A072873 -id:A072873 - cp's OEIS frontend

A365632 The number of divisors of n that are terms of A072873.

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Sep 14 2023

The sum of these divisors is A365633(n) and the largest of them is A327939(n).

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

Cf. A072873, A327939, A365633, A365634.

f[p_, e_] := 1 + Floor[e/p]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i,2]\f[i,1]);}

Multiplicative with a(p^e) = 1 + floor(e/p).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} p^p/(p^p-1) = 1.3850602852... .

Data corrected by Amiram Eldar, Sep 20 2023

A365637 a(n) is the smallest number k such that k*n is a term of A072873.

Original entry on oeis.org

1, 2, 9, 1, 625, 18, 117649, 2, 3, 1250, 25937424601, 9, 23298085122481, 235298, 5625, 1, 48661191875666868481, 6, 104127350297911241532841, 625, 1058841, 51874849202, 907846434775996175406740561329, 18, 125, 46596170244962, 1, 117649, 88540901833145211536614766025207452637361

Offset: 1

Amiram Eldar, Sep 14 2023

Amiram Eldar, Table of n, a(n) for n = 1..388

Cf. A072873, A365636.

f[p_, e_] := p^(Ceiling[e/p]*p - e); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^(f[i,1] * ceil(f[i,2] / f[i,1]) - f[i,2]));}

Multiplicative with a(p^e) = p^(p*ceiling(e/p) - e).
a(n) = A365636(n)/n.
a(n) >= 1 with equality if and only if n is in A072873.

Data, formulas and codes corrected by Amiram Eldar, Feb 15 2024

A368336 The number of divisors of the largest term of A072873 that divides of n.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 4, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 1, 1, 1, 7, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 5, 4, 1, 1, 3, 1, 1, 1

Offset: 1

Amiram Eldar, Dec 21 2023

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

Cf. A000005, A048103, A072873, A327939, A365632, A368331, A368335.

f[p_, e_] := (e - Mod[e, p] + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,2] - f[i,2]%f[i,1] + 1);}

a(n) = A000005(A327939(n)).
Multiplicative with a(p^e) = e - (e mod p) + 1.
a(n) >= 1, with equality if and only if n is in A048103.
a(n) <= A000005(n), with equality if and only if n is in A072873.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + p/(p^p-1)) = 1.86196549645040699446... .

A365633 The sum of divisors of n that are terms of A072873.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 7, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 4, 3, 1, 1, 1, 7, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 7, 1, 1, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 1, 1, 1, 15, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 7, 4, 1, 1, 3, 1, 1, 1

Offset: 1

Amiram Eldar, Sep 14 2023

The number of these divisors is A365632(n) and the largest of them is A327939(n).

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

Cf. A072873, A327939, A332653, A365632.

f[p_, e_] := (p^(Floor[e/p] + 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^(1+f[i,2] \ f[i,1])-1)/(f[i,1] - 1));}

Multiplicative with a(p^e) = (p^(floor(e/p)+1) - 1)/(p - 1).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (A332653(p)/(p^(p-1)-1) - 1/(p*(p-1))) = 2.253624924813... .

A365636 a(n) is the smallest multiple of n that is a term of A072873.

Original entry on oeis.org

1, 4, 27, 4, 3125, 108, 823543, 16, 27, 12500, 285311670611, 108, 302875106592253, 3294172, 84375, 16, 827240261886336764177, 108, 1978419655660313589123979, 12500, 22235661, 1141246682444, 20880467999847912034355032910567, 432, 3125, 1211500426369012, 27, 3294172

Offset: 1

Amiram Eldar, Sep 14 2023

Amiram Eldar, Table of n, a(n) for n = 1..388

Cf. A072873, A365637.

f[p_, e_] := p^(Ceiling[e/p]*p); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^(f[i,1] * ceil(f[i,2] / f[i,1])));}

Multiplicative with a(p^e) = p^(p*ceiling(e/p)).
a(n) = n * A365637(n).
a(n) >= n with equality if and only if n is in A072873.
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p/(p^p-1)) = 1.86196549645040699446... .

Data, formulas and codes corrected by Amiram Eldar, Feb 15 2024

A267757 Number of terms of A072873 less than or equal to 10^n.

Original entry on oeis.org

2, 5, 9, 15, 25, 36, 52, 73, 98, 129, 167, 213, 270, 338, 421, 517, 632, 768, 920, 1102, 1311, 1547, 1824, 2143, 2501, 2911, 3379, 3906, 4493, 5164, 5920, 6757, 7704, 8765, 9943, 11247, 12706, 14324, 16103, 18078, 20268, 22666, 25315, 28239, 31437, 34941, 38785, 43002

Offset: 1

Robert G. Wilson v, Jan 20 2016

nonn

Robert G. Wilson v, Table of n, a(n) for n = 1..108

Cf. A072873 (numbers n such that sum( e(i)/p(i) ) is an integer, where the prime factorization of n is Product( p(i)^e(i) )).

mx = 10^108; lst = Sort@ Flatten@ Table[
2^a*3^b*5^c*7^d*11^e*13^f*17^g*19^h*23^i*29^j*31^k*37^l*41^m*43^n*47^o*53^p,
{a, 0, Log[ 2, mx],2},
{b, 0, Log[ 3, mx/ 2^a], 3},
{c, 0, Log[ 5, mx/(2^a*3^b)], 5},
{d, 0, Log[ 7, mx/(2^a*3^b*5^c)], 7},
{e, 0, Log[11, mx/(2^a*3^b*5^c*7^d)], 11},
{f, 0, Log[13, mx/(2^a*3^b*5^c*7^d*11^e)], 13},
{g, 0, Log[17, mx/(2^a*3^b*5^c*7^d*11^e*13^f)], 17},
{h, 0, Log[19, mx/(2^a*3^b*5^c*7^d*11^e*13^f*17^g)], 19},
{i, 0, Log[23, mx/(2^a*3^b*5^c*7^d*11^e*13^f*17^g*19^h)], 23},
{j, 0, Log[29, mx/(2^a*3^b*5^c*7^d*11^e*13^f*17^g*19^h*23^i)], 29},
{k, 0, Log[31, mx/(2^a*3^b*5^c*7^d*11^e*13^f*17^g*19^h*23^i*29^j)], 31},
{l, 0, Log[37, mx/(2^a*3^b*5^c*7^d*11^e*13^f*17^g*19^h*23^i*29^j*31^k)], 37},
{m, 0, Log[41, mx/(2^a*3^b*5^c*7^d*11^e*13^f*17^g*19^h*23^i*29^j*31^k*37^l)], 41},
{n, 0, Log[43, mx/(2^a*3^b*5^c*7^d*11^e*13^f*17^g*19^h*23^i*29^j*31^k*37^l*41^m)], 43},
{o, 0, Log[47, mx/(2^a*3^b*5^c*7^d*11^e*13^f*17^g*19^h*23^i*29^j*31^k*37^l*41^m*43^n)], 47},
{p, 0, Log[53, mx/(2^a*3^b*5^c*7^d*11^e*13^f*17^g*19^h*23^i*29^j*31^k*37^l*41^m*43^n*47^o)], 53},
{q, 0, Log[59, mx/(2^a*3^b*5^c*7^d*11^e*13^f*17^g*19^h*23^i*29^j*31^k*37^l*41^m*43^n*47^o*53^p)], 59}]; Table[ Length@ Select[lst, # <= 10^n &], {n, 108}]

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

Original entry on oeis.org

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

A083345 Numerator of r(n) = Sum(e/p: n=Product(p^e)); a(n) = n' / gcd(n,n'), where n' is the arithmetic derivative of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 5, 1, 3, 2, 7, 1, 4, 1, 9, 8, 2, 1, 7, 1, 6, 10, 13, 1, 11, 2, 15, 1, 8, 1, 31, 1, 5, 14, 19, 12, 5, 1, 21, 16, 17, 1, 41, 1, 12, 13, 25, 1, 7, 2, 9, 20, 14, 1, 3, 16, 23, 22, 31, 1, 23, 1, 33, 17, 3, 18, 61, 1, 18, 26, 59, 1, 13, 1, 39, 11, 20, 18, 71, 1, 11, 4, 43, 1, 31, 22

