A106490 -id:A106490 - cp's OEIS frontend

A349281 a(n) is the number of prime powers (not including 1) that are (1+e)-divisors of n.

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, 4, 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, 1, 3, 3

Offset: 1

Amiram Eldar, Nov 13 2021

(1+e)-divisors are defined in A049599.
First differs from A106490 at n = 64.
The total number of prime powers (not including 1) that divide n is A001222(n).
If p|n and p^e is the highest power of p that divides n, then the powers of p that are (1+e)-divisors of n are of the form p^d where d|e.

			8 has 3 (1+e)-divisors, 1, 2 and 8. Two of these divisors, 2 and 8 = 2^3 are prime powers. Therefore, a(8) = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A000961, A001222, A046099, A049599, A077761, A106490, A246655, A349258.

f[p_,e_] := DivisorSigma[0, e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a,100]

A349281(n) = vecsum(apply(e->numdiv(e),factor(n)[,2])); \\ Antti Karttunen, Nov 13 2021

Additive with a(p^e) = A000005(e).
a(n) <= A001222(n), with equality if and only if n is cubefree (A046099).
a(n) <= A049599(n)-1, with equality if and only if n is a prime power (including 1, A000961).
Sum_{k=1..n} a(n) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{p prime} f(1/p) = 0.51780076119050171903..., where f(x) = -x + (1-x) * Sum_{k>=1} x^k/(1-x^k). - Amiram Eldar, Sep 29 2023

A381401 a(n) is the number of (possibly non-distinct) prime elements in the multiset of bases and exponents in the prime factorization of n.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Feb 24 2025

			a(144) = 3 because the prime factorization of 144 is 2^4*3^2 and the multiset of these bases and exponents is {2, 2, 3, 4}, containing 3 primes.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A106490, A349281, A381398, A381399.

A381401[n_] := Count[FactorInteger[n], _?PrimeQ, {2}];
Array[A381401, 100]

A253783 a(1) = 0; for n>1: a(n) = A075167(1+A071178(n)) + (A061395(n) - A061395(A051119(n))) + a(A051119(n)).

Original entry on oeis.org

0, 2, 3, 3, 4, 4, 5, 3, 4, 5, 6, 5, 7, 6, 5, 4, 8, 5, 9, 6, 6, 7, 10, 5, 5, 8, 4, 7, 11, 6, 12, 4, 7, 9, 6, 6, 13, 10, 8, 6, 14, 7, 15, 8, 6, 11, 16, 6, 6, 6, 9, 9, 17, 5, 7, 7, 10, 12, 18, 7, 19, 13, 7, 5, 8, 8, 20, 10, 11, 7, 21, 6, 22, 14, 6, 11, 7, 9, 23, 7, 5, 15, 24, 8, 9, 16, 12, 8, 25, 7, 8, 12, 13, 17, 10, 6, 26, 7, 8, 7, 27, 10, 28

Offset: 1

Antti Karttunen, Jan 16 2015

nonn

An auxiliary recurrence for computing A075167.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A051119, A061395, A075167, A071178.

Cf. also A106490.

a(1) = 0; for n>1: a(n) = A075167(1+A071178(n)) + (A061395(n) - A061395(A051119(n))) + a(A051119(n)).

A338669 The prime tower factorization of a(n) is obtained by replacing the rightmost prime number by 1 in the prime tower factorization of n; a(1) = 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 4, 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 8, 5, 2, 3, 4, 1, 6, 1, 2, 3, 2, 5, 12, 1, 2, 3, 8, 1, 6, 1, 4, 9, 2, 1, 16, 7, 10, 3, 4, 1, 6, 5, 8, 3, 2, 1, 12, 1, 2, 9, 4, 5, 6, 1, 4, 3, 10, 1, 24, 1, 2, 15, 4, 7, 6, 1, 16, 9, 2, 1, 12

Offset: 1

Rémy Sigrist, Apr 23 2021

nonn

The prime tower factorization of a number is defined in A182318.

Sequence A338668 gives the rightmost prime number.

			See Links section.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Illustration of initial terms

Cf. A106490, A182318, A338668.

a(n) = { if (n==1, 1, my (f=factor(n), w=#f~, p=f[w,1], e=f[w,2]); if (e==1, n/p, n*p^(a(e)-e))) }

a(n) = 1 iff n = 1 or n is a prime number.
A106490(a(n)) = 1 + A106490(n) for any n > 1.
a^k(n) = 1 for k = A106490(n) (where a^k denotes the k-th iterate of a).

A334766 Lexicographically earliest sequence of distinct positive integers such that for any n > 0, the prime tower factorization of a(n+1) can be obtained from that of a(n) by adding or removing exactly one prime number.

Original entry on oeis.org

1, 2, 4, 12, 6, 3, 9, 18, 36, 144, 48, 16, 80, 20, 10, 5, 15, 30, 60, 180, 90, 45, 225, 75, 25, 50, 100, 300, 150, 450, 900, 3600, 720, 240, 1200, 400, 2800, 560, 112, 28, 14, 7, 21, 42, 84, 252, 126, 63, 315, 105, 35, 70, 140, 420, 210, 630, 1260, 5040, 1008

Offset: 1

Rémy Sigrist, May 10 2020

nonn

The prime tower factorization of a number is defined in A182318.
For any n > 0, a(n+1) is either a multiple or a divisor of a(n).
For any prime number p, the sequence contains a multiple of p.

			The first terms, alongside their prime tower factorizations, are:
  n   a(n)  Prime tower factorization of a(n)
  --  ----  ---------------------------------
   1     1  1
   2     2  2
   3     4  2^2
   4    12  2^2   * 3
   5     6  2     * 3
   6     3          3
   7     9          3^2
   8    18  2     * 3^2
   9    36  2^2   * 3^2
  10   144  2^2^2 * 3^2
  11    48  2^2^2 * 3
  12    16  2^2^2
  13    80  2^2^2       * 5
  14    20  2^2         * 5
  15    10  2           * 5
  16     5                5

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A334766

Cf. A106490, A182318, A298480.

PARI
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See Links section.
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abs(A106490(a(n+1)) - A106490(a(n))) = 1.

cp's OEIS Frontend

A349281 a(n) is the number of prime powers (not including 1) that are (1+e)-divisors of n.

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A381401 a(n) is the number of (possibly non-distinct) prime elements in the multiset of bases and exponents in the prime factorization of n.

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A253783 a(1) = 0; for n>1: a(n) = A075167(1+A071178(n)) + (A061395(n) - A061395(A051119(n))) + a(A051119(n)).

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A338669 The prime tower factorization of a(n) is obtained by replacing the rightmost prime number by 1 in the prime tower factorization of n; a(1) = 1.

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A334766 Lexicographically earliest sequence of distinct positive integers such that for any n > 0, the prime tower factorization of a(n+1) can be obtained from that of a(n) by adding or removing exactly one prime number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula