A077761 -id:A077761 - cp's OEIS frontend

A376885 The number of factors of n of the form p^(k!) counted with multiplicity, where p is a prime and k >= 1, when the factorization is uniquely done using the factorial-base representation of the exponents in the prime factorization of n.

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

Offset: 1

Amiram Eldar, Oct 08 2024

Let n = Product p^e be the canonical prime factorization of n. The factorization of n that is based on the factorial-base representation of the exponents is done by factoring each prime power p^e into prime powers, p^e = Product_{k} (p^(k!))^d_k where e = Sum_{k>=1} d_k * k! is the factorial-base representation of e. So, the factors in this factorization are prime powers with exponents that are factorial numbers. Each factor in the factorization, p^(k!), can have a multiplicity d in the range [1, k], so this factorization of n is a product of numbers of the form (p^(k!))^d.
The number of factors counted with multiplicity (the sum of the multiplicities d in (p^(k!))^d) is given by this sequence (analogous to A001222 for the canonical prime factorization).
The number of distinct factors p^(k!) of n is A376886(n) (analogous to A001221).
The number of divisors of n that can be constructed from partial sets of these factors (with multiplicities that are not larger than those in n) is A376887(n) (analogous to A000005), and their sum is A376888(n) (analogous to A000203).

			For n = 8 = 2^3, the representation of 3 in factorial base is 11, i.e., 3 = 1! + 2!, so 8 = (2^(1!))^1 * (2^(2!))^1 and a(8) = 1 + 1 = 2.
For n = 16 = 2^4, the representation of 4 in factorial base is 20, i.e., 4 = 2 * 2!, so 16 = (2^(2!))^2 and a(16) = 2.

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

Cf. A000142, A007623, A034968, A077761.
Cf. A000005, A000203, A001221, A001222, A376886, A376887, A376888.
Similar sequences: A064547, A318464.

fdigsum[n_] := Module[{k = n, m = 2, r, s = 0}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, s += r; m++]; s]; f[p_, e_] := fdigsum[e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

fdigsum(n) = {my(k = n, m = 2, r, s = 0); while([k, r] = divrem(k, m); k != 0 || r != 0, s += r; m++); s;}
a(n) = {my(e = factor(n)[, 2]); sum(i = 1, #e, fdigsum(e[i]));}

Additive with a(p^e) = A034968(e).

Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{p prime} f(1/p) = 0.18682321026088865388..., where f(x) = -x + (1-x) * Sum_{k>=1} A034968(k)* x^k.

A069903 Number of distinct prime factors of n-th triangular number.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Apr 10 2002

			A000217(11) = 11*(11+1)/2 = 66 = 2*3*11, therefore a(11) = 3.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000217, A001221, A059957, A069901, A069902, A069904, A077761.

PrimeNu[#]&/@Accumulate[Range[90]] (* Harvey P. Dale, Oct 06 2016 *)

a(n) = omega(n*(n+1)/2); \\ Michel Marcus, Feb 05 2021

a(n)=onega(n/gcd(n,2))+omega((n+1)/gcd(n+1)) \\ Charles R Greathouse IV, Sep 21 2024

a(n) = A001221(A000217(n)).

Sum_{k=1..n} a(k) = 2 * n * (log(log(n)) + B - 1/4) + O(n/log(n)), where B is Mertens's constant (A077761). - Amiram Eldar, Sep 21 2024

A238949 Degree of divisor lattice D(n).

Original entry on oeis.org

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

Sung-Hyuk Cha, Mar 07 2014

Antti Karttunen, Table of n, a(n) for n = 1..10000 (terms 1..200 from Sung-Hyuk Cha)
Sung-Hyuk Cha, Edgar G. DuCasse, and Louis V. Quintas, Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures, arXiv:1405.5283 [math.NT], 2014.
Index entries for sequences computed from exponents in factorization of n.

Cf. A001221, A056170, A057427, A077761, A085548.

Prepend[Table[Total[FactorInteger[n][[All, 2]] /. x_ /; x > 1 -> 2], {n, 2, 85}], 0] (* Geoffrey Critzer, Mar 02 2015 *)

a(n) = {my(f = factor(n)); sum(i=1, #f~, 1 + (f[i,2] > 1));} \\ Michel Marcus, Mar 03 2015

(define (A238949 n) (if (= 1 n) 0 (+ 1 (A057427 (+ -1 (A067029 n))) (A238949 (A028234 n))))) ;; Antti Karttunen, Jul 23 2017

a(n) = A001221(n) + A056170(n) as given in the Cha, DuCasse, Quintas reference. - Geoffrey Critzer, Mar 02 2015
Additive with a(p^e) = 1+A057427(e-1). - Antti Karttunen, Jul 23 2017
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{p prime} 1/p^2 (A085548). - Amiram Eldar, Feb 13 2024

