A106492 -id:A106492 - cp's OEIS frontend

A106490 Total number of bases and exponents in Quetian Superfactorization of n, excluding the unity-exponents at the tips of branches.

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

Offset: 1

Antti Karttunen, May 09 2005 based on Leroy Quet's message ('Super-Factoring' An Integer) posted to SeqFan-mailing list on Dec 06 2003

nonn

Quetian Superfactorization proceeds by factoring a natural number to its unique prime-exponent factorization (p1^e1 * p2^e2 * ... pj^ej) and then factoring recursively each of the (nonzero) exponents in similar manner, until unity-exponents are finally encountered.

			a(64) = 3, as 64 = 2^6 = 2^(2^1*3^1) and there are three non-1 nodes in that superfactorization. Similarly, for 360 = 2^(3^1) * 3^(2^1) * 5^1 we get a(360) = 5. a(65536) = a(2^(2^(2^(2^1)))) = 4.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
A. Karttunen, Scheme-program for computing this sequence.
Index entries for sequences computed from exponents in factorization of n

Cf. A028234, A064372, A067029, A106444, A106491, A106492, A106493.
Cf. A276230 (gives first k such that a(k) = n, i.e., this sequence is a left inverse of A276230).
After n=1 differs from A038548 for the first time at n=24, where A038548(24)=4, while a(24)=3.

a:= proc(n) option remember; `if`(n=1, 0,
      add(1+a(i[2]), i=ifactors(n)[2]))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 07 2014

a[n_] := a[n] = If[n == 1, 0, Sum[1 + a[i[[2]]], {i,FactorInteger[n]}]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

A067029(n) = if(n<2, 0, factor(n)[1,2]);
A028234(n) = my(f = factor(n)); if (#f~, f[1, 1] = 1); factorback(f); /* after Michel Marcus */
a(n) = if(n<2, 0, 1 + a(A067029(n)) + a(A028234(n)));
for(n=1, 150, print1(a(n),", ")) \\ Indranil Ghosh, Mar 23 2017, after formula by Antti Karttunen

Additive with a(p^e) = 1 + a(e).
a(1) = 0; for n > 1, a(n) = 1 + a(A067029(n)) + a(A028234(n)). - Antti Karttunen, Mar 23 2017
Other identities. For all n >= 1:
a(A276230(n)) = n.
a(n) = A106493(A106444(n)).
a(n) = A106491(n) - A064372(n).

A106491 Total number of bases and exponents in Quetian Superfactorization of n, including the unity-exponents at the tips of branches.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 3, 3, 4, 2, 5, 2, 4, 4, 4, 2, 5, 2, 5, 4, 4, 2, 5, 3, 4, 3, 5, 2, 6, 2, 3, 4, 4, 4, 6, 2, 4, 4, 5, 2, 6, 2, 5, 5, 4, 2, 6, 3, 5, 4, 5, 2, 5, 4, 5, 4, 4, 2, 7, 2, 4, 5, 5, 4, 6, 2, 5, 4, 6, 2, 6, 2, 4, 5, 5, 4, 6, 2, 6, 4, 4, 2, 7, 4, 4, 4, 5, 2, 7, 4, 5, 4, 4, 4, 5, 2, 5, 5, 6, 2, 6

Offset: 1

Antti Karttunen, May 09 2005 based on Leroy Quet's message ('Super-Factoring' An Integer) posted to SeqFan-mailing list on Dec 06 2003

nonn

			a(64) = 5, as 64 = 2^6 = 2^(2^1*3^1) and there are 5 nodes in that superfactorization. Similarly, for 360 = 2^(3^1) * 3^(2^1) * 5^1 we get a(360) = 8. See comments at A106490.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
A. Karttunen, Scheme-program for computing this sequence.
Index entries for sequences computed from exponents in factorization of n

Cf. A000961, A028234, A064372, A067029, A106444, A106490, A106492, A106494.

a:= proc(n) option remember; `if`(n=1, 1,
      add(1+a(i[2]), i=ifactors(n)[2]))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 07 2014

a[n_] := a[n] = If[n == 1, 1, Sum[1 + a[i[[2]]], {i, FactorInteger[n]}]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

A067029(n) = if(n<2, 0, factor(n)[1,2]);
A028234(n) = my(f = factor(n)); if (#f~, f[1, 1] = 1); factorback(f); /* after Michel Marcus */
a(n) = if(n<2, 1, if(A028234(n)==1, 1 + a(A067029(n)), 1 + a(A067029(n)) + a(A028234(n))));
for(n=1, 150, print1(a(n),", ")) \\ Indranil Ghosh, Mar 23 2017, after formula by Antti Karttunen

From Antti Karttunen, Mar 23 2017: (Start)
a(1) = 1, and for n > 1, if A028234(n) = 1, a(n) = 1 + a(A067029(n)), otherwise a(n) = 1 + a(A067029(n)) + a(A028234(n)).
If n is a prime power p^k (a term of A000961), a(n) = 1 + a(k).
(End)
Other identities. For all n >= 1:
a(n) = A106490(n) + A064372(n).
a(n) = A106494(A106444(n)).

A338038 a(n) is the sum of the primes and exponents in the prime factorization of n, but ignoring 1-exponents.

Original entry on oeis.org

0, 2, 3, 4, 5, 5, 7, 5, 5, 7, 11, 7, 13, 9, 8, 6, 17, 7, 19, 9, 10, 13, 23, 8, 7, 15, 6, 11, 29, 10, 31, 7, 14, 19, 12, 9, 37, 21, 16, 10, 41, 12, 43, 15, 10, 25, 47, 9, 9, 9, 20, 17, 53, 8, 16, 12, 22, 31, 59, 12, 61, 33, 12, 8, 18, 16, 67, 21, 26, 14, 71, 10

Offset: 1

Michel Marcus, Oct 08 2020

nonn

First differs from A106492 for n=64.

			For n = 18 = 2*3^2, a(18) = 2 + (3+2) = 7.

Robert Israel, Table of n, a(n) for n = 1..10000
Chris Bispels, Muhammet Boran, Steven J. Miller, Eliel Sosis, and Daniel Tsai, v-Palindromes: An Analogy to the Palindromes, arXiv:2405.05267 [math.HO], 2024.
Daniel Tsai, A recurring pattern in natural numbers of a certain property, arXiv:2010.03151 [math.NT], 2020.
Daniel Tsai, A recurring pattern in natural numbers of a certain property, Integers (2021) Vol. 21, Article #A32.

Cf. A008474, A001694, A106492, A338039.

f:= proc(n) local t;
  add(t[1]+t[2],t=subs(1=0,ifactors(n)[2]));
end proc:
map(f, [$1..100]); # Robert Israel, Oct 13 2020

a[1] = 0; a[n_] := Plus @@ First /@ (f = FactorInteger[n]) + Plus @@ Select[Last /@ f, # > 1 &]; Array[a, 100] (* Amiram Eldar, Oct 08 2020 *)

a(n) = my(f=factor(n)); vecsum(f[,1]) + sum(k=1, #f~, if (f[k,2]!=1, f[k,2]));

a(n) = A008474(n) for powerful numbers (A001694).

cp's OEIS Frontend

A106490 Total number of bases and exponents in Quetian Superfactorization of n, excluding the unity-exponents at the tips of branches.

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A106491 Total number of bases and exponents in Quetian Superfactorization of n, including the unity-exponents at the tips of branches.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A338038 a(n) is the sum of the primes and exponents in the prime factorization of n, but ignoring 1-exponents.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula