A000040 -id:A000040 - cp's OEIS frontend

A245704 Permutation of natural numbers: a(1) = 1, a(A014580(n)) = A000040(a(n)), a(A091242(n)) = A002808(a(n)), where A000040(n) = n-th prime, A002808(n) = n-th composite number, and A014580(n) and A091242(n) are binary codes for n-th irreducible and n-th reducible polynomial over GF(2), respectively.

1, 2, 3, 4, 6, 8, 5, 9, 12, 15, 7, 10, 13, 16, 21, 25, 14, 18, 19, 22, 26, 33, 38, 24, 11, 28, 30, 34, 39, 49, 23, 55, 36, 20, 42, 45, 37, 50, 56, 69, 47, 35, 77, 52, 32, 60, 17, 64, 54, 70, 78, 94, 66, 51, 29, 105, 74, 48, 41, 84, 53, 27, 88, 76, 95, 106, 73, 125, 91, 72, 44, 140, 97, 100, 68, 58, 115, 75, 40

Offset: 1

Antti Karttunen, Aug 02 2014

nonn

All the permutations A091203, A091205, A106443, A106445, A106447, A235042 share the same property that the binary representations of irreducible GF(2) polynomials (A014580) are mapped bijectively to the primes (A000040) but while they determine the mapping of corresponding reducible polynomials (A091242) to the composite numbers (A002808) by a simple multiplicative rule, this permutation employs index-recursion also in that case.

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A245703.
Cf. A000040, A002808, A007097, A010051, A014580, A091225, A091226, A091230, A091242, A091245.
Similar or related permutations: A245701, A245822, A227413, A091203, A091205, A106443, A106445, A106447, A235042, A244987, A245450.

allocatemem(123456789);
default(primelimit, 2^22)
A091226 = vector(2^22);
A002808(n)={ my(k=-1); while( -n + n += -k + k=primepi(n), ); n}; \\ This function from M. F. Hasler
isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
j=0; n=2; while((n < 2^22), if(isA014580(n), A091226[n] = A091226[n-1]+1, A091226[n] = A091226[n-1]); n++)
A091245(n) = ((n-A091226[n])-1);
A245704(n) = if(1==n, 1, if(isA014580(n), prime(A245704(A091226[n])), A002808(A245704(A091245(n)))));
for(n=1, 10001, write("b245704.txt", n, " ", A245704(n)));

;; With memoization-macro definec.
(definec (A245704 n) (cond ((= 1 n) n) ((= 1 (A091225 n)) (A000040 (A245704 (A091226 n)))) (else (A002808 (A245704 (A091245 n))))))

a(1) = 1, after which, if A091225(n) is 1 [i.e. n is in A014580], then a(n) = A000040(a(A091226(n))), otherwise a(n) = A002808(a(A091245(n))).
As a composition of related permutations:
a(n) = A227413(A245701(n)).
a(n) = A245822(A091205(n)).
Other identities. For all n >= 1, the following holds:
a(A091230(n)) = A007097(n). [Maps iterates of A014580 to the iterates of primes. Permutation A091205 has the same property].
A010051(a(n)) = A091225(n). [After a(1)=1, maps binary representations of irreducible GF(2) polynomials (= A014580) to primes and the corresponding representations of reducible polynomials to composites].

A246378 Permutation of natural numbers: a(1) = 1, a(2n) = nthcomposite(a(n)), a(2n+1) = nthprime(a(n)), where nthcomposite = A002808, nthprime = A000040.

Original entry on oeis.org

1, 4, 2, 9, 7, 6, 3, 16, 23, 14, 17, 12, 13, 8, 5, 26, 53, 35, 83, 24, 43, 27, 59, 21, 37, 22, 41, 15, 19, 10, 11, 39, 101, 75, 241, 51, 149, 114, 431, 36, 89, 62, 191, 40, 103, 82, 277, 33, 73, 54, 157, 34, 79, 58, 179, 25, 47, 30, 67, 18, 29, 20, 31, 56, 167, 134, 547, 102, 379, 304, 1523, 72, 233

Offset: 1

Antti Karttunen, Aug 27 2014

Contains an infinite number of infinite cycles. See comments at A246377.

Antti Karttunen, Table of n, a(n) for n = 1..4096
Index entries for sequences that are permutations of the natural numbers

Inverse: A246377.
Similar or related permutations: A237126, A054429, A227413, A236854, A246375, A246380, A246682, A163511.
Cf. A000035, A000040, A002808, A010051, A071904.

A002808(n) = { my(k=-1); while( -n + n += -k + k=primepi(n), ); n }; \\ This function from M. F. