A323075 -id:A323075 - cp's OEIS frontend

A323164 a(n) = A000720(A323075(n)).

0, 1, 2, 2, 3, 2, 4, 3, 4, 2, 5, 4, 6, 3, 5, 4, 7, 2, 8, 5, 5, 4, 9, 6, 5, 3, 8, 5, 10, 4, 11, 7, 9, 2, 10, 8, 12, 5, 8, 5, 13, 4, 14, 9, 11, 6, 15, 5, 14, 3, 10, 8, 16, 5, 11, 10, 8, 4, 17, 11, 18, 7, 14, 9, 16, 2, 19, 10, 15, 8, 20, 12, 21, 5, 10, 8, 19, 5, 22, 13, 11, 4, 23, 14, 15, 9, 17, 11, 24, 6, 22, 15, 14, 5, 19, 14, 25, 3, 19, 10, 26, 8, 27, 16, 20

Offset: 1

Antti Karttunen, Jan 08 2019

nonn

One less than the restricted growth sequence transform of A323075.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000079, A000720, A000918, A323075, A323165 (ordinal transform).

A060681[n_] := n - If[1 == n, n, n/Min[FactorInteger[n][[All, 1]]]];
A323075[n_] := With[{nn = 1 + A060681[n]}, If[nn == n, n, A323075[nn]]];
a[n_] = PrimePi[A323075[n]];
Array[a, 105] (* Jean-François Alcover, Jan 10 2022, from PARI code *)

A060681(n) = (n-if(1==n,n,n/vecmin(factor(n)[,1])));
A323075(n) = { my(nn = 1+A060681(n)); if(nn==n,n,A323075(nn)); };
A323164(n) = primepi(A323075(n));

a(n) = A000720(A323075(n)).

For all odd primes p, a(2^k * p - 2^(k+1) + 2) = a(A000079(k) * p - A000918(k+1)) = A000720(p) for k >= 0. - After David A. Corneth's similar observation for A323075.

A323165 Ordinal transform of A323075 (and also of A323164).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 08 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A060681, A323075, A323164.

f[n_] := n - n/FactorInteger[n][[1, 1]]; (* f is A060681 *)
g[n_] := g[n] = If[n == 1+f[n], n, g[1+f[n]]]; (* g is A323075 *)
b[_] = 0;
a[n_] := a[n] = With[{t = g[n]}, b[t] = b[t]+1];
Array[a, 105] (* Jean-François Alcover, Dec 20 2021 *)

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A060681(n) = (n-if(1==n,n,n/vecmin(factor(n)[,1])));
A323075(n) = { my(nn = 1+A060681(n)); if(nn==n,n,A323075(nn)); };
v323165 = ordinal_transform(vector(up_to,n,A323075(n)));
A323165(n) = v323165[n];

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)).

cp's OEIS Frontend

A323164 a(n) = A000720(A323075(n)).

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A323165 Ordinal transform of A323075 (and also of A323164).

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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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Programs

Mathematica

PARI

Formula