A122810 -id:A122810 - cp's OEIS frontend

A122815 Terms in A122375 that differ from corresponding term in A122810.

2, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 4, 4, 5, 4, 5, 5, 4, 6, 4, 6, 4, 6, 5, 5, 4, 6, 5, 6, 6, 4, 6, 5, 4, 6, 5, 6, 4, 5, 7, 6, 4, 6, 4, 6, 5, 5, 4, 7, 6, 4, 7, 7, 5, 4, 7, 6, 4, 5, 7, 6, 4, 6, 4, 7, 4, 5, 5, 6, 6, 4, 7, 5, 4, 7, 5, 7, 4, 7, 6, 6, 5, 8, 4, 4, 6, 6, 4, 7, 5, 4, 4, 4, 6, 6, 5, 8, 7, 5, 6, 4, 8

Offset: 1

Ray Chandler, Sep 22 2006

nonn

Cf. A122375, A122810, A122812, A122816.

a(n) = A122375(A122812(n)).

A122816 Terms in A122810 that differ from corresponding term in A122375.

Original entry on oeis.org

3, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 5, 5, 6, 5, 6, 6, 5, 7, 5, 7, 5, 7, 6, 6, 5, 7, 6, 7, 7, 5, 7, 6, 5, 8, 6, 7, 5, 6, 8, 7, 5, 7, 5, 7, 6, 6, 5, 8, 7, 5, 8, 8, 6, 5, 8, 7, 5, 6, 8, 7, 5, 7, 5, 8, 5, 6, 6, 7, 7, 5, 8, 6, 5, 8, 6, 8, 5, 8, 7, 7, 6, 9, 5, 5, 7, 7, 5, 8, 6, 5, 5, 5, 7, 7, 6, 9, 8, 6, 7, 5, 9

Offset: 1

Ray Chandler, Sep 22 2006

nonn

Cf. A122375, A122810, A122812, A122815.

a(n) = A122810(A122812(n)).

A122812 Numbers k where A046523(A005179(k)) differs from A046523(A038547(k)).

Original entry on oeis.org

8, 24, 48, 64, 72, 80, 108, 112, 128, 144, 160, 162, 176, 192, 208, 216, 224, 243, 256, 272, 288, 304, 320, 324, 352, 368, 384, 416, 432, 448, 464, 480, 486, 496, 512, 544, 576, 592, 608, 640, 648, 656, 672, 688, 704, 729, 736, 752, 768, 832, 848, 864, 896

Offset: 1

Ray Chandler, Sep 22 2006

nonn

Where the prime signature of the least number with exactly k divisors differs from the prime signature of the least odd number with exactly k divisors.
Also where A122375(k) differs from A122810(k).
Also where A122376(k) differs from A122811(k).

Amiram Eldar, Table of n, a(n) for n = 1..2000

Cf. A005179, A038547, A046523, A122375, A122376, A122810-A122819.

A122375 Number of distinct prime factors of the smallest number with exactly n divisors.

Original entry on oeis.org

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, 3, 3, 1, 3, 1, 4, 2, 2, 2, 4, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 4, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 4, 1, 2, 3, 4, 2, 3, 1, 3, 2, 3, 1, 4, 1, 2, 3, 3, 2, 3, 1, 4, 4, 2, 1, 4, 2, 2, 2, 4, 1, 4, 2, 3, 2, 2, 2, 5, 1, 3, 3, 4, 1, 3, 1, 4, 3

Offset: 1

Lekraj Beedassy, Aug 30 2006

nonn

a(n) = 1 iff n is prime.

			a(14) = 2 as A005179(14) = 192 which has 2 distinct prime divisors (2 and 3). - _David A. Corneth_, Dec 10 2021

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 2000 terms from Antti Karttunen)

Cf. A001221, A005179, A038457, A122810, A130279.

A005179 = Cases[Import["https://oeis.org/A005179/b005179.txt", "Table"], {, }][[All, 2]];
a[n_] := PrimeNu[A005179[[n]]];
a /@ Range[2000] (* Jean-François Alcover, Mar 06 2020 *)

a(n) = omega(A005179(n)), where omega(n) = A001221(n).

a(n) = A001221(A130279(n)). - Reinhard Zumkeller, May 21 2007

Edited and extended by Ray Chandler, Sep 11 2006

A366989 The number of prime powers p^q dividing n, where p is prime and q is either 1 or prime (A334393 without the first term 1).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Oct 31 2023

First differs from A122810 at n = 48, and from A318322 at n = 64.

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

Cf. A000720, A001221, A005117, A077761, A334393, A366988, A366991, A366992, A366994.

Cf. A122810, A318322.

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

a(n) = {my(f = factor(n)); sum(i = 1, #f~, 1 + primepi(f[i, 2]));}

Additive with a(p^e) = A000720(e) + 1.
a(n) = 1 is and only if n is squarefree (A005117) > 1.
a(n) = A366988(n) + A001221(n).
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761), C = Sum_{p prime} P(p) = 0.67167522222173297323..., and P(s) is the prime zeta function.

A318322 Additive with a(p^e) = A007306(e).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 31 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A007306, A318306, A318316.

Differs from A122810 for the first time at n=48, where a(48) = 4, while A122810(48) = 5.

A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A007306(n) = if(!n,1,A002487(n+n-1));
A318322(n) = vecsum(apply(e -> A007306(e),factor(n)[,2]));

from functools import reduce
from sympy import factorint
def A318322(n): return sum(sum(reduce(lambda x,y:(x[0],sum(x)) if int(y) else (sum(x),x[1]),bin((e<<1)-1)[-1:2:-1],(1,0))) for e in factorint(n).values()) # Chai Wah Wu, May 18 2023

a(n) = A007814(A318316(n)).

cp's OEIS Frontend

A122815 Terms in A122375 that differ from corresponding term in A122810.

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A122816 Terms in A122810 that differ from corresponding term in A122375.

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A122812 Numbers k where A046523(A005179(k)) differs from A046523(A038547(k)).

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A122375 Number of distinct prime factors of the smallest number with exactly n divisors.

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A366989 The number of prime powers p^q dividing n, where p is prime and q is either 1 or prime (A334393 without the first term 1).

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A318322 Additive with a(p^e) = A007306(e).

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