More terms from Antti Karttunen, Jul 23 2017

A318464 Additive with a(p^e) = A007895(e), where A007895(n) gives the number of terms in Zeckendorf representation of n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 30 2018

nonn

From Amiram Eldar, Aug 09 2024: (Start)
The number of factors of n of the form p^Fibonacci(k), where p is a prime and k >= 2, when the factorization is uniquely done using the Zeckendorf representation of the exponents in the prime factorization of n.
Equivalently, the number of Zeckendorf-infinitary divisors of n (defined in A318465) that are prime powers (A246655). (End)

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

Cf. A007895, A077761, A246655, A318465, A318469.

z[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; a[n_] := Total[z /@ FactorInteger[n][[;; , 2]]]; a[1] = 0; Array[a, 100] (* Amiram Eldar, May 15 2023 *)

A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A007895(n) = { my(s=0); while(n>0, s++; n -= fibonacci(1+A072649(n))); (s); }
A318464(n) = vecsum(apply(e -> A007895(e),factor(n)[,2]));

a(n) = A007814(A318465(n)).
a(n) = A001222(A318469(n)).
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{k>=2} (A007895(k)-A007895(k-1)) * P(k) = 0.05631817952062180045..., where P(s) is the prime zeta function. - Amiram Eldar, Oct 09 2023

A376365 The number of distinct prime factors of the cubefree numbers.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Sep 21 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
Sourabhashis Das, Wentang Kuo, and Yu-Ru Liu, Distribution of omega(n) over h-free and h-full numbers, arXiv:2409.10430 [math.NT], 2024. See Theorem 1.1.

Cf. A001221, A004709, A059956, A072047, A077761, A376361, A376363, A376366.

f[k_] := Module[{e = If[k == 1, {}, FactorInteger[k][[;; , 2]]]}, If[AllTrue[e, # < 3 &], Length[e], Nothing]]; Array[f, 150]

lista(kmax) = {my(e, is); for(k = 1, kmax, e = factor(k)[, 2]; is = 1; for(i = 1, #e, if(e[i] > 2, is = 0; break)); if(is, print1(#e, ", ")));}

a(n) = A001221(A004709(n)).

Sum_{A004709(k) <= x} a(k) = (6/Pi^2) * x * (log(log(x)) + B - C) + O(x/log(x)), where B is Mertens's constant (A077761) and C = Sum_{p prime} (p-1)/(p*(p^3-1)) = 0.10770743252352371604... (Das et al., 2024).

A008481 If n = Product (p_j^k_j) then a(n) = Sum partition(k_j).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 5, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 3, 3, 1, 3, 1, 7, 2, 2, 2, 4, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 6, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 4, 1, 2, 3, 11, 2, 3, 1, 3, 2, 3, 1, 5, 1, 2, 3, 3, 2, 3, 1, 6, 5, 2, 1, 4, 2, 2, 2

Offset: 1

Olivier Gérard

a(n) is a function of the prime signature of n (cf. A025487). - Matthew Vandermast, Jun 24 2012

Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 10000 terms from Vincenzo Librandi)
Index entries for sequences computed from exponents in factorization of n.

Cf. A000041, A007814, A025487, A077761, A318312.

Differs from A318473 for the first time at n=32, where a(32)=7, while A318473(32)=8.

a:= n-> add(combinat[numbpart](i[2]), i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 30 2018

Prepend[ Array[ Plus @@ (PartitionsP /@ Last[ Transpose[ FactorInteger[ # ] ] ])&, 100, 2 ], 0 ]
(* Second program: *)
Array[Total[PartitionsP /@ FactorInteger[#][[All, -1]] - Boole[# == 1]] &, 87] (* Michael De Vlieger, Sep 02 2018 *)

A008481(n) = vecsum(apply(e -> numbpart(e),factor(n)[,2])); \\ Antti Karttunen, Aug 30 2018

From Antti Karttunen, Aug 30 2018: (Start)
Additive with a(p^e) = A000041(e).
a(n) = A007814(A318312(n)). (End)
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{p prime} f(1/p) = 1.03089282973521424158..., where f(x) = -1 + (1-x) * Product_{k>=1} (1 + x^k)/(1 - x^(2*k)). - Amiram Eldar, Sep 29 2023

Term a(1) corrected from 1 to 0 (for an empty sum) by Antti Karttunen, Aug 30 2018

A329354 a(n) = Sum_{d|n} d*omega(d).

Original entry on oeis.org

0, 2, 3, 6, 5, 17, 7, 14, 12, 27, 11, 45, 13, 37, 38, 30, 17, 62, 19, 71, 52, 57, 23, 101, 30, 67, 39, 97, 29, 162, 31, 62, 80, 87, 82, 162, 37, 97, 94, 159, 41, 220, 43, 149, 137, 117, 47, 213, 56, 152, 122, 175, 53, 197, 126, 217, 136, 147, 59, 410, 61, 157, 187, 126, 148, 336, 67, 227, 164, 342, 71, 362, 73, 187, 213, 253, 172, 394, 79

Offset: 1

Antti Karttunen, Nov 15 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A001221, A013939, A062799, A180253, A323599, A328260, A329375.

