A077761 -id:A077761 - cp's OEIS frontend

A125030 a(n) = sum of exponents in the prime factorization of n that are noncomposite.

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

Offset: 1

Leroy Quet, Nov 16 2006

			a(720) = 3, 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, whose sum is 3).

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

Cf. A077761, A001222, A125029, A125071.

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

A125030(n) = vecsum(apply(e -> if((1==e)||isprime(e),e,0), factorint(n)[, 2])); \\ Antti Karttunen, Jul 07 2017

From Amiram Eldar, Sep 30 2023: (Start)
Additive with a(p^e) = e if e is composite, and 0 otherwise.
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = - P(2) + Sum_{p prime} p * (P(p) - P(p+1)) = 0.52262278983683613884..., where P(s) is the prime zeta function. (End)

Extended by Ray Chandler, Nov 19 2006

A125073 a(n) = sum of the exponents in the prime factorization of n which are triangular numbers.

Original entry on oeis.org

0, 1, 1, 0, 1, 2, 1, 3, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 4, 0, 2, 3, 1, 1, 3, 1, 0, 2, 2, 2, 0, 1, 2, 2, 4, 1, 3, 1, 1, 1, 2, 1, 1, 0, 1, 2, 1, 1, 4, 2, 4, 2, 2, 1, 2, 1, 2, 1, 6, 2, 3, 1, 1, 2, 3, 1, 3, 1, 2, 1, 1, 2, 3, 1, 1, 0, 2, 1, 2, 2, 2, 2, 4, 1, 2, 2, 1, 2, 2, 2, 1, 1, 1, 1, 0, 1, 3, 1, 4, 3

Offset: 1

Leroy Quet, Nov 18 2006

			The prime factorization of 360 is 2^3 *3^2 *5^1. There are two exponents in this factorization which are triangular numbers, 1 and 3. So a(360) = 1 + 3 = 4.

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

Cf. A010054, A077761, A125072, A125030, A125071.

f[n_] := Plus @@ Select[Last /@ FactorInteger[n], IntegerQ[Sqrt[8# + 1]] &];Table[f[n], {n, 110}] (* Ray Chandler, Nov 19 2006 *)

A010054(n) = issquare(8*n + 1); \\ This function from Michael Somos, Apr 27 2000.
A125073(n) = vecsum(apply(e -> (A010054(e)*e), factorint(n)[, 2])); \\ Antti Karttunen, Jul 08 2017

Additive with a(p^e) = A010054(e)*e. - Antti Karttunen, Jul 08 2017

Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = -P(2) + Sum_{k>=2} (k*(k+1)/2) * (P(k*(k+1)/2) - P(k*(k+1)/2 + 1)) = -0.10099019472003733178..., where P(s) is the prime zeta function. - Amiram Eldar, Sep 28 2023

Extended by Ray Chandler, Nov 19 2006

A318306 Additive with a(p^e) = A002487(e).

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

Offset: 1

Antti Karttunen, Aug 29 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. A002487, A077761, A318307.

Cf. also A046644.

A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A318306(n) = vecsum(apply(e -> A002487(e),factor(n)[,2]));

from functools import reduce
from sympy import factorint
def A318306(n): return sum(sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(e)[-1:2:-1],(1,0))) for e in factorint(n).values()) # Chai Wah Wu, May 18 2023

a(n) = A007814(A318307(n)).

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) = 0.15790080909728804399..., where f(x) = -x + x * (1-x) * Product{k>=0} (1 + x^(2^k) + x^(2^(k + 1))). - Amiram Eldar, Feb 11 2024

A318440 a(n) = A046645(n) - A007814(n); the 2-adic valuation of A299150.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 2, 3, 1, 3, 1, 2, 2, 1, 1, 2, 3, 1, 4, 2, 1, 2, 1, 3, 2, 1, 2, 4, 1, 1, 2, 2, 1, 2, 1, 2, 4, 1, 1, 4, 3, 3, 2, 2, 1, 4, 2, 2, 2, 1, 1, 3, 1, 1, 4, 4, 2, 2, 1, 2, 2, 2, 1, 4, 1, 1, 4, 2, 2, 2, 1, 4, 7, 1, 1, 3, 2, 1, 2, 2, 1, 4, 2, 2, 2, 1, 2, 4, 1, 3, 4, 4, 1, 2, 1, 2, 3

Offset: 1

Antti Karttunen, Sep 02 2018

After two initial terms, all terms are positive.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000120, A007814, A046645, A077761, A299150, A318656.

