A051674 -id:A051674 - cp's OEIS frontend

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

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

A328233 Numbers n such that the arithmetic derivative of A276086(n) is prime.

Original entry on oeis.org

3, 7, 9, 33, 37, 38, 211, 213, 218, 241, 242, 246, 247, 249, 2313, 2317, 2319, 2341, 2342, 2346, 2521, 2523, 2526, 2529, 2550, 2553, 2559, 30031, 30038, 30039, 30061, 30062, 30063, 30066, 30069, 30242, 30243, 30249, 30270, 30278, 30279, 32341, 32342, 32347, 32370, 32373, 32377, 32379, 32551, 32553, 510513, 510518, 510519

Offset: 1

Antti Karttunen, Oct 09 2019

nonn

Numbers n for which A327860(n) = A003415(A276086(n)) is a prime.
Numbers n such that A276086(n) is in A157037.
Terms come in distinct "batches", where in each batch they are "slightly more" than the nearest primorial (A002110) below. This is explained by the fact that for A276086(n) to be a squarefree (which is the necessary condition for A157037), n's primorial base expansion (A049345) must not contain digits larger than 1. Thus this is a subsequence of A276156.
Numbers n such that A327860(A276086(n)) = A003415(A276087(n)) is a prime [A276087(n) is in A157037] are much rarer: 2, 4, 30, 212, 421, 30045, 510511, 512820, 9729723, ...
For all terms k in this sequence, A327969(k) <= 4, and particularly A327969(k) = 2 when k is a prime. Otherwise, when k is not a prime, but A003415(k) is, A327969(k) = 3, while for other cases (when k is neither prime nor in A157037), we have A327969(k) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..269
Index entries for sequences related to primorial base

Cf. A002110, A003415, A051674, A157037, A276086, A327860, A327969, A327978, A328232, A328240.

Subsequence of A276156, of A328116, and of A328242.

A327860(n) = { my(m=1, i=0, s=0, pr=1, nextpr); while((n>0), i=i+1; nextpr = prime(i)*pr; if((n%nextpr), my(e=((n%nextpr)/pr)); m *= (prime(i)^e); s += (e / prime(i)); n-=(n%nextpr)); pr=nextpr); (s*m); };
isA328233(n) = isprime(A327860(n));

A328321 Numbers n for which A328311(n) = 1 + A051903(A003415(n)) - A051903(n) is strictly positive.

Original entry on oeis.org

4, 6, 10, 12, 14, 15, 16, 20, 21, 22, 26, 27, 28, 30, 33, 34, 35, 36, 38, 39, 42, 44, 46, 48, 50, 51, 52, 54, 55, 57, 58, 60, 62, 64, 65, 66, 68, 69, 70, 74, 76, 77, 78, 80, 82, 84, 85, 86, 87, 91, 92, 93, 94, 95, 99, 100, 102, 105, 106, 108, 110, 111, 112, 114, 115, 116, 118, 119, 122, 123, 124, 129, 130, 132, 133

Offset: 1

Antti Karttunen, Oct 13 2019

nonn

Numbers n for which A051903(A003415(n)) >= A051903(n), i.e., numbers such that taking their arithmetic derivative does not decrease their "degree", A051903, the maximal exponent in prime factorization.

			10 = 2*5 has maximal exponent (A051903) 1, and its arithmetic derivative A003415(10) = 2+5 = 7 also has maximal exponent 1, thus 10 is included in this sequence.
15 = 3*5 has maximal exponent 1, and its arithmetic derivative A003415(15) = 3+5 = 8 = 2^3 has maximal exponent 3, thus 15 is included in this sequence.
For 8 = 2^3, its arithmetic derivative A003415(8) = 12 = 2^2 * 3, and as 2 < 3 (highest exponent of 12 is less than that of 8), 8 is NOT included here, and from this we also see that A100716 is not a subsequence of this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A003415, A051903, A100716, A328302, A328310, A328311.

Cf. A328320 (complement), A051674, A157037, A328304, A328305 (subsequences).

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A328311(n) = if(n<=1,0,1+(A051903(A003415(n)) - A051903(n)));
isA328321(n) = (A328311(n)>0);

A083347 Numbers k such that Sum(e/p: k=Product(p^e)) < 1.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 98, 99, 101, 102

Offset: 1

Reinhard Zumkeller, Apr 25 2003

nonn

Numbers k whose arithmetic derivative (A003415) k' < k. - T. D. Noe, Apr 24 2011

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

Cf. A003415, A072873, A051674, A083348.

