A094289 -id:A094289 - cp's OEIS frontend

A051674 a(n) = prime(n)^prime(n).

4, 27, 3125, 823543, 285311670611, 302875106592253, 827240261886336764177, 1978419655660313589123979, 20880467999847912034355032910567, 2567686153161211134561828214731016126483469, 17069174130723235958610643029059314756044734431

Offset: 1

Asher Auel

Numbers k such that bigomega(k)^(bigomega(k)) = k, where bigomega = A001222. - Lekraj Beedassy, Aug 21 2004
Positive k such that k' = k, where k' is the arithmetic derivative of k. - T. D. Noe, Oct 12 2004
David Beckwith proposes (in the AMM reference): "Let n be a positive integer and let p be a prime number. Prove that (p^p) | n! implies that (p^(p + 1)) | n!". - Jonathan Vos Post, Feb 20 2006
Subsequence of A100716; A003415(m*a(n)) = A129283(m)*a(n), especially A003415(a(n)) = a(n). - Reinhard Zumkeller, Apr 07 2007
A168036(a(n)) = 0. - Reinhard Zumkeller, May 22 2015

			a(1) = 2^2 = 4.
a(2) = 3^3 = 27.
a(3) = 5^5 = 3125.

J.-M. De Koninck & A. Mercier, 1001 Problemes en Theorie Classique Des Nombres, Problem 740 pp. 95; 312, Ellipses Paris 2004.

T. D. Noe, Table of n, a(n) for n = 1..40
David Beckwith, Problem 11158, American Mathematical Monthly, Vol. 112, No. 5 (May 2005), p. 468.
Jurij Kovic, The Arithmetic Derivative and Antiderivative, Journal of Integer Sequences, Vol. 15 (2012), #12.3.8.

Cf. A000040, A000312, A003415 (arithmetic derivative of n), A129150, A129151, A129152, A048102, A072873 (multiplicative closure), A104126.
Subsequence of A100717; A203908(a(n)) = 0.
Subsequence of A097764.
Cf. A168036, A094289 (decimal expansion of Sum(1/p^p)).

a051674_list = map (\p -> p ^ p) a000040_list
-- Reinhard Zumkeller, Jan 21 2012

[p^p: p in PrimesUpTo(30)]; // Vincenzo Librandi, Mar 27 2014

A051674:=n->ithprime(n)^ithprime(n): seq(A051674(n), n=1..10); # Wesley Ivan Hurt, Jun 25 2016

Array[Prime[ # ]^Prime[ # ] &, 12] (* Vladimir Joseph Stephan Orlovsky, May 01 2008 *)
#^#&/@Prime[Range[10]] (* Harvey P. Dale, May 17 2024 *)

a(n)=n=prime(n);n^n \\ Charles R Greathouse IV, Mar 20 2013

from gmpy2 import mpz
[mpz(prime(n))**mpz(prime(n)) for n in range(1,100)] # Chai Wah Wu, Jul 28 2014

a(n) = A000312(A000040(n)). - Altug Alkan, Sep 01 2016

Sum_{n>=1} 1/a(n) = A094289. - Amiram Eldar, Oct 13 2020

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

Original entry on oeis.org

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

A076265 a(n) = Product_{i=1..n} prime(i)^prime(i).

Original entry on oeis.org

4, 108, 337500, 277945762500, 79301169838123235887500, 24018350267611933650627567399079537500, 19868946365457062696924774946056904675112420776003728137500

Offset: 1

Jeff Burch, Nov 23 2002

Denominator of Sum_{i=1..n} 1/(p(i)^p(i)), where p(i) = i-th prime. The numerators are in A117579. E.g., 1/4, 31/108, 96983/337500, 79870008269/277945762500, ... - Jonathan Vos Post, Mar 29 2006
Equally, denominator of Sum_{k=1..n}(-1)^(k+1) * 1/p(k)^p(k), where p(k) = prime(k). - Alexander Adamchuk, Aug 22 2006
C = Sum_{k>=1} (-1)^(k+1)/(prime(k)^prime(k)) = 1/2^2 - 1/3^3 + 1/5^5 - 1/7^7 + 1/11^11 - 1/13^13 + ... A122147 is the decimal expansion of C = 0.213281748700785698255627... - Alexander Adamchuk, Aug 22 2006
Hyperprimorials, from primorials by analogy with hyperfactorials. See A006939. - Matthew Campbell, Jul 30 2015

			A122148(n)/a(n) begins 1/4, 23/108, 71983/337500, ... - _Alexander Adamchuk_, Aug 22 2006

Cf. A051674, A122147, A122148, A094289, A117579, A076265, A000040.