Cf. also A305439.

f[p_, e_] := (1 + Mod[p, 2])*e - DigitCount[e, 2, 1]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 28 2023 *)

A007814(n) = valuation(n,2);
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A046645(n) = vecsum(apply(e -> A005187(e),factor(n)[,2]));
A318440(n) = A046645(n) - A007814(n);

a(n) = A046645(n) - A007814(n).
a(n) = A007814(A299150(n)).
Additive with a(p^e) = (1 + (p mod 2))*e - A000120(e). - Amiram Eldar, Apr 28 2023
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = -1 + Sum_{p prime} f(1/p) = 0.410258867603361890498..., where f(x) = -x + Sum_{k>=0} (2^(k+1)-1)*x^(2^k)/(1+x^(2^k)). - Amiram Eldar, Sep 30 2023

A318473 Additive with a(p^e) = A000045(e+1).

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, 8, 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, 13, 2, 3, 1, 3, 2, 3, 1, 5, 1, 2, 3, 3, 2, 3, 1, 6, 5, 2, 1, 4, 2, 2, 2, 4, 1, 4, 2, 3, 2, 2, 2, 9, 1, 3, 3, 4, 1, 3, 1, 4, 3

Offset: 1

Antti Karttunen, Aug 29 2018

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

Cf. A000045, A077761, A318471, A318474.

Differs from A008481 for the first time at n=32, where a(32)=8, while A008481(32)=7.

a[n_] := Total@ Fibonacci[FactorInteger[n][[;; , 2]] + 1]; a[1] = 0; Array[a, 100] (* Amiram Eldar, May 15 2023 *)

A318473(n) = vecsum(apply(e -> fibonacci(1+e),factor(n)[,2]));

a(n) = A007814(A318474(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} Fibonacci(k-1) * P(k) = 1.30985781707683753402..., where P(s) is the prime zeta function. - Amiram Eldar, Oct 09 2023

A332423 If n = Product (p_j^k_j) then a(n) = Sum ((-1)^(k_j + 1) * k_j).

Original entry on oeis.org

0, 1, 1, -2, 1, 2, 1, 3, -2, 2, 1, -1, 1, 2, 2, -4, 1, -1, 1, -1, 2, 2, 1, 4, -2, 2, 3, -1, 1, 3, 1, 5, 2, 2, 2, -4, 1, 2, 2, 4, 1, 3, 1, -1, -1, 2, 1, -3, -2, -1, 2, -1, 1, 4, 2, 4, 2, 2, 1, 0, 1, 2, -1, -6, 2, 3, 1, -1, 2, 3, 1, 1, 1, 2, -1, -1, 2, 3, 1, -3

Offset: 1

Ilya Gutkovskiy, Feb 12 2020

Sum of odd exponents in prime factorization of n minus the sum of even exponents in prime factorization of n.

			a(2700) = a(2^2 * 3^3 * 5^2) = -2 + 3 - 2 = -1.