Cf. A168036.

a083347 n = a083347_list !! (n-1)
a083347_list = filter ((< 0) . a168036) [1..]
-- Reinhard Zumkeller, May 22 2015, May 10 2011

Select[Range@ 102, If[Abs@ # < 2, 0, # Total[#2/#1 & @@@ FactorInteger@ Abs@ #]] < # &] (* Michael De Vlieger, Feb 02 2019 *)

A083345(a(n)) < A083346(a(n));
A083345(A051674(n)) = A083346(A051674(n));
A083345(A083348(n)) > A083346(A083348(n)).
A168036(a(n)) < 0. - Reinhard Zumkeller, May 22 2015

A088730 Numbers of the form p^p - 1, where p is a prime.

Original entry on oeis.org

3, 26, 3124, 823542, 285311670610, 302875106592252, 827240261886336764176, 1978419655660313589123978, 20880467999847912034355032910566, 2567686153161211134561828214731016126483468

Offset: 1

Cino Hilliard, Nov 23 2003

nonn

Sum of reciprocals = 0.3721161884983118696170302604..

			a(1) = 3 because the first prime is 2 and 2^2 - 1 = 3.
a(2) = 26 because the second prime is 3 and 3^3 - 1 = 26.
a(3) = 3124 because the fifth prime is 5 and 5^5 - 1 = 3124.

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

Cf. A051674, A088807, A125135 (factorizations).

[p^p-1: p in PrimesUpTo(20)]; // Vincenzo Librandi, Mar 27 2014

Table[Prime[n]^Prime[n] - 1, {n, 10}] (* Alonso del Arte, May 22 2013 *)
#^#-1&/@Prime[Range[10]] (* Harvey P. Dale, Jun 10 2013 *)

a(n)={my(p=prime(n)); p^p-1;} \\ Joerg Arndt, May 27 2013

a(n) = A051674(n) - 1. - R. J. Mathar, Jul 15 2007

More terms from Ray Chandler, Feb 21 2004

A114850 (n-th semiprime)^(n-th semiprime).

Original entry on oeis.org

256, 46656, 387420489, 10000000000, 11112006825558016, 437893890380859375, 5842587018385982521381124421, 341427877364219557396646723584, 88817841970012523233890533447265625

Offset: 1

Jonathan Vos Post, Feb 20 2006

Semiprime analog of A051674. This is also a subset of A113877 "semiprimes to semiprime powers."

Harvey P. Dale, Table of n, a(n) for n = 1..121

Cf. A001358, A051674, A113877.

#^#&/@Select[Range[30],PrimeOmega[#]==2&] (* Harvey P. Dale, Nov 07 2016 *)

a(n) = A001358(n)^A001358(n).

Corrected by Don Reble, Nov 22 2006

A129152 The n-th arithmetic derivative of 5^6.

Original entry on oeis.org

15625, 18750, 34375, 37500, 87500, 187500, 475000, 1212500, 2437500, 6362500, 12737500, 25487500, 50987500, 101987500, 206975000, 530037500, 1060087500, 3890025000, 15175012500, 45525375000, 177026512500, 596222100000, 2708984250000, 12765250350000

Offset: 0

Reinhard Zumkeller, Apr 01 2007

nonn

In general, the trajectory of p^(p+1) under A003415 is equal to p^p times the trajectory of p under A129283: n -> n + n'. Here we have the case p = 5 (see A129286 for a(n)/5^5), see A129150 and A129151 for p = 2 and 3. - M. F. Hasler, Nov 28 2019

Cf. A129150, A129151, A068327.

Cf. A099309, A051674, A100716.

a129152 n = a129152_list !! n
a129152_list = iterate a003415 15625  -- 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 = 5^6; Join[{s}, Table[s = dn[s], {18}]] (* T. D. Noe, Mar 07 2013 *)

A129152_upto(N)=vector(N,n,N=if(n>1,A003415(N),5^6)) \\ gives a(0..N-1). To get a(1..N) put A003415() around if() instead inside.  M. F. Hasler, Nov 28 2019

a(n+1) = A003415(a(n)), a(0) = 5^6 = 15625.

a(n) = A129286(n)*5^5; A129251(a(n)) > 0. - Reinhard Zumkeller, Apr 07 2007

A098699 Anti-derivative of n: or the first occurrence of n in A003415, or zero if impossible.

Original entry on oeis.org

1, 2, 0, 0, 4, 6, 9, 10, 15, 14, 21, 0, 8, 22, 33, 26, 12, 0, 65, 34, 51, 18, 57, 0, 20, 46, 69, 27, 115, 0, 161, 30, 16, 62, 93, 0, 155, 0, 217, 45, 111, 42, 185, 82, 24, 50, 129, 0, 44, 94, 141, 63, 235, 0, 329, 75, 52, 0, 265, 70, 36, 66, 177, 122, 183, 0, 305, 0, 40, 134

