A064918 -id:A064918 - cp's OEIS frontend

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

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

A323076 Number of iterations of map x -> 1+(x-(largest divisor d < x)), starting from x=n, needed to reach a fixed point, which is always either a prime or 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 1, 1, 3, 0, 1, 0, 2, 1, 2, 0, 4, 0, 1, 2, 2, 0, 1, 3, 3, 1, 2, 0, 3, 0, 1, 1, 5, 1, 1, 0, 2, 2, 3, 0, 3, 0, 1, 1, 2, 0, 4, 1, 4, 2, 2, 0, 3, 2, 1, 3, 4, 0, 1, 0, 2, 1, 2, 1, 6, 0, 2, 1, 2, 0, 1, 0, 3, 3, 3, 1, 4, 0, 1, 3, 4, 0, 1, 2, 2, 1, 2, 0, 3, 1, 1, 2, 5, 2, 2, 0, 5, 1, 3, 0, 3, 0, 1, 1

Offset: 1

Antti Karttunen, Jan 04 2019

nonn

Differs from A064918 at n = 25, 48, 51, 69, 75, 81, 85, 94, 95, 99, 100, 111, 115, 121, ...

Antti Karttunen, Table of n, a(n) for n = 1..20669
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A060681, A064918, A323075 (the fixed points reached), A323077, A323079.

Cf. also A039651.

{0}~Join~Array[-2 + Length@ NestWhileList[1 + (# - Divisors[#][[-2]]) &, #, UnsameQ, All] &, 104, 2] (* Michael De Vlieger, Jan 04 2019 *)

A060681(n) = (n-if(1==n,n,n/vecmin(factor(n)[,1])));
A323076(n) = { my(nn = 1+A060681(n)); if(nn==n,0,1+A323076(nn)); };

If n == (1+A060681(n)), then a(n) = 0, otherwise a(n) = 1 + a(1+A060681(n)).

A064917 a(n) is the result of beginning with n and iterating k -> A064916(k) until a prime is reached.

Original entry on oeis.org

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

Offset: 2

Reinhard Zumkeller, Oct 14 2001

nonn

Well-defined since A064916(n) < n for nonprimes.

a(p) = p for all primes p.

			a(6) = 3 as A064916(6) = 4 and A064916(4) = 3.

Harry J. Smith, Table of n, a(n) for n = 2..1000
Michael Gilleland, Some Self-Similar Integer Sequences

Cf. A064916, A064918, A064921.

lpf(n)= { local(f); f=factor(n); return(f[1, 1]) } { for (n=2, 1000, m=n; while (!isprime(m), L=lpf(m); m=m / L + L - 1); write("b064917.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

cp's OEIS Frontend

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

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A323076 Number of iterations of map x -> 1+(x-(largest divisor d < x)), starting from x=n, needed to reach a fixed point, which is always either a prime or 1.

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A064917 a(n) is the result of beginning with n and iterating k -> A064916(k) until a prime is reached.

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A064922 Number of iterations in A064920 to reach a prime.

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