A104126 -id:A104126 - 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

A054743 If n = Product p_i^e_i then p_i < e_i (where e_i > 0) for all i.

Original entry on oeis.org

1, 8, 16, 32, 64, 81, 128, 243, 256, 512, 648, 729, 1024, 1296, 1944, 2048, 2187, 2592, 3888, 4096, 5184, 5832, 6561, 7776, 8192, 10368, 11664, 15552, 15625, 16384, 17496, 19683, 20736, 23328, 31104, 32768, 34992, 41472, 46656

Offset: 1

James Sellers, Apr 22 2000

Closed under multiplication. Use A104126 to construct A192135 by putting A104126(n) * prime(n)^k in a list up to some chosen bound. Create this sequence by multiplying any k elements of A192135 with distinct prime factors in a list (k>1). The last list along with A192135 is this sequence when sorted. - David A. Corneth, Jun 07 2016

			8 appears in the list because 8 = 2^3 and 2<3.
Construction of elements up to 1000: 1. Put 2^3 and 3^5 in a list; {8, 81} (The terms of A104126 up to 1000.) 2. For each element, put products the last list with their distinct prime factors up to 1000. Gives: {8, 16, 32, 64, 128, 256, 512, 81, 243, 729} (Terms from A192135 up to 1000). 3. Put products of k powers of distinct primes in a new list up to 1000: {648} (k>1). Unite {648} with {8, 16, 32, 64, 128, 256, 512, 81, 243, 729}. {8, 16, 32, 64, 128, 256, 512, 81, 243, 729, 648}. Sort the list. This gives: {8, 16, 32, 64, 81, 128, 243, 256, 512, 648, 729}, which are the elements below 1000 in this sequence. - _David A. Corneth_, Jun 07 2016

Robert Israel, Table of n, a(n) for n = 1..10001

Cf. A048103, A054744, A104126, A192135.

Cf. A207481 (same construction, but with p_i>=e_i),

N:= 10^10: # to get all terms <= N
p:= 1:
S:= {1}:
do
  p:= nextprime(p);
  if p^(p+1) > N then break fi;
  pp:= [seq(p^j, j=p+1 .. ilog[p](N))];
  S:= S union select(`<=`,{seq(seq(s*q,s=S),q=pp)},N);
od:
sort(convert(S,list)); # Robert Israel, Jun 07 2016

okQ[n_] := AllTrue[FactorInteger[n], #[[1]] < #[[2]]&];
Join[{1}, Select[Range[50000], okQ]] (* Jean-François Alcover, Jun 08 2016 *)

lista(nn) = {for (n=1, nn, f = factor(n); ok = 1; for (i=1, #f~, if (f[i, 1] >= f[i, 2], ok = 0; break;);); if (ok, print1(n, ", ")););} \\ Michel Marcus, Jun 15 2013

Sum_{n>=1} 1/a(n) = Product_{p prime} 1 + 1/((p-1)*p^p) = 1.27325025767774256043... - Amiram Eldar, Nov 24 2020

1 prepended by Alec Jones, Jun 07 2016

A104128 a(n) = p + p^(p+1), where p = prime(n).

Original entry on oeis.org

10, 84, 15630, 5764808, 3138428376732, 3937376385699302, 14063084452067724991026, 37589973457545958193355620, 480250763996501976790165756943064, 74462898441675122902293018227199467668020630, 529144398052420314716929933900838757437386767392

Offset: 1

Cino Hilliard, Mar 06 2005

Sum of reciprocals rapidly converges to 0.11196891489794721930017828981362..

Ray Chandler, Table of n, a(n) for n = 1..76

Table[p + p^(p+1), {p, Prime[Range[12]]}] (* Robert G. Wilson v, Feb 27 2012 *)

A104128(n) := block(
        return(A000040(n)^(1+A000040(n))+A000040(n))
)$ /* R. J. Mathar, Feb 27 2012 */

A104128(n)=(n=prime(n))+n^(n+1) \\ M. F. Hasler, Feb 26 2012

a(n) = A000040(n) + A104126(n).

Definition corrected and incorrect program removed by R. J. Mathar, Feb 27 2012

More terms from Harvey P. Dale, Feb 27 2012

A381215 Numbers k such that the difference between the largest and smallest element of the set of bases and exponents (including exponents = 1) in the prime factorization of k is 1.

Original entry on oeis.org

2, 8, 9, 36, 72, 81, 108, 216, 625, 15625, 117649, 5764801, 25937424601, 3138428376721, 23298085122481, 3937376385699289, 48661191875666868481, 14063084452067724991009, 104127350297911241532841, 37589973457545958193355601, 907846434775996175406740561329

Offset: 1

Paolo Xausa, Feb 19 2025

			72 is a term because 72 = 2^3*3^2, the set of these bases and exponents is {2, 3} and 3 - 2 = 1.

Paolo Xausa, Table of n, a(n) for n = 1..155

Positions of ones in A381214.

Cf. A036878, A081812, A104126, A381212, A381317.

Join[{2, 8, 9, 36, 72, 81, 108, 216}, Flatten[Map[#^{# - 1, # + 1} &, Prime[Range[3, 10]]]]]

For n >= 9, a(n) = A381317(n-4).

A381317 Numbers of the form p^(p +- 1), where p is prime.