Offset: 1

Reinhard Zumkeller, Apr 25 2003

Least common multiple of n and its arithmetic derivative, divided by n, i.e. a(n) = lcm(n,n')/n = A086130(n)/A000027(n). - Giorgio Balzarotti, Apr 14 2011
From Antti Karttunen, Nov 12 2024: (Start)
Positions of multiples of any natural number in this sequence (like A369002, A369644, A369005, or A369007) form always a multiplicative semigroup: if m and n are in that sequence, then so is m*n.
Proof: a(x) = x' / gcd(x,x') = A003415(x) / A085731(x) by definition. Let v_p(x) be the p-adic valuation of x, with p prime. Let e = v_p(c), the p-adic valuation of natural number c whose multiples we are searching for. For v_p(a(x)) >= e > 0 and v_p(a(y)) >= e > 0 to hold we must have v_p(x') = v_p(x)+h and v_p(y') = v_p(y)+k, for some h >= e, k >= e for p^e to divide a(x) and a(y).
  Then, as a(xy) = (xy)' / gcd(xy,(xy)') = (x'y + y'x) / gcd(xy, (x'y + y'x)), we have, for the top side, v_p((xy)') = min(v_p(x')+v_p(y), v_p(y')+v_p(x)) = min(v_p(x) + h + v_p(y), v_p(y) + k + v_p(x)) = v_p(xy) + min(h,k), and for the bottom side we get v_p(gcd(xy, (x'y + y'x))) = min(v_p(xy), v_p(xy) + min(h,k)) = v_p(xy), so v_p(a(xy)) = min(h,k) >= e, thus p^e | a(xy). For a composite c that is not a prime power, c | a(xy) holds if the above equations hold for all p^e || c.
(End)

			Fractions begin with 0, 1/2, 1/3, 1, 1/5, 5/6, 1/7, 3/2, 2/3, 7/10, 1/11, 4/3, ...
For n = 12, 2*2*3 = 2^2 * 3^1 --> r(12) = 2/2 + 1/3 = (6+2)/6, therefore a(12) = 4, A083346(12) = 3.
For n = 18, 2*3*3 = 2^1 * 3^2 --> r(18) = 1/2 + 2/3 = (3+4)/6, therefore a(18) = 7, A083346(18) = 6.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A083346 (denominator), A000027, A072873, A083347, A083348, A085731, A086130, A136141, A342001, A342002 [= a(A276086(n))].
Cf. A369001 (anti-parity), A377874 (parity).
Cf. A369002 (positions of even terms), A369003 (of odd terms), A369644 (of multiples of 3), A369005 (of multiples of 4), A373265 (of terms of the form 4m+2), A369007 (of multiples of 27), A369008, A369068 (Möbius transform), A369069.
Cf. A373363, A373366, A373377, A373485.

Array[Numerator@ Total[FactorInteger[#] /. {p_, e_} /; e > 0 :> e/p] - Boole[# == 1] &, 85] (* Michael De Vlieger, Feb 25 2018 *)

A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~,i,f[i,2]/f[i,1]))); }; \\ Antti Karttunen, Feb 25 2018

The fraction a(n)/A083346(n) is totally additive with a(p) = 1/p. - Franklin T. Adams-Watters, May 17 2006
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A083346(k) = Sum_{p prime} 1/(p*(p-1)) = 0.773156... (A136141). - Amiram Eldar, Sep 29 2023
a(n) = A003415(n) / A085731(n) = A342001(n) / A369008(n). - Antti Karttunen, Jan 16 2024

Secondary definition added by Antti Karttunen, Nov 12 2024

A085731 Greatest common divisor of n and its arithmetic derivative.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 4, 3, 1, 1, 4, 1, 1, 1, 16, 1, 3, 1, 4, 1, 1, 1, 4, 5, 1, 27, 4, 1, 1, 1, 16, 1, 1, 1, 12, 1, 1, 1, 4, 1, 1, 1, 4, 3, 1, 1, 16, 7, 5, 1, 4, 1, 27, 1, 4, 1, 1, 1, 4, 1, 1, 3, 64, 1, 1, 1, 4, 1, 1, 1, 12, 1, 1, 5, 4, 1, 1, 1, 16, 27, 1, 1, 4, 1, 1, 1, 4, 1, 3, 1, 4, 1, 1, 1, 16

Offset: 1

Reinhard Zumkeller, Jul 20 2003

a(n) = 1 iff n is squarefree (A005117), cf. A068328.

This sequence is very probably multiplicative. - Mitch Harris, Apr 19 2005

T. D. Noe, Table of n, a(n) for n = 1..10000
Victor Ufnarovski and Bo Ahlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003, #03.3.4.

Cf. A003415, A005117, A048103, A068328, A083346, A083347, A086130, A129251, A189100, A189036, A189103, A190116, A276086, A327858, A327938, A328572, A340070.