Table[Denominator[Sum[1/Prime[k]^Prime[k],{k,1,n}]],{n,1,10}] (* Alexander Adamchuk, Aug 22 2006 *)
Denominator[Accumulate[1/#^#&/@Prime[Range[10]]]] (* Harvey P. Dale, Jan 24 2013 *)

a(n)=prod(i=1,n,prime(i)^prime(i)) \\ Charles R Greathouse IV, Aug 05 2015

log a(n) ~ (n^2 log^2 n)/2. - Charles R Greathouse IV, Sep 14 2015

Entry revised by N. J. A. Sloane, Apr 10 2006

Edited by N. J. A. Sloane, Aug 04 2008 at the suggestion of R. J. Mathar

A100124 Decimal expansion of Sum_{n>0} 1/prime(n)!.

Original entry on oeis.org

6, 7, 5, 1, 9, 8, 4, 3, 7, 9, 1, 1, 1, 1, 4, 3, 4, 1, 9, 0, 0, 5, 6, 1, 6, 0, 7, 5, 9, 1, 3, 5, 7, 2, 9, 9, 5, 3, 9, 2, 7, 6, 7, 8, 8, 5, 6, 5, 1, 3, 2, 6, 5, 1, 5, 6, 0, 3, 4, 1, 0, 6, 4, 5, 1, 6, 8, 8, 5, 8, 6, 1, 4, 8, 5, 4, 2, 4, 4, 3, 3, 4, 4, 1, 1, 4, 6, 2, 7, 2, 2, 8, 0, 2, 7, 8, 9, 5, 7, 1

Offset: 0

Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 11 2004

Mingarelli shows that this constant is irrational. - Charles R Greathouse IV, Nov 05 2013
Convergence follows because A100124 < e - 2 = 0.71828... = 1/2! + 1/3! + 1/4! + 1/5! because e - 2 contains every term in A100124. The relation to e suggests a different question: is this constant not just irrational but also transcendental? - Timothy Varghese, May 07 2014
This is e times the probability that a prime is chosen from a Poisson distribution with lambda = 1. - Charles R Greathouse IV, Dec 07 2014

			0.67519843791111434190056160759135729953927678856513265156034106451688586148...

Angelo B. Mingarelli, Abstract factorials, arXiv:0705.4299 [math.NT], 2007-2012.

Cf. A000720, A010051, A039716, A094289.

RealDigits[Sum[1/Prime[n]!, {n, 1, 20}], 10, 100][[1]] (* Amiram Eldar, Nov 25 2020 *)

default(realprecision,100); sum(n=1,100,1/(prime(n)!),0.)

prec=exp(lambertw(default(realprecision)/exp(1)*log(10))+1)+9; P=s=.5;p=2;forprime(q=3,prec,P/=prod(i=p+1,q,i);s+=P;p=q); s \\ Charles R Greathouse IV, Nov 05 2013

Equals Sum_{k>0} A010051(k)/k!. - R. J. Cano, Jan 25 2017
From Amiram Eldar, Nov 25 2020: (Start)
Equals Sum_{k>=1} 1/A039716(k).
Equals Sum_{k>=1} pi(k)/((k+1)*(k-1)!), where pi = A000720. (End)

A100127 Decimal expansion of Sum_{n>0} prime(n)/n!.

Original entry on oeis.org

4, 7, 3, 8, 6, 3, 8, 7, 0, 2, 6, 8, 6, 2, 1, 9, 9, 9, 2, 7, 1, 7, 7, 9, 4, 7, 0, 3, 4, 8, 1, 2, 4, 9, 5, 5, 3, 6, 0, 1, 8, 2, 3, 8, 4, 6, 6, 4, 2, 1, 0, 6, 6, 4, 2, 4, 8, 0, 9, 6, 1, 9, 3, 4, 8, 4, 3, 7, 5, 4, 0, 8, 7, 7, 9, 6, 9, 7, 1, 6, 7, 4, 8, 9, 2, 2, 7, 2, 2, 1, 3, 7, 6, 0, 2, 3, 3, 7, 7, 4, 5, 6, 7, 7, 1

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 11 2004

This constant is irrational (Erdős, 1958). - Amiram Eldar, Apr 13 2020

			4.73863870268621999271779470348124955360182384664210664248096193484375408779697167489...