Table[Sum[d*PrimeNu[d], {d, Divisors[n]}], {n, 1, 100}] (* Vaclav Kotesovec, Aug 18 2021 *)

PARI
```
A329354(n) = sumdiv(n,d,omega(d)*d);
```

a(n) = Sum_{d|n} d*A001221(d).
a(n) = A180253(n) - A323599(n).
a(n) = A328260(n) + A329375(n).
a(n) = Sum_{d|n} (n/d) * sopf(d). - Wesley Ivan Hurt, May 24 2021
Dirichlet g.f.: primezeta(s-1) * zeta(s-1) * zeta(s). - Ilya Gutkovskiy, Aug 18 2021
Conjecture: Sum_{k=1..n} a(k) ~ Pi^2 * n^2 * (log(log(n)) + A077761) / 12. - Vaclav Kotesovec, Mar 03 2023

A376886 The number of distinct factors of n of the form p^(k!), where p is a prime and k >= 1, when the factorization is uniquely done using the factorial-base representation of the exponents in the prime factorization of n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Oct 08 2024

See A376885 for details about this factorization.
First differs from A371090 at n = 2^18 = 262144.
Differs from A064547 at n = 64, 128, 192, 256, 320, 384, 448, 512, ... .
Differs from A058061 at n = 128, 384, 512, 640, 896, ... .

			For n = 8 = 2^3, the representation of 3 in factorial base is 11, i.e., 3 = 1! + 2!, so 8 = (2^(1!))^1 * (2^(2!))^1 and a(8) = 1 + 1 = 2.
For n = 16 = 2^4, the representation of 4 in factorial base is 20, i.e., 4 = 2 * 2!, so 16 = (2^(2!))^2 and a(16) = 1.

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

Cf. A007623, A058061, A060130, A077761, A371090.

Similar sequences: A064547, A318464, A376885.

fdignum[n_] := Module[{k = n, m = 2, r, s = 0}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, If[r > 0, s++]; m++]; s]; f[p_, e_] := fdignum[e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

fdignum(n) = {my(k = n, m = 2, r, s = 0); while([k, r] = divrem(k, m); k != 0 || r != 0, if(r > 0, s ++); m++); s;}
a(n) = {my(e = factor(n)[, 2]); sum(i = 1, #e, fdignum(e[i]));}

Additive with a(p^e) = A060130(e).

Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{p prime} f(1/p) = 0.12589120926760155013..., where f(x) = -x + (1-x) * Sum_{k>=1} A060130(k) * x^k.

A092523 Number of distinct prime factors of n-th odd number.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Apr 07 2004

Ray Chandler, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Distinct Prime Factors.
Eric Weisstein's World of Mathematics, Odd Number.

Cf. A001221, A077761, A091304, A099812.

Table[PrimeNu[n],{n,1,211,2}] (* Harvey P. Dale, May 25 2015 *)

a(n) = omega(2*n-1); \\ Amiram Eldar, Sep 21 2024

a(n) = A001221(2*n-1).

Sum_{k=1..n} a(k) = n * (log(log(n)) + B - 1/2) + O(n/log(n)), where B is Mertens's constant (A077761). - Amiram Eldar, Sep 21 2024

A125029 a(n) = number of exponents in the prime factorization of n that are noncomposite.

Original entry on oeis.org

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

Offset: 1

Leroy Quet, Nov 16 2006

			a(720) = 2, since the prime factorization of 720 is 2^4 * 3^2 * 5^1 and two of the exponents in this factorization are noncomposites (the exponents 2 and 1).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A001221, A077761, A080339, A125030, A125070.

a[n_] := Length @ Select[Last /@ FactorInteger[n], # == 1 || PrimeQ[ # ] &]; a[1] = 0; Table[a[n], {n, 110}] (* Ray Chandler, Nov 19 2006 *)

A125029(n) = vecsum(apply(e -> if((1==e)||isprime(e),1,0), factorint(n)[, 2])); \\ Antti Karttunen, Jul 07 2017

From Amiram Eldar, Sep 30 2023: (Start)
Additive with a(p^e) = A080339(e).
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B - C), where B is Mertens's constant (A077761) and C = P(3) - Sum_{p prime >= 3} (P(p) - P(p+1)) = 0.05377157198303445809..., where P(s) is the prime zeta function. (End)

Extended by Ray Chandler, Nov 19 2006

cp's OEIS Frontend

A376885 The number of factors of n of the form p^(k!) counted with multiplicity, where p is a prime and k >= 1, when the factorization is uniquely done using the factorial-base representation of the exponents in the prime factorization of n.

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A069903 Number of distinct prime factors of n-th triangular number.

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A238949 Degree of divisor lattice D(n).

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A318464 Additive with a(p^e) = A007895(e), where A007895(n) gives the number of terms in Zeckendorf representation of n.

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A376365 The number of distinct prime factors of the cubefree numbers.

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A008481 If n = Product (p_j^k_j) then a(n) = Sum partition(k_j).

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A329354 a(n) = Sum_{d|n} d*omega(d).

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