A051674 -id:A051674 - cp's OEIS frontend

A068327 Arithmetic derivative of n^n.

0, 0, 4, 27, 1024, 3125, 233280, 823543, 201326592, 2324522934, 70000000000, 285311670611, 142657607172096, 302875106592253, 100008061430022144, 3503151123046875000, 590295810358705651712, 827240261886336764177, 826274569581227289083904, 1978419655660313589123979

Offset: 0

Reinhard Zumkeller, Feb 27 2002

p prime: a(p) = A003415(p^p) = p^p.

A003415(A051674(n)) = A051674(n).

			a(10) = A003415(10^10) = A003415(2^10 * 5^10) = 10^10 * (10/2 + 10/5) = 10^10 * (5 + 2) = 70000000000 by formula in A003415.

Alois P. Heinz, Table of n, a(n) for n = 0..380 (first 100 terms from T. D. Noe)

Cf. A000312, A003415.

a:= n-> n^(n+1)*add(i[2]/i[1], i=ifactors(n)[2]):
seq(a(n), n=0..30);  # Alois P. Heinz, Jun 09 2015
# alternative
A068327 := proc(n)
        A003415(n^n) ;
end proc:
seq( A068327(n),n=0..10) ; # R. J. Mathar, Oct 19 2021

a312[n_] := Sum[ StirlingS2[n, k]*n!/(n - k)!, {k, 0, n}]; a3415[n_] := With[ {fi = FactorInteger[n]}, n*Total[ fi[[All, 2]] / fi[[All, 1]] ] ]; a3415[0] = a3415[1] = 0; a[n_] := a3415[ a312[n] ]; Table[ a[n], {n, 1, 16}] (* Jean-François Alcover, Mar 27 2013 *)

from sympy import factorint
def A068327(n): return sum((n**(n+1)*e//p for p,e in factorint(n).items())) if n > 1 else 0 # Chai Wah Wu, Jun 12 2022

a(n) = A003415(A000312(n)).

a(n) = n^n * A003415(n) = A000312(n) * A003415(n). - Alois P. Heinz, Jun 09 2015

A083348 Numbers k such that r(k) = Sum(e/p: k = Product(p^e)) > 1.

Original entry on oeis.org

8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 44, 48, 52, 54, 56, 60, 64, 68, 72, 76, 80, 81, 84, 88, 90, 92, 96, 100, 104, 108, 112, 116, 120, 124, 126, 128, 132, 135, 136, 140, 144, 148, 150, 152, 156, 160, 162, 164, 168, 172, 176, 180, 184, 188, 189, 192, 196, 198

Offset: 1

Reinhard Zumkeller, Apr 25 2003

nonn

The number of terms not exceeding 10^m, for m = 1, 2, ..., are 1, 29, 318, 3174, 31763, 317813, 3177179, 31774009, 317745099, 3177373819, ... . Apparently, the asymptotic density of this sequence exists and equals 0.3177... . - Amiram Eldar, Jun 24 2022

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A003415, A072873, A051674, A083345, A083346, A083347, A168036, A369048 (characteristic function), A369049.

Subsequence of A100717.

a083348 n = a083348_list !! (n-1)
a083348_list = filter ((> 0) . a168036) [1..]
-- Reinhard Zumkeller, May 22 2015, May 10 2011

ad[n_] := Switch[n, 0 | 1, 0, _, If[PrimeQ[n], 1, Sum[Module[ {p, e}, {p, e} = pe; n e/p], {pe, FactorInteger[n]}]]];
Select[Range[1000], ad[#] > # &] (* Jean-François Alcover, May 04 2023 *)

A083345(a(n)) > A083346(a(n)).
A083345(A051674(n)) = A083346(A051674(n)).
A083345(A083347(n)) < A083346(A083347(n)).
A168036(a(n)) > 0. - Reinhard Zumkeller, May 22 2015

A100717 Numbers k having a prime divisor p such that p^p is the highest power of p that divides k.

