A078897 -id:A078897 - cp's OEIS frontend

A078899 Number of times the greatest prime factor of n is the greatest prime factor for numbers <=n; a(1)=1.

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

Offset: 1

Reinhard Zumkeller, Dec 12 2002

For n>1: a(n)=1 iff n is prime;

a(n) = n/p for n<=p*(p+1) and p = greatest prime factor of n.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A006530, A078898, A078897.

p:= proc() 0 end:
a:= proc(n) option remember; local t;
      t:= max(numtheory[factorset](n)[]);
      p(t):= p(t)+1
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 09 2015

p[_] = 0; a[1] = 1;
a[n_] := a[n] = Module[{t}, t = FactorInteger[n][[-1, 1]]; p[t] = p[t]+1];
Array[a, 100] (* Jean-François Alcover, Jun 09 2018, after Alois P. Heinz *)

Ordinal transform of A006530 (Gpf). - Franklin T. Adams-Watters, Aug 28 2006

A078896 Number of times the smallest prime factor of n is a factor in all numbers <= n; a(1) = 1.

Original entry on oeis.org

1, 1, 1, 3, 1, 4, 1, 7, 4, 8, 1, 10, 1, 11, 6, 15, 1, 16, 1, 18, 9, 19, 1, 22, 6, 23, 13, 25, 1, 26, 1, 31, 15, 32, 8, 34, 1, 35, 18, 38, 1, 39, 1, 41, 21, 42, 1, 46, 8, 47, 23, 49, 1, 50, 13, 53, 27, 54, 1, 56, 1, 57, 30, 63, 15, 64, 1, 66, 32, 67, 1, 70, 1, 71, 35, 73, 12, 74, 1, 78, 40

Offset: 1

Reinhard Zumkeller, Dec 12 2002

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A020639, A078897, A078898.

With[{s = Array[Flatten@ Map[ConstantArray[#1, #2] & @@ # &, FactorInteger[#]] &, 81]}, Array[Count[Take[s, #1], #2, 2] & @@ {#, s[[#, 1]]} &, Length@ s]] (* Michael De Vlieger, Dec 16 2017 *)

a(n) = if (n==1, 1, my(p = factor(n)[1, 1]); sum(i=1, n, valuation(i, p))); \\ Michel Marcus, Dec 27 2014

For n>1, a(n) = 1 iff n is prime.
a(n) = A078897(n) iff n is a prime power (A000961).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Sum_{p prime} (1/(p*(p-1))) * Product_{primes q < p} (1-1/q) = 0.6125177915489... . - Amiram Eldar, May 14 2025

A252890 Number of times the greatest prime factor of n^2 + 1 is a factor in all numbers <= n.

Original entry on oeis.org

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

Offset: 1

Michel Lagneau, Dec 24 2014

nonn

The greatest prime factor is counted with multiplicity (see the example).

a(n)=1 iff n^2 + 1 is prime.

			a(7)=4 because 7^2 + 1 = 50 and 5 is 4 times a factor:
2^2+1 = 5;
3^2+1 = 10 = 2*5;
7^2+1 = 50 = 2*5*5 (two times).

Michel Lagneau, Table of n, a(n) for n = 1..10000

Cf. A089120, A014442, A078897.

with(numtheory): with(padic,ordp):
f:= proc(n) local p ,q, n0;
  q:=factorset(n^2+1);n0:=nops(q);p:= q[n0];
  add(ordp(k^2+1, p), k=1..n);
end proc:
seq(f(n), n=1.. 100);
# Using code from Robert Israel adapted for this sequence. See A078897.

cp's OEIS Frontend

A078899 Number of times the greatest prime factor of n is the greatest prime factor for numbers <=n; a(1)=1.

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A078896 Number of times the smallest prime factor of n is a factor in all numbers <= n; a(1) = 1.

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A252890 Number of times the greatest prime factor of n^2 + 1 is a factor in all numbers <= n.

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Maple