=k*m; k++); write("b064924.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 29 2009

a(n) = A007917(n) * (A064722(n) + 1)

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A064921 Iterate A064920 until a prime is reached.

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A064922 Number of iterations in A064920 to reach a prime.

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A064923 a(n) = Min { k | A064920(k) = n }.

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A064916 a(n) = n/lpf(n) + lpf(n) - 1, where lpf = A020639 = least prime factor.

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A064924 If n is prime then a(n) = n; for the subsequent nonprime positions a(n + k) = (k+1)*n; then at the next prime position a new subsequence begins.

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