Cf. A001222, A037916, A067255, A071321, A077761, A195017, A316523, A319273, A332422, A350386, A350387.

a[n_] := Plus @@ ((-1)^(#[[2]] + 1) #[[2]] & /@ FactorInteger[n]); a[1] = 0; Table[a[n], {n, 1, 80}]

a(n) = vecsum(apply(x -> (-1)^(x+1) * x, factor(n)[, 2])); \\ Amiram Eldar, Oct 09 2023

From Amiram Eldar, Oct 09 2023: (Start)
Additive with a(p^e) = (-1)^(e+1) * e.
a(n) = A350387(n) - A350386(n).
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B - C), where B is Mertens's constant (A077761) and C = Sum_{p prime} (3*p+1)/(p*(p+1)^2) = 0.81918453457738985491 ... . (End)

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

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, 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

A366078 The number of distinct prime factors of the cubefree part of n (A360539).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Sep 28 2023

The number of exponents smaller than 3 in the prime factorization of n.

The number of prime factors of the cubefree part of n (A360539), counted with multiplicity is A366077(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk and Matilde Lalín, Sums of omega(n) and Omega(n) over the k-free parts and k-full parts of some particular sequences, Integers, Vol. 22 (2022), Article #A113.

Cf. A001221, A004709, A005117, A036966, A077761, A085541, A295659, A360539, A366077.

f[p_, e_] := If[e < 3, 1, 0]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecsum(apply(x -> if(x < 3, 1, 0), factor(n)[, 2]));

a(n) = A001221(A360539(n)).
a(n) = A001221(n) - A295659(n).
Additive with a(p^e) = 1 if e <= 2, and a(p^e) = 0 for e >= 3.
a(n) >= 0, with equality if and only if n is cubefull (A036966).
a(n) <= A001221(n), with equality if and only if n is cubefree (A004709).
a(n) <= A366077(n), with equality if and only if n is squarefree (A005117).
Sum_{k=1..m} a(k) = n * (log(log(n)) + B - C) + O(n/log(n)), where B is Mertens's constant (A077761) and C = Sum_{p prime} 1/p^3 = 0.174762... (A085541).

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.

A376364 The number of unitary divisors that are cubes of primes applied to the cubefull numbers.

Original entry on oeis.org

0, 1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 1, 1, 0, 0, 1, 0, 1, 2, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 2, 0, 2, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 0, 2, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 2, 1, 0, 0, 0, 1, 1, 0, 1, 1, 3, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 2, 0, 2, 0, 0, 1, 2, 1, 0, 0, 0, 1, 1, 2, 2, 1, 0, 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, On the number of prime factors with a given multiplicity over h-free and h-full numbers, Journal of Number Theory, Vol. 267 (2025), pp. 176-201; arXiv preprint, arXiv:2409.11275 [math.NT], 2024. See Theorem 1.3.
Index entries for sequences related to powerful numbers.

Cf. A036966, A077761, A295883, A362974, A376362, A376363.

f[k_] := Module[{e = If[k == 1, {}, FactorInteger[k][[;; , 2]]]}, If[AllTrue[e, # > 2 &], Count[e, 3], Nothing]]; Array[f, 60000]

lista(kmax) = {my(e, is); for(k = 1, kmax, e = factor(k)[, 2]; is = 1; for(i = 1, #e, if(e[i] < 3, is = 0; break)); if(is, print1(#select(x -> x == 3, e), ", ")));}

a(n) = A295883(A036966(n)).

Sum_{A036966(k) <= x} a(k) = c * sqrt(x) * (log(log(x)) + B - log(3) - L(3, 6)) + O(x^(1/3)/log(x)), where c = A362974, B is Mertens's constant (A077761), L(h, r) = Sum_{p prime} 1/(p^(r/h - 1) * (p - p^(1 - 1/h) + 1)), and L(3, 6) = 0.67060646664392140547... (Das et al., 2025).

cp's OEIS Frontend

A125030 a(n) = sum of exponents in the prime factorization of n that are noncomposite.

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A125073 a(n) = sum of the exponents in the prime factorization of n which are triangular numbers.

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A318306 Additive with a(p^e) = A002487(e).

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A318440 a(n) = A046645(n) - A007814(n); the 2-adic valuation of A299150.

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A318473 Additive with a(p^e) = A000045(e+1).

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A332423 If n = Product (p_j^k_j) then a(n) = Sum ((-1)^(k_j + 1) * k_j).

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

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