Original entry on oeis.org

4, 12, 20, 27, 28, 36, 44, 52, 54, 60, 68, 76, 84, 92, 100, 108, 116, 124, 132, 135, 140, 148, 156, 164, 172, 180, 188, 189, 196, 204, 212, 216, 220, 228, 236, 244, 252, 260, 268, 270, 276, 284, 292, 297, 300, 308, 316, 324, 332, 340, 348, 351, 356, 364, 372

Offset: 1

Leroy Quet, Dec 10 2004

nonn

For each prime p, the sequence includes all k*p^p for k such that gcd(k,p)=1. - T. D. Noe

The asymptotic density of this sequence is 1 - Product_{p prime} (1 - 1/p^p + 1/p^(p+1)) = 0.14682429539560371215... . - Amiram Eldar, Jun 25 2022

			54 is included because 3^3, but not 3^4, divides 54.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A100716, A203908.

Subsequences: A051674, A048102 \ {1}.

a100717 n = a100717_list !! (n-1)
a100717_list = filter ((== 0) . a203908) [1..]
-- Reinhard Zumkeller, Dec 24 2013

fQ[n_] := Union[ Table[ #[[1]] == #[[2]]] & /@ FactorInteger[n]][[ -1]] == True; Select[ Range[2, 375], fQ[ # ] &] (* Robert G. Wilson v, Dec 14 2004 *)

A203908(a(n)) = 0. - Reinhard Zumkeller, Dec 24 2013

More terms from T. D. Noe and Robert G. Wilson v, Dec 14 2004

A125137 a(n) = p^p + 1, where p = prime(n).

Original entry on oeis.org

5, 28, 3126, 823544, 285311670612, 302875106592254, 827240261886336764178, 1978419655660313589123980, 20880467999847912034355032910568, 2567686153161211134561828214731016126483470, 17069174130723235958610643029059314756044734432, 10555134955777783414078330085995832946127396083370199442518

Offset: 1

N. J. A. Sloane, Jan 21 2007

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..77

See A125136 for factorizations. Cf. A088730, A125135.

Cf. A104131, A114846.

[p^p+1: p in PrimesUpTo(40)]; // Vincenzo Librandi, Mar 27 2014

Table[Prime[n]^Prime[n] + 1, {n , 20}] (* Vincenzo Librandi, Mar 27 2014 *)

a(n) = A051674(n)+1. - R. J. Mathar, Apr 23 2007

A342090 Numbers with at least one prime power p^e in their prime factorization such that p|e.

Original entry on oeis.org

4, 12, 16, 20, 27, 28, 36, 44, 48, 52, 54, 60, 64, 68, 76, 80, 84, 92, 100, 108, 112, 116, 124, 132, 135, 140, 144, 148, 156, 164, 172, 176, 180, 188, 189, 192, 196, 204, 208, 212, 216, 220, 228, 236, 240, 244, 252, 256, 260, 268, 270, 272, 276, 284, 292, 297, 300

Offset: 1

Amiram Eldar, Feb 27 2021

nonn

Numbers with a unitary divisor of the form p^(m*p) where p is a prime and m > 0.
The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 1, 19, 188, 1883, 18825, 188244, 1882429, 18824297, 188242957, 1882429628, ...
The asymptotic density of this sequence is 1 - Product_{p prime} 1 - (p - 1)/(p*(p^p - 1)) = 0.18824296270011399086...

			4 = 2^2 is a term since 2 divides 2.
8 = 2^3 is not a term since 2 does not divide 3.

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

Subsequence of A013929.
Subsequences: A000302, A000312, A009971, A017113, A051674.
Cf. A072873, A369070 (characteristic function).

q[n_] := AnyTrue[FactorInteger[n], Divisible[Last[#], First[#]] &]; Select[Range[2, 300], q]

Wrong term 1 removed by Amiram Eldar, Jan 16 2024

A061789 a(n) = Sum_{k=1..n} prime(k)^prime(k).

Original entry on oeis.org

4, 31, 3156, 826699, 285312497310, 303160419089563, 827240565046755853740, 1979246896225360344977719, 20880469979094808259715377888286, 2567686153182091604540923022990731504371755

Offset: 1

Amarnath Murthy, May 25 2001

a(n) is prime for n = 2, 4, and 24, and no other n up to at least 600. - Zak Seidov and Robert Israel, Apr 11 2025

			a(3) = 2^2 + 3^3 + 5^5 = 3156.