Offset: 0

Robert G. Wilson v, Sep 21 2004

nonn

With Goldbach's conjecture, any even integer n = 2k > 2 can be written as sum of two primes, n = p + q, and therefore admits N = pq as (not necessarily smallest) anti-derivative, so a(2k) > 0, and a(2k) <= pq <= k^2. [Remark inspired by L. Polidori.] - M. F. Hasler, Apr 09 2015

a(n) <= n^2/4 for n > 1. This is because if A003415(x) = n > 1, x = a*b for some a,b > 1, and then n = A003415(x) = a*A003415(b) + A003415(a)*b >= a + x/a >= 2*sqrt(x), i.e. x <= (n/2)^2. - Robert Israel, May 29 2023

T. D. Noe, Table of n, a(n) for n = 0..5000

Cf. A003415, A051674, zeros in A098700.

ader:= proc(n) local t;
  n * add(t[2]/t[1], t = ifactors(n)[2])
end proc:
N:= 100: # for a(0) .. a(N)
V:= Array(0..N): count:= 0:
for x from 1 to N^2/4 while count < 100 do
  v:= ader(x);
  if v > 0 and v <= 100 and V[v] = 0 then
    count:= count+1; V[v]:= x;
  fi;
od:
convert(V,list); # Robert Israel, May 29 2023

a[1] = 0; a[n_] := Block[{f = Transpose[ FactorInteger[ n]]}, If[ PrimeQ[n], 1, Plus @@ (n*f[[2]]/f[[1]])]]; b = Table[0, {70}]; b[[1]] = 1; Do[c = a[n]; If[c < 70 && b[[c + 1]] == 0, b[[c + 1]] = n], {n, 10^3}]; b

A098699(n)=for(k=1,(n\2)^2+2,A003415(k)==n&&return(k)) \\ M. F. Hasler, Apr 09 2015

from sympy import factorint
def A098699(n):
    if n < 2:
        return n+1
    for m in range(1,(n**2>>2)+1):
        if sum((m*e//p for p,e in factorint(m).items())) == n:
            return m
    return 0 # Chai Wah Wu, Sep 12 2022

a(n) = n for { 4, 27, 3125, 823543, ... } = { p^p; p prime } = A051674.

A104126 a(n) = prime(n)^(prime(n)+1).

Original entry on oeis.org

8, 81, 15625, 5764801, 3138428376721, 3937376385699289, 14063084452067724991009, 37589973457545958193355601, 480250763996501976790165756943041

Offset: 1

Cino Hilliard, Mar 06 2005

Sum of reciprocals rapidly converges to 0.1374098524791901212366977116..

A182938(a(n)) = 0. [Reinhard Zumkeller, Feb 18 2012]

Reinhard Zumkeller, Table of n, a(n) for n = 1..75

Cf. A051674, A000040.

a104126 n = p ^ (p + 1) where p = a000040 n
-- Reinhard Zumkeller, Feb 18 2012

#^(#+1)&/@Prime[Range[10]] (* Harvey P. Dale, Dec 12 2021 *)

ptopp1(n) = { local(x,y,z,sr=0); forprime(x=1,n, y=x^(x+1); z=(x+1)^x; sr+=1./y; print1(y","); ); print(); print(sr) }

Offset corrected by Reinhard Zumkeller, Feb 18 2012

A129151 The n-th arithmetic derivative of 3^4.

Original entry on oeis.org

81, 108, 216, 540, 1188, 2484, 5076, 10260, 23112, 57996, 135648, 475632, 1586736, 4760640, 20409408, 89259840, 374899968, 1880140032, 9400707072, 64402394112, 395614900224, 2769304412160, 22930714939392, 162970999640064, 1188480788434944, 8320496444780544

Offset: 0

Reinhard Zumkeller, Apr 01 2007

nonn

In general, the trajectory of p^(p+1) under A003415 is equal to p^p times the trajectory of p under A129283: n -> n + n'. Here we have the case p = 3 (see A129285 for a(n)/3^3), see A129150 and A129152 for p = 2 and 5. - M. F. Hasler, Nov 28 2019

Cf. A129150, A129152, A068327.

Cf. A099309, A051674, A100716.

a129151 n = a129151_list !! n
a129151_list = iterate a003415 81  -- 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 = 3^4; Join[{s}, Table[s = dn[s], {25}]] (* T. D. Noe, Mar 07 2013 *)

a(n+1) = A003415(a(n)), a(0) = 3^4 = 81.

a(n) = A129285(n)*3^3; A129251(a(n)) > 0. - Reinhard Zumkeller, Apr 07 2007

cp's OEIS Frontend

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

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A328233 Numbers n such that the arithmetic derivative of A276086(n) is prime.

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A328321 Numbers n for which A328311(n) = 1 + A051903(A003415(n)) - A051903(n) is strictly positive.

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A083347 Numbers k such that Sum(e/p: k=Product(p^e)) < 1.

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A088730 Numbers of the form p^p - 1, where p is a prime.

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Magma

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A114850 (n-th semiprime)^(n-th semiprime).

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A129152 The n-th arithmetic derivative of 5^6.

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A098699 Anti-derivative of n: or the first occurrence of n in A003415, or zero if impossible.