Original entry on oeis.org

2, 8, 9, 81, 625, 15625, 117649, 5764801, 25937424601, 3138428376721, 23298085122481, 3937376385699289, 48661191875666868481, 14063084452067724991009, 104127350297911241532841, 37589973457545958193355601, 907846434775996175406740561329, 480250763996501976790165756943041

Offset: 1

Paolo Xausa, Feb 20 2025

Paolo Xausa, Table of n, a(n) for n = 1..150

Union of A036878 and A104126.

Subsequence of A381215.

Flatten[Map[#^{# - 1, # + 1} &, Prime[Range[10]]]]

A166326 a(n) = prime(n)^(prime(n)+1) - (prime(n)+1)^prime(n).

Original entry on oeis.org

-1, 17, 7849, 3667649, 2395420006033, 3143661612445145, 11877172892329028459041, 32347093457545958193355601, 424678439961073471604787362241217

Offset: 1

Vladimir Joseph Stephan Orlovsky, Oct 11 2009

sign

			a(1) = 2^3 - 3^2 = -1. a(2) = 3^4 - 4^3 = 17. a(3) = 5^6 - 6^5 = 7849.

Cf. A000040, A007925, A140893, A104126, A104127.

Array[Prime[ # ]^(Prime[ # ]+1)-(Prime[ # ]+1)^Prime[ # ]&,16]

a(n) = A104126(n) - A104127(n) = A007925(A000040(n)). - R. J. Mathar, Oct 14 2009

Keyword:sign set by R. J. Mathar, Oct 14 2009

A352081 Numbers of the form k*p^k, where k>1 and p is a prime.

Original entry on oeis.org

8, 18, 24, 50, 64, 81, 98, 160, 242, 324, 338, 375, 384, 578, 722, 896, 1029, 1058, 1215, 1682, 1922, 2048, 2500, 2738, 3362, 3698, 3993, 4374, 4418, 4608, 5618, 6591, 6962, 7442, 8978, 9604, 10082, 10240, 10658, 12482, 13778, 14739, 15309, 15625, 15842, 18818

Offset: 1

Amiram Eldar, Apr 16 2022

nonn

Each term in this sequence has a single presentation in the form k*p^k.

			8 is a term since 8 = 2*2^2.
18 is a term since 18 = 2*3^2.
24 is a term since 24 = 3*2^3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Peter Lindqvist and Jaak Peetre, On the remainder in a series of Mertens, Expositiones Mathematicae, Vol. 15 (1997), pp. 467-478. See eq. (3).
Eric Weisstein's World of Mathematics, Mertens Constant. See eq. (5).

Subsequences: A036289 \ {0, 2}, A036290 \ {0, 3}, A036291 \ {0, 5}, A036293 \ {0, 7}, A073113 \ {2}, A079704, A100042, A104126.

Cf. A001620, A077761, A143524.

addP[p_, n_] := Module[{k = 2, s = {}, m}, While[(m = k*p^k) <= n, k++; AppendTo[s, m]]; s]; seq[max_] := Module[{m = Floor[Sqrt[max/2]], s = {}, ps}, ps = Select[Range[m], PrimeQ]; Do[s = Join[s, addP[p, max]], {p, ps}]; Sort[s]]; seq[2*10^4]

Sum_{n>=1} 1/a(n) = -A143524 = gamma - B_1, where gamma is Euler's constant (A001620), and B_1 is Mertens's constant (A077761).

A257404 Numbers of the form p * q^p where p and q are primes, in increasing order.

Original entry on oeis.org

8, 18, 24, 50, 81, 98, 160, 242, 338, 375, 578, 722, 896, 1029, 1058, 1215, 1682, 1922, 2738, 3362, 3698, 3993, 4418, 5618, 6591, 6962, 7442, 8978, 10082, 10658, 12482, 13778, 14739, 15309, 15625, 15842, 18818

Offset: 1

William Brian Repko, Apr 22 2015

nonn

			(2,2):8, (2,3):18, (3,2):24, (2,5):50, (3,3):81, (2,7):98.

Cf. some subsequences: A079704, A104126.

primes = [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97];
results = [];
max = 2 * Math.pow(primes[primes.length-1],2);
for (i = 0; i < primes.length; i++) {
    for (j = 0; j  < primes.length; j++) {
        p = primes[i];
        q = primes[j];
        n = p * Math.pow(q,p);
        if (n <= max) {
            // add it
            results.push(n);
        } else {
            // break out of this loop
            break;
        }
    }
}
// sort results and print them
results.sort(function(a, b){return a-b}).valueOf();

max=10^5; p=q=2; Sort[Reap[While[2*q^2 <= max, While[(n=p*q^p) <= max, Sow@n; p=NextPrime@p]; p=2; q=NextPrime@q ]][[2,1]]] (* Giovanni Resta, May 19 2015 *)

is(n)={bittest(6,#n=factor(n)~)||return;#n==1&&return(n[1,1]+1==n[2,1]);(n[2,1]==1&&n[2,2]==n[1,1])||(n[2,2]==1&&n[1,2]==n[2,1])} \\ M. F. Hasler, May 04 2015

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

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A054743 If n = Product p_i^e_i then p_i < e_i (where e_i > 0) for all i.

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A104128 a(n) = p + p^(p+1), where p = prime(n).

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A381215 Numbers k such that the difference between the largest and smallest element of the set of bases and exponents (including exponents = 1) in the prime factorization of k is 1.

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A381317 Numbers of the form p^(p +- 1), where p is prime.

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A166326 a(n) = prime(n)^(prime(n)+1) - (prime(n)+1)^prime(n).

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A352081 Numbers of the form k*p^k, where k>1 and p is a prime.

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