A051674 -id:A051674 - cp's OEIS frontend

A168036 Difference between n' and n, where n' is the arithmetic derivative of n (A003415).

0, -1, -1, -2, 0, -4, -1, -6, 4, -3, -3, -10, 4, -12, -5, -7, 16, -16, 3, -18, 4, -11, -9, -22, 20, -15, -11, 0, 4, -28, 1, -30, 48, -19, -15, -23, 24, -36, -17, -23, 28, -40, -1, -42, 4, -6, -21, -46, 64, -35, -5, -31, 4, -52, 27, -39, 36, -35, -27, -58, 32, -60, -29

Offset: 0

Paolo P. Lava, Nov 17 2009

Let k = n'-n. For k = -1 n is a primary pseudoperfect number (A054377), apart from n=1; For k=0 n is p^p, being p a prime number (A051674); For k = 1 n is a Giuga number (A007850).

T. D. Noe, Table of n, a(n) for n = 0..10000

Cf. A007850, A051674, A054377, A136141.

Cf. A003415, A051674, A083347, A083348.

a168036 n = a003415 n - n  -- Reinhard Zumkeller, May 22 2015

with(numtheory);
A168036:=proc(q)
local n,p;
for n from 0 to q do
  print(n*add(op(2,p)/op(1,p),p=ifactors(n)[2])-n); od; end:
A168036(1000); # Paolo P. Lava, Nov 05 2012

np[k_] := Module[{f, n, m, p}, If[k < 2, np[k] = 0; Return[0], If[PrimeQ[k], np[k] = 1; Return[1], f = FactorInteger[k, 2]; m = f[[1, 1]]; n = k/m; p = m np[n] + n np[m]; np[k] = p; Return[p]]]];
Table[np[n] - n, {n, 0, 100}] (* Robert Price, Mar 14 2020 *)

a(A083347(n)) < 0; a(A051674(n)) = 0; a(A083348(n)) > 0. - Reinhard Zumkeller, May 22 2015

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = -1 + Sum_{p prime} 1/(p*(p-1)) = A136141 - 1 = -0.226843... . - Amiram Eldar, Dec 08 2023

A099309 Numbers n whose k-th arithmetic derivative is nonzero for all k. Complement of A099308.

Original entry on oeis.org

4, 8, 12, 15, 16, 20, 24, 26, 27, 28, 32, 35, 36, 39, 40, 44, 45, 48, 50, 51, 52, 54, 55, 56, 60, 63, 64, 68, 69, 72, 74, 75, 76, 80, 81, 84, 86, 87, 88, 90, 91, 92, 95, 96, 99, 100, 102, 104, 106, 108, 110, 111, 112, 115, 116, 117, 119, 120, 122, 123, 124, 125, 128, 132

Offset: 1

T. D. Noe, Oct 12 2004

nonn

Numbers of the form n = m*p^p (where p is prime), i.e., multiples of some term in A051674, have n' = (m + m')*p^p, which is again of the same form, but strictly larger iff m > 1. Therefore successive derivatives grow to infinity in this case, and they are constant when m = 1. There are other terms in this sequence, but I conjecture that they all eventually lead to a term of this form, e.g., 26 -> 15 -> 8 etc. - M. F. Hasler, Apr 09 2015

See A003415.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A003415 (arithmetic derivative of n), A099307 (least k such that the k-th arithmetic derivative of n is zero), A099308 (numbers whose k-th arithmetic derivative is zero for some k).
Cf. A341999 (characteristic function),
Positions of zeros in A256750, A351078, A351079 (after their initial zeros), also in A328308, A328312.
Subsequences include: A100716, A327929, A327934, A328251, A359547 (intersection with A048103).

is(n)=until(4>n=factorback(n~)*sum(i=1,#n,n[2,i]/n[1,i]), for(i=1,#n=factor(n)~,n[1,i]>n[2,i]||return(1))) \\ M. F. Hasler, Apr 09 2015

A048102 Numbers k such that if k = Product p_i^e_i then p_i = e_i for all i.