Harry J. Smith, Table of n, a(n) for n = 1..76

Cf. A051674.

p:= 1: s:= 0: S:= NULL:
for k from 1 to 30 do
p:= nextprime(p);
s:= s + p^p;
S:= S,s
od:
S; # Robert Israel, Apr 11 2025

P3[n_] := Sum[Prime[i]^Prime[i], {i, 1, n}]; Table[P3[n], {n, 1, 10}] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)
Accumulate[#^#&/@Prime[Range[10]]] (* Harvey P. Dale, Apr 10 2015 *)

a=n=0; forprime (p=2, 383, write("b061789.txt", n++, " ", a+=p^p)) \\ Harry J. Smith, Jul 28 2009

from itertools import accumulate, count, islice
from sympy import prime
def A061789_gen(): # generator of terms
    yield from accumulate(((p:=prime(k))**p for k in count(1)))
A061789_list = list(islice(A061789_gen(),20)) # Chai Wah Wu, Jun 17 2022

Partial sums of A051674. - R. J. Mathar, Apr 26 2007

Corrected and extended by Jason Earls, May 26 2001

A074583 Numbers k such that sopfr(k) = S(k), where sopfr = A001414 and S = A002034.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239

Offset: 1

Jason Earls, Aug 24 2002

These are the prime powers p^e with e <= p. - Reinhard Zumkeller, Dec 15 2003

Complement to A192135 with respect to A000961;

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Subsequence of A000961; A000040, A000430, and A051674 are subsequences.

Cf. A095874, A192135, A192188, A343983.

import Data.Set (singleton, deleteFindMin, insert)
a074583 n = a074583_list !! (n-1)
a074583_list = 1 : f (singleton 2) a000040_list where
  f s ps'@(p:p':ps)
    | m == p      = p : f (insert (p*p) $ insert p' s') (p':ps)
    | m < spf^spf = m : f (insert (m*spf) s') ps'
    | otherwise   = m : f s' ps'
      where spf = a020639 m  -- smallest prime factor of m, cf. A020639
            (m, s') = deleteFindMin s
-- Simpler version:
a074583_list = map a000961 a192188_list
-- Reinhard Zumkeller, Jun 05 2011, Jun 26 2011

sopfr[n_] := Total[Times @@@ FactorInteger[n]];
S[n_] := Module[{m = 1}, While[!IntegerQ[m!/n], m++]; m];
Select[Range[1000], sopfr[#] == S[#]&] (* Jean-François Alcover, Nov 09 2017 *)

isok(n) = my(f=factor(n)); n==1 || (#f~==1 && f[1, 1]>=f[1, 2]); \\ Seiichi Manyama, May 07 2021

a(n) = A000961(A192188(n)); A095874(a(n)) = A192188(n). - Reinhard Zumkeller, Jun 26 2011

A082949 Numbers of the form p^q * q^p, with distinct primes p and q.

Original entry on oeis.org

72, 800, 6272, 30375, 247808, 750141, 1384448, 37879808, 189267968, 235782657, 1313046875, 3502727631, 4437573632, 451508436992, 634465620819, 2063731785728, 7863818359375, 7971951402153, 188153927303168, 453238525390625, 1145440056788109

Offset: 1

Reinhard Zumkeller, May 26 2003

nonn

A001221(a(n)) = 2;
A001222(a(n)) = A001414(a(n)) = A020639(a(n)) + A006530(a(n)) = A051904(a(n)) + A051903(a(n));
A020639(a(n)) = A051904(a(n));
A006530(a(n)) = A051903(a(n)).

			2^7 * 7^2 = 128*49 = 6272, therefore 6272 is in the sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A053810, A006881, A051674.
Cf. A098096, numbers of the form 2^p * p^2.
Cf. A001221, A001222, A001414, A020639, A006530, A051903, A051904.
Cf. A151800.

import Data.Set (singleton, deleteFindMin, insert)
a082949 n = a082949_list !! (n-1)
a082949_list = f $ singleton (2 ^ 3 * 3 ^ 2, 2, 3) where
   f s = y : f (if p' < q then insert (p' ^ q * q ^ p', p', q) s'' else s'')
         where s'' = insert (p ^ q' * q' ^ p, p, q') s'
               p' = a151800 p; q' = a151800 q
               ((y, p, q), s') = deleteFindMin s
-- Reinhard Zumkeller, Feb 07 2015