Paul Erdős, Sur certaines séries à valeur irrationnelle (in French), Enseignement Math. (2) 4 (1958), pp. 93-100.

Cf. A094289, A100124, A100126.

RealDigits[ Sum[ Prime[n]/n!, {n, 75}], 10, 105][[1]] (* Robert G. Wilson v, Jan 18 2015 *)

default(realprecision,100);sum(n=1,100,prime(n)/(n!),0.)

A100126 Decimal expansion of Sum_{n>0} n/(prime(n)!).

Original entry on oeis.org

8, 5, 9, 1, 2, 7, 1, 1, 0, 3, 5, 1, 0, 8, 8, 4, 3, 0, 3, 6, 8, 2, 2, 8, 2, 6, 2, 3, 0, 7, 5, 3, 7, 9, 0, 8, 2, 5, 8, 5, 3, 4, 5, 7, 7, 0, 6, 1, 7, 7, 6, 1, 7, 5, 6, 6, 1, 6, 7, 6, 1, 7, 7, 5, 8, 6, 8, 3, 7, 3, 2, 7, 9, 8, 0, 3, 7, 3, 5, 8, 8, 9, 7, 5, 1, 3, 2, 2, 0, 2, 2, 4, 8, 2, 6, 2, 5, 6, 3, 9

Offset: 0

Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 11 2004

			0.8591271103510884303682282623075379082585345770617761756616761775868373279803735889751322022482625639...

Cf. A094289, A100124.

RealDigits[Total[Table[n/Prime[n]!,{n,20}]],10,120][[1]] (* Harvey P. Dale, Oct 26 2015 *)

default(realprecision,100);sum(n=1,100,n/(prime(n)!),0.)

A122147 Decimal expansion of Sum[ (-1)^(k+1) * 1/p(k)^p(k) ], where p(k) = Prime[k].

Original entry on oeis.org

2, 1, 3, 2, 8, 1, 7, 4, 8, 7, 0, 0, 7, 8, 5, 6, 9, 8, 2, 5, 5, 6, 2, 7, 4, 8, 1, 3, 6, 9, 8, 4, 8, 4, 3, 6, 0, 2, 7, 7, 2, 7, 9, 7, 2, 5, 3, 2, 2, 4, 6, 4, 1, 0, 0, 7, 1, 4, 2, 2, 2, 2, 0, 1, 2, 3, 8, 3, 9, 5, 6, 7, 6, 0, 0, 3, 7, 2, 6, 9, 0, 0, 5, 6, 3, 7, 1, 2, 2, 0, 1, 1, 8, 6, 1, 8, 8, 2, 3, 4, 4, 1, 5, 5, 5

Offset: 0

Alexander Adamchuk, Aug 22 2006

C = Sum[ (-1)^(k+1) * 1/Prime[k]^Prime[k], {k,1,Infinity} ] = 1/2^2 - 1/3^3 + 1/5^5 - 1/7^7 + 1/11^11 - 1/13^13 + ... Partial sums are A122148[n] / A076265[n] = Sum[ (-1)^(k+1) * 1/Prime[k]^Prime[k], {k,1,n} ] = 1/4, 23/108, 71983/337500, ...

			C = 0.2132817487007856982556274813698484360277279725322464100714222201238395676003\
726900563712201186188234415559844581411471306301650311286030077813464608267160\
801494597797561591251174806253914566160177882...

Cf. A051674, A122148, A076265, A094289, A117579, A076265, A000040.

A122148 Numerator of Sum[ (-1)^(k+1) * 1/p(k)^p(k), {k,1,n}], where p(k) = Prime[k].

Original entry on oeis.org

1, 23, 71983, 59280758269, 16913492177093188294859, 5122675745984257357873512804013239827, 4237683625666802603266159755806379107958975382128522814879

Offset: 1

Alexander Adamchuk, Aug 22 2006

C = Sum[ (-1)^(k+1) * 1/Prime[k]^Prime[k], {k,1,Infinity} ] = 1/2^2 - 1/3^3 + 1/5^5 - 1/7^7 + 1/11^11 - 1/13^13 + ... A122147[n] is a decimal expansion of C = 0.213281748700785698255627...

			a[n] / A076265[n] begins 1/4, 23/108, 71983/337500, ...