Cf. A072873 (positions of records), A268398 (partial sums).

a085731 n = gcd n $ a003415 n -- Reinhard Zumkeller, May 10 2011

d[0] = d[1] = 0; d[n_] := d[n] = n*Total[Apply[#2/#1 &, FactorInteger[n], {1}]]; a[n_] := GCD[n, d[n]]; Table[a[n], {n, 1, 96}] (* Jean-François Alcover, Feb 21 2014 *)
f[p_, e_] := p^If[Divisible[e, p], e, e - 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 31 2023 *)

a(n) = {my(f = factor(n)); for (i=1, #f~, if (f[i,2] % f[i,1], f[i,2]--);); factorback(f);} \\ Michel Marcus, Feb 14 2016

a(n) = GCD(n, A003415(n)).
Multiplicative with a(p^e) = p^e if p divides e; a(p^e) = p^(e-1) otherwise. - Eric M. Schmidt, Oct 22 2013
From Antti Karttunen, Feb 28 2021: (Start)
Thus a(A276086(n)) = A328572(n), by the above formula and the fact that A276086 is a permutation of A048103.
a(n) = n / A083346(n) = A190116(n) / A086130(n). (End)

A324524 Numbers where every prime index divides its multiplicity in the prime factorization. Numbers divisible by a power of prime(k)^k for each prime index k.

Original entry on oeis.org

1, 2, 4, 8, 9, 16, 18, 32, 36, 64, 72, 81, 125, 128, 144, 162, 250, 256, 288, 324, 500, 512, 576, 648, 729, 1000, 1024, 1125, 1152, 1296, 1458, 2000, 2048, 2250, 2304, 2401, 2592, 2916, 4000, 4096, 4500, 4608, 4802, 5184, 5832, 6561, 8000, 8192, 9000, 9216

Offset: 1

Gus Wiseman, Mar 07 2019

nonn

These are a kind of self-describing numbers (cf. A001462, A304679).
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The prime signature of a number is the multiset of multiplicities (or exponents) in its prime factorization.
Also Heinz numbers of integer partitions in which every part divides its multiplicity (counted by A001156). The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
Also products of elements of A062457.

			The sequence of terms together with their prime indices begins as follows. For example, we have 18: {1,2,2} because 18 = prime(1) * prime(2) * prime(2).
    1: {}
    2: {1}
    4: {1,1}
    8: {1,1,1}
    9: {2,2}
   16: {1,1,1,1}
   18: {1,2,2}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   64: {1,1,1,1,1,1}
   72: {1,1,1,2,2}
   81: {2,2,2,2}
  125: {3,3,3}
  128: {1,1,1,1,1,1,1}
  144: {1,1,1,1,2,2}
  162: {1,2,2,2,2}
  250: {1,3,3,3}
  256: {1,1,1,1,1,1,1,1}

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

Cf. A001156, A033461, A056239, A062457, A066328, A072873, A112798, A118914 (prime signature), A124010, A181819, A276078, A304679.
Cf. A109298, A324525, A324570, A324571, A324572, A324587, A324588.
Range of values of A090884.
Sequences related to self-description: A000002, A001462, A079000, A079254, A276625, A304360.

q:= n-> andmap(i-> irem(i[2], numtheory[pi](i[1]))=0, ifactors(n)[2]):
select(q, [$1..10000])[];  # Alois P. Heinz, Mar 08 2019

Select[Range[1000],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>Divisible[k,PrimePi[p]]]&]
v = Join[{1}, Prime[(r = Range[10])]^r]; n = Length[v]; vmax = 10^4; s = {1}; Do[v1 = v[[k]]; rmax = Floor[Log[v1, vmax]]; s1 = v1^Range[0, rmax]; s2 = Select[Union[Flatten[Outer[Times, s, s1]]], # <= vmax &]; s = Union[s, s2], {k, 2, n}]; Length[s] (* Amiram Eldar, Sep 30 2020 *)

Closed under multiplication.

Sum_{n>=1} 1/a(n) = Product_{k>=1} 1/(1-prime(k)^(-k)) = 2.26910478689594012492... - Amiram Eldar, Sep 30 2020

cp's OEIS Frontend

A365632 The number of divisors of n that are terms of A072873.

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A365637 a(n) is the smallest number k such that k*n is a term of A072873.

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A368336 The number of divisors of the largest term of A072873 that divides of n.

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A365633 The sum of divisors of n that are terms of A072873.

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A365636 a(n) is the smallest multiple of n that is a term of A072873.

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A267757 Number of terms of A072873 less than or equal to 10^n.

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

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