Original entry on oeis.org

1, 4, 27, 108, 3125, 12500, 84375, 337500, 823543, 3294172, 22235661, 88942644, 2573571875, 10294287500, 69486440625, 277945762500, 285311670611, 1141246682444, 7703415106497, 30813660425988, 302875106592253, 891598970659375, 1211500426369012, 3566395882637500

Offset: 1

N. J. A. Sloane

			3^3*5^5 = 84375.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1038 terms from T. D. Noe)
Daniel Mondot, Table of n, a(n) for n = 1..10000 with factorizations

Cf. A027748, A048103, A048104, A051674, A072873, A124010.

import Data.Set (empty, fromList, deleteFindMin, union)
import qualified Data.Set as Set (null, map)
a048102 n = a048102_list !! (n-1)
a048102_list = 1 : f empty [1] a051674_list where
  f s ys pps'@(pp:pps)
    | Set.null s = f (fromList (map (* pp) ys)) (pp:ys) pps
    | pp < m     = f (s `union` Set.map (* pp) s `union`
                      fromList (map (* pp) ys)) ys pps
    | otherwise  = m : f s' (m:ys) pps'
    where (m,s') = deleteFindMin s
-- Reinhard Zumkeller, Jan 21 2012

isok(n) = my(f = factor(n)); for (k=1, #f~, if (f[k,1] != f[k,2], return(0))); 1; \\ Michel Marcus, Apr 29 2016

A027748(a(n),k) = A124010(a(n),k) for k = 1 .. A001221(a(n)). - Reinhard Zumkeller, Jan 21 2012

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/p^p) = 1.2967126856... - Amiram Eldar, Oct 13 2020

More terms from Naohiro Nomoto, Jun 28 2001

A097764 Numbers of the form (kp)^p for prime p and k=1,2,3,....

Original entry on oeis.org

4, 16, 27, 36, 64, 100, 144, 196, 216, 256, 324, 400, 484, 576, 676, 729, 784, 900, 1024, 1156, 1296, 1444, 1600, 1728, 1764, 1936, 2116, 2304, 2500, 2704, 2916, 3125, 3136, 3364, 3375, 3600, 3844, 4096, 4356, 4624, 4900, 5184, 5476, 5776, 5832, 6084, 6400

Offset: 1

T. D. Noe, Aug 24 2004

The polynomial x^n - n is reducible over the integers for n in this sequence.
A result of Vahlen shows that the polynomial x^n - n is reducible over the integers for n in this sequence and no other n.
The representation (k*p)^p is generally not unique, e.g. a(120) = 46656 = (108*2)^2 = (12*3)^3. - Reinhard Zumkeller, Feb 14 2015
This is also numbers of the form (km)^m for any m > 1, not just primes. Let m be > 1; then m has a prime factor, so let m=pj, p a prime and j an integer > 0. Then (km)^m = (kpj)^pj = (k^j p^j j^j)^p = ((k^j p^(j-1) j^j) p) ^ p. - Franklin T. Adams-Watters, Sep 13 2015

T. D. Noe, Table of n, a(n) for n = 1..1000
A. Schinzel, Problems and results on polynomials.

Cf. A084746 (least k such that n^k-k is prime).
Cf. A097792 (numbers of the form 4k^4 or (kp)^p).
Cf. A000040, A051674, A255134 (first differences).

import Data.Set (singleton, deleteFindMin, insert)
a097764 n = a097764_list !! (n-1)
a097764_list = f 0 (singleton (4, 2, 2)) $
                 tail $ zip a051674_list a000040_list where
   f m s ppps'@((pp, p) : ppps)
     | pp < qq   = f m (insert (pp, p, 2) s) ppps
     | qq == m   = f m (insert ((k * q) ^ q, q, k + 1) s') ppps'
     | otherwise = qq : f qq (insert ((k * q) ^ q, q, k + 1) s') ppps'
     where ((qq, q, k), s') = deleteFindMin s
-- Reinhard Zumkeller, Feb 14 2015

nMax=10000; lst={}; n=1; While[p=Prime[n]; p^p<=nMax, k=1; While[(k*p)^p<=nMax, AppendTo[lst, (k*p)^p]; k++ ]; n++ ]; Union[lst]

is(n)=my(b,e=ispower(n,,&b),f); if(e==0, return(0)); f=factor(e)[,1]; for(i=1,#f, if(b%f[i]==0, return(1))); 0 \\ Charles R Greathouse IV, Aug 29 2016

A054412 Numbers k such that, in the prime factorization of k, the product of exponents equals the product of prime factors.