Take[Union[Select[Flatten[Table[If[p != q, Prime[p]^Prime[q]*Prime[q]^Prime[p]], {p, 100}, {q, 100}]], IntegerQ]], 30] (* Alonso del Arte, Oct 28 2005 *)
Select[Range[10! ],Length[FactorInteger[ # ]]==2&&FactorInteger[ # ][[1,1]]==FactorInteger[ # ][[2,2]]&&FactorInteger[ # ][[1,2]]==FactorInteger[ # ][[2,1]]&] (* Vladimir Joseph Stephan Orlovsky, Apr 23 2010 *)
With[{nn=30},Take[Union[First[#]^Last[#] Last[#]^First[#]&/@ Subsets[ Prime[Range[nn]],{2}]],nn]] (* Harvey P. Dale, Aug 19 2012 *)

term(p,q)=p^q*q^p;
l=listcreate(465); for(m=1,30, for(n=m+1,31, listput(l,term(prime(m), prime(n))))); listsort(l) \\ Rick L. Shepherd, Sep 07 2003

Corrected and extended by Rick L. Shepherd, Sep 07 2003

A117579 Numerator of Sum[i=1..n] 1/(p(i)^p(i)), p(i) = i-th prime.

Original entry on oeis.org

1, 31, 96983, 79870008269, 22787845491220720044859, 6901871132161346809864777612017764827, 5709505682874900155174610004469973097336266239002423739879, 11295798267103963562742898223286548990219261148710007871289771185589362412305596041

Offset: 1

Jonathan Vos Post, Mar 29 2006

			1/4, 31/108, 96983/337500, 79870008269/277945762500, 22787845491220720044859/79301169838123235887500,
6901871132161346809864777612017764827/24018350267611933650627567399079537500

Denominators = A076265.

Cf. A000040, A051674.

Numerator[Accumulate[1/#^#&/@Prime[Range[10]]]] (* Harvey P. Dale, Jan 24 2013 *)

a(n) = Numerator of Sum[i=1..n] 1/(p(i)^p(i)). a(n) = Numerator of Sum[i=1..n] 1/(A000040(i)^A000040(i)). a(n) = Numerator of Sum[i=1..n] 1/A051674(i).

Corrected and extended by Harvey P. Dale, Jan 24 2013

A192015 Arithmetic derivative of prime powers: a(n) = A003415(A000961(n)).

Original entry on oeis.org

0, 1, 1, 4, 1, 1, 12, 6, 1, 1, 32, 1, 1, 1, 10, 27, 1, 1, 80, 1, 1, 1, 1, 14, 1, 1, 1, 192, 1, 1, 1, 1, 108, 1, 1, 1, 1, 1, 1, 1, 1, 22, 75, 1, 448, 1, 1, 1, 1, 1, 1, 1, 1, 26, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 405, 1, 1024, 1, 1, 1, 1, 1, 1, 1, 34

Offset: 1

Reinhard Zumkeller, Jun 26 2011

nonn

a(A000040(n)) = 1; a(A002808(n)) > 1;
A001787, A027471, A100484, A079705 and A051674 are subsequences;
A001787 and A024622 give record values and where they occur;
A192016(n) = A003415(a(n)).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Power

a192015 = a003415 . a000961  -- Reinhard Zumkeller, Apr 16 2014

Join[{0}, Reap[For[n = 1, n <= 300, n++, f = FactorInteger[n]; If[Length[f] == 1, Sow[n*Total[Apply[#2/#1&, f, {1}]]]]]][[2, 1]]] (* Jean-François Alcover, Feb 21 2014 *)

from sympy import primepi, integer_nthroot, factorint
def A192015(n):
    if n == 1: return 0
    def f(x): return int(n+x-1-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return sum((m*e//p for p,e in factorint(m).items())) # Chai Wah Wu, Aug 15 2024

a(n) = A025474(n) * A025473(n)^(A025474(n) - 1).

cp's OEIS Frontend

A068327 Arithmetic derivative of n^n.

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A083348 Numbers k such that r(k) = Sum(e/p: k = Product(p^e)) > 1.

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A100717 Numbers k having a prime divisor p such that p^p is the highest power of p that divides k.

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A125137 a(n) = p^p + 1, where p = prime(n).

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A342090 Numbers with at least one prime power p^e in their prime factorization such that p|e.

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A061789 a(n) = Sum_{k=1..n} prime(k)^prime(k).

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A074583 Numbers k such that sopfr(k) = S(k), where sopfr = A001414 and S = A002034.

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