A129252 -id:A129252 - cp's OEIS frontend

A129251 Number of distinct prime factors p of n such that p^p is a divisor of n.

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

Offset: 1

Reinhard Zumkeller, Apr 07 2007

Average value is A094289 = 0.28735...; attains record values on A076265, in particular a(A076265(n)) = n.

			Since 15 = 3^1 * 5^1, a(15) = 0. But 16 = 2^4 is divisible by 2^2, so a(16) = 1. - _Michael B. Porter_, Aug 18 2016

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Horst Alzer and Man Kam Kwong, On Sándor's Inequality for the Riemann Zeta Function, J. Int. Seq. (2023) Vol. 26, Article 23.3.6.

Cf. A129252, A020639, A028234, A001221, A051674, A067029, A094289.
Cf. A048103 (indices of zeros), A100716 (nonzeros).
Differs from A276077 for the first time at n=625, where a(625) = 0, while A276077(625) = 1.

{0}~Join~Table[Count[FactorInteger[n][[All, 1]], ?(Mod[n, #^#] == 0 &)], {n, 2, 120}] (* _Michael De Vlieger, Oct 30 2019 *)

a(n)=my(s,t,v); forprime(p=2,, v=valuation(n,p); if(v, n/=p^v; if(v>=p, s++), if(p^p>n, return(s)))) \\ Charles R Greathouse IV, Sep 14 2015

(define (A129251 n) (if (= 1 n) 0 (+ (A129251 (A028234 n)) (if (zero? (modulo n (expt (A020639 n) (A020639 n)))) 1 0))))
;; Antti Karttunen, Aug 18 2016

(define (A129251 n) (if (= 1 n) 0 (+ (A129251 (A028234 n)) (if (>= (A067029 n) (A020639 n)) 1 0))))
;; Antti Karttunen, Aug 18 2016

a(A048103(n)) = 0, a(A100716(n)) > 0.
a(n) << sqrt(log n)/log log n. - Charles R Greathouse IV, Sep 14 2015
From Antti Karttunen, Aug 18 2016: (Start)
These formulas use Iverson bracket, which gives 1 as its value if the condition given inside [ ] is true and 0 otherwise:
a(1) = 0, for n > 1, a(n) = a(A028234(n)) + [A067029(n) >= A020639(n)].
Or, for n > 1, a(n) = a(A028234(n)) + [0 = n mod (A020639(n)^A020639(n))]. (End)
a(n) = Sum_{d|n} [rad(d) = Omega(d)*[omega(d) = 1]], where [ ] is the Iverson bracket. - Wesley Ivan Hurt, Feb 09 2022
Additive with a(p^e) = 1 if e >= p, and 0 otherwise. - Amiram Eldar, Nov 07 2022

Data section filled up to 120 terms by Antti Karttunen, Aug 18 2016

A327936 Multiplicative with a(p^e) = p if e >= p, otherwise 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2

Offset: 1

Antti Karttunen, Oct 01 2019

			For n = 12 = 2^2 * 3^1, only prime factor p = 2 satisfies p^p | 12, thus a(12) = 2.
For n = 108 = 2^2 * 3^3, both prime factors p = 2 and p = 3 satisfy p^p | 108, thus a(108) = 2*3 = 6.

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

Cf. A001221, A129251, A327937, A327938, A327939.

Differs from A129252 for the first time at n=108.

Array[Apply[Times, FactorInteger[#] /. {p_, e_} /; IntegerQ@ p :> If[e >= p, p, 1]] &, 120] (* Michael De Vlieger, Oct 01 2019 *)

A327936(n) = { my(f = factor(n)); for(k=1, #f~, f[k,2] = (f[k,2]>=f[k,1])); factorback(f); };

Multiplicative with a(p^e) = p if e >= p, otherwise 1.
A001221(a(n)) = A129251(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + (p-1)/p^p) = 1.3443209052633459342... . - Amiram Eldar, Nov 07 2022

A380528 Smallest prime p such that p^p is a divisor of A380459(n), or 1 if no such factor exists, where A380459(n) = Product_{d|n} A276086(n/d)^A349394(d).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 09 2025

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to primorial base.

Cf. A129252, A276086, A349394, A380459, A380468 (positions of 1's), A380529 [= a(A005117(n))], A380530 (positions of records).

A129252(n) = { my(pp); forprime(p=2, , pp = p^p; if(!(n%pp), return(p)); if(pp > n, return(1))); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A349394(n) = { my(p=0, e); if((e=isprimepower(n, &p)), p^(e-1), 0); };
A380459(n) = { my(m=1); fordiv(n, d, m *= A276086(d)^A349394(n/d)); (m); };
A380528(n) = A129252(A380459(n));

a(n) = A129252(A380459(n)).

A380529 Smallest prime p such that p^p is a divisor of A380459(A005117(n)), or 1 if no such factor exists, where A380459(n) = Product_{d|n} A276086(n/d)^A349394(d) and A005117 lists the squarefree numbers.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 09 2025

nonn

Antti Karttunen, Table of n, a(n) for n = 1..39845
Index entries for sequences related to primorial base.

Cf. A005117, A129252, A276086, A349394, A380459, A380468, A380470, A380528.

for(n=1,oo,if(issquarefree(n),print1(A380528(n),", ")));

a(n) = A380528(A005117(n)) = A129252(A380459(A005117(n))).

cp's OEIS Frontend

A129251 Number of distinct prime factors p of n such that p^p is a divisor of n.

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A327936 Multiplicative with a(p^e) = p if e >= p, otherwise 1.

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A380528 Smallest prime p such that p^p is a divisor of A380459(n), or 1 if no such factor exists, where A380459(n) = Product_{d|n} A276086(n/d)^A349394(d).

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A380529 Smallest prime p such that p^p is a divisor of A380459(A005117(n)), or 1 if no such factor exists, where A380459(n) = Product_{d|n} A276086(n/d)^A349394(d) and A005117 lists the squarefree numbers.

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