Original entry on oeis.org

1, 4, 27, 72, 108, 192, 800, 1458, 3125, 5120, 6272, 12500, 21600, 30375, 36000, 48600, 77760, 84375, 114688, 116640, 121500, 138240, 169344, 225000, 247808, 337500, 384000, 395136, 600000, 653184, 750141, 823543, 857304, 979776, 1384448, 1474560, 1500000

Offset: 1

Leroy Quet, May 09 2000

nonn

For p prime, numbers of the form p^p satisfy the condition, hence A051674 is a subsequence. - Michel Marcus, May 19 2014

Also, numbers of the form p^q * q^p, with distinct primes p and q, satisfy the condition, hence A082949 is a subsequence. - Bernard Schott, Apr 11 2020

			192 is included because 192 =2^6 *3^1 and 2*3 = 6*1.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A054412

Cf. A051674, A054411, A082949.

peppfQ[n_]:=Module[{f=Transpose[FactorInteger[n]]},Times@@First[f] == Times@@Last[f]]; Select[Range[1.5*10^6],peppfQ] (* Harvey P. Dale, Oct 14 2015 *)

isok(n) = my(f = factor(n)); prod(i=1, #f~, f[i,2]) == prod(i=1, #f~, f[i,1]); \\ Michel Marcus, May 19 2014

PARI
```
\\ See Links section.
```

More terms from James Sellers, May 23 2000

New name and three more terms from Michel Marcus, May 19 2014

A053089 a(n) = prime(n)^prime(n+1).

Original entry on oeis.org

8, 243, 78125, 1977326743, 34522712143931, 8650415919381337933, 239072435685151324847153, 257829627945307727248226067259, 3091058643093537522799545838540043339063, 2159424054808578564166497528588784562372597429

Offset: 1

Enoch Haga, Mar 18 2000

			a(3)=78125 because 5 is the 3rd prime, 7 is the 4th prime and 5^7=78125.

Sergio Silva, Teste Numerico. [link is broken]

Cf. A000040, A051674.

#[[1]]^#[[2]]&/@Partition[Prime[Range[10]],2,1] (* Harvey P. Dale, Jul 10 2015 *)

a(n) = my(p=prime(n)); p^nextprime(p+1); \\ Michel Marcus, Feb 27 2020

a(n) = A000040(n)^A000040(n+1). - Omar E. Pol, Feb 27 2020

One additional term (a(10)) from Harvey P. Dale, Jul 10 2015

A008478 Integers of the form Product p_j^k_j = Product k_j^p_j; p_j in A000040.

Original entry on oeis.org

1, 4, 16, 27, 72, 108, 432, 800, 3125, 6272, 12500, 21600, 30375, 50000, 84375, 121500, 169344, 225000, 247808, 337500, 486000, 750141, 823543, 1350000, 1384448, 3000564, 3294172, 6690816, 12002256, 13176688, 19600000, 22235661, 37380096, 37879808, 59295096, 88942644

Offset: 1

Olivier Gérard

nonn

Fixed points of A008477.

a(3) = 16 is the only term of the form p^q with p <> q. - Bernard Schott, Mar 28 2021

			16 = 2^4 = 4^2.
27 = 3^3.
108 = 2^2*3^3.
6272 = 2^7*7^2.
121500 = 2^2 * 3^5*5^3.

Cf. A000040, A008477.

Some subsequences: p_i^p_i (A051674), Product_i {p_i^p_i} (A048102), Product_(j,k)(p_j^p_k * p_k^p_j) with p_j < p_k (A082949) (see examples).

f[n_] := Product[{p, e} = pe; e^p, {pe, FactorInteger[n]}];
Reap[For[n = 1, n <= 10^8, n++, If[f[n] == n, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Mar 29 2021 *)

for(n=2,10^8,if(n==prod(i=1,omega(n), component(component(factor(n),2),i)^component(component(factor(n),1),i)),print1(n,",")))

More terms from David W. Wilson

a(34)-a(36) from Jean-François Alcover, Mar 29 2021

A094289 Decimal expansion of Sum(1/p^p) where p is prime.

Original entry on oeis.org

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

Offset: 0

Cino Hilliard, Jul 31 2004

This constant approximately equals 5226294/18187381. - Yalcin Aktar, Nov 05 2006

The asymptotic mean of A129251. - Amiram Eldar, Nov 07 2022

			0.287358251306224179736418045878932206955908802685881709299499368947089...

Cf. A073009, A083648, A051674 (prime(n)^prime(n)), A129251.

digits = 105; n0 = 10; dn = 10; Clear[f]; f[n_] := f[n] = RealDigits[ Sum[ 1/Prime[k]^Prime[k], {k, 1, n}], 10, digits+5] // First; f[n = n0]; f[n = n+dn]; While[Print["n = ", n]; f[n] != f[n-dn], n = n+dn]; Take[f[n], digits] (* Jean-François Alcover, Nov 22 2013 *)

ptothep(n) = { local(x,s,a); default(realprecision,200); s=0; forprime(x=2,n,s+=1./x^x); a=Vec(Str(s)); for(x=3,n,print1(eval(a[x]),",")) }

Offset corrected by R. J. Mathar, Feb 05 2009

A129150 The n-th arithmetic derivative of 2^3.