Cf. A051674, A122147, A094289, A117579, A076265, A000040.

Table[Numerator[Sum[(-1)^(k+1)*1/Prime[k]^Prime[k],{k,1,n}]],{n,1,10}]

a(n) = Numerator[ Sum[ (-1)^(k+1) * 1/Prime[k]^Prime[k], {k,1,n} ] ].

A351411 Number of divisors of n not of the form p^p, p prime.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 3, 3, 4, 2, 5, 2, 4, 4, 4, 2, 6, 2, 5, 4, 4, 2, 7, 3, 4, 3, 5, 2, 8, 2, 5, 4, 4, 4, 8, 2, 4, 4, 7, 2, 8, 2, 5, 6, 4, 2, 9, 3, 6, 4, 5, 2, 7, 4, 7, 4, 4, 2, 11, 2, 4, 6, 6, 4, 8, 2, 5, 4, 8, 2, 11, 2, 4, 6, 5, 4, 8, 2, 9, 4, 4, 2, 11, 4, 4, 4, 7, 2, 12, 4, 5, 4

Offset: 1

Wesley Ivan Hurt, Feb 10 2022

			a(108) = 10; 2 of the 12 divisors of 108 are of the form p^p (p prime), namely 4 = 2^2 and 27 = 3^3; therefore a(108) = 12-2 = 10.

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

Cf. A000005 (tau), A001221 (omega), A001222 (Omega), A007947 (rad).

Cf. A051674, A129251.

f1[p_, e_] := e + 1; f2[p_, e_] := If[e < p, 0, 1]; a[1] = 1; a[n_] := Times @@ f1 @@@ (f = FactorInteger[n]) - Plus @@ f2 @@@ f; Array[a, 100] (* Amiram Eldar, Oct 01 2023 *)

a(n) = {my(f = factor(n)); vecprod(apply(x -> x+1, f[, 2])) - sum(i = 1, #f~, f[i, 2] >= f[i, 1]); } \\ Amiram Eldar, Oct 01 2023

a(n) = tau(n) - Sum_{d|n} [rad(d) = Omega(d)*[omega(d) = 1]], where [ ] is the Iverson bracket.
a(n) = A000005(n) - A129251(n).
Sum_{k=1..n} a(k) ~ n * (log(n) + c), where c = A147533 - A094289 = -0.1329269215... . Amiram Eldar, Oct 01 2023

A100128 Decimal expansion of Sum_{n>0} 1/(n*prime(n)^n).

Original entry on oeis.org

5, 5, 8, 3, 2, 7, 6, 2, 2, 2, 3, 1, 7, 6, 9, 1, 5, 8, 6, 1, 4, 2, 7, 9, 1, 9, 6, 8, 0, 3, 6, 0, 4, 6, 9, 5, 1, 7, 5, 2, 5, 2, 1, 2, 8, 2, 0, 8, 7, 4, 2, 0, 1, 7, 3, 9, 0, 2, 2, 2, 9, 1, 7, 4, 2, 3, 5, 8, 5, 4, 7, 6, 3, 3, 1, 6, 3, 7, 6, 1, 7, 6, 5, 2, 0, 0, 9, 3, 6, 9, 6, 7, 2, 6, 5, 2, 1, 0, 4, 0

Offset: 0

Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 11 2004

			0.5583276222317691586142791968036046951752521282087420173902229174235854763316376176520093696726521040...

Cf. A094289, A100124, A100126, A100127.

PARI
```
suminf(k=1, 1/(k*prime(k)^k))
```

cp's OEIS Frontend

A051674 a(n) = prime(n)^prime(n).

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

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A076265 a(n) = Product_{i=1..n} prime(i)^prime(i).

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A100124 Decimal expansion of Sum_{n>0} 1/prime(n)!.

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A100127 Decimal expansion of Sum_{n>0} prime(n)/n!.

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A100126 Decimal expansion of Sum_{n>0} n/(prime(n)!).

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A122147 Decimal expansion of Sum[ (-1)^(k+1) * 1/p(k)^p(k) ], where p(k) = Prime[k].

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