Original entry on oeis.org

8, 12, 16, 32, 80, 176, 368, 752, 1520, 3424, 8592, 20096, 70464, 235072, 705280, 3023616, 13223680, 55540736, 278539264, 1392697344, 9541095424, 58609614848, 410267320320, 3397142953984, 24143851798528, 176071227916288, 1232666139967488, 9523075842834432

Offset: 0

Reinhard Zumkeller, Apr 01 2007

nonn

Conjecture: a strictly increasing sequence. - J. Lowell, Sep 10 2008
The sequence is strictly increasing because (4*n)' = 4*n + 4*n'. - David Radcliffe, Aug 19 2014
8 is the smallest integer that has a nontrivial trajectory (not going to 0 nor reduced to a fixed point as 4) under A003415, but 15 = A090636(1) has 8 as second term in its trajectory. 20 is the next larger such integer with a distinct trajectory, but has two larger predecessors, cf. A090635. - M. F. Hasler, Nov 27 2019
In general, the trajectory of p^(p+1) under A003415 has a common factor p^p, and divided by p^p it gives the trajectory of p under A129283: n -> n + n'. Here we have the case p = 2 (see A129284 for a(n)/2^2), cf. A129151 and A129152 for p = 3 and 5. - M. F. Hasler, Nov 28 2019

Charles R Greathouse IV, Table of n, a(n) for n = 0..100

Cf. A129151, A129152, A068327, A099309, A051674, A100716, A129284.

Row n = 8 of A258651.

a129150 n = a129150_list !! n
a129150_list = iterate a003415 8  -- Reinhard Zumkeller, Apr 29 2012

dn[0] = 0; dn[1] = 0; dn[n_?Negative] := -dn[-n]; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Total[n*f[[2]]/f[[1]]]]]; s = 2^3; Join[{s}, Table[s = dn[s], {28}]] (* T. D. Noe, Mar 07 2013 *)

A129150(n,a=8)={if(n<0, vector(-n,n, if(n>1, a=A003415(a), a)), for(n=1,n, a=A003415(a)); a)}  \\ For n<0 return the vector a[0..-n-1]. - M. F. Hasler, Nov 27 2019

a(n+1) = A003415(a(n)), a(0) = 2^3 = 8.
a(n) = A090636(n+2).
A129251(a(n)) > 0. - Reinhard Zumkeller, Apr 07 2007
a(n) = 4*A129284(n). - M. F. Hasler, Nov 27 2019

a(21)-a(27) from Paolo P. Lava, Apr 16 2012

A062481 a(n) = n^prime(n).

Original entry on oeis.org

1, 8, 243, 16384, 48828125, 13060694016, 232630513987207, 144115188075855872, 8862938119652501095929, 100000000000000000000000000000, 191943424957750480504146841291811, 8505622499821102144576131684114829934592, 4695452425098908797088971409337422035076128813

Offset: 1

Labos Elemer, Jul 09 2001

Harry J. Smith, Table of n, a(n) for n = 1..50

Cf. A000040, A000312, A051674, A096250.

Table[n^Prime[n],{n,20}] (* Harvey P. Dale, Jun 12 2014 *)

a(n)={n^prime(n)} \\ Harry J. Smith, Aug 08 2009

from sympy import prime
def a(n): return n**prime(n)
print([a(n) for n in range(1, 14)]) # Michael S. Branicky, Jun 15 2022

From Amiram Eldar, Nov 18 2020: (Start)
a(n) = n^A000040(n).
Sum_{n>=1} 1/a(n) = A096250. (End)
a(n) == n (mod prime(n)). [Fermat's little theorem] - Nicolas Bělohoubek, Jun 14 2022

More terms from Harvey P. Dale, Jun 12 2014

cp's OEIS Frontend

A168036 Difference between n' and n, where n' is the arithmetic derivative of n (A003415).

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A099309 Numbers n whose k-th arithmetic derivative is nonzero for all k. Complement of A099308.

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A048102 Numbers k such that if k = Product p_i^e_i then p_i = e_i for all i.

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A097764 Numbers of the form (kp)^p for prime p and k=1,2,3,....

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A054412 Numbers k such that, in the prime factorization of k, the product of exponents equals the product of prime factors.

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A053089 a(n) = prime(n)^prime(n+1).

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A008478 Integers of the form Product p_j^k_j = Product k_j^p_j; p_j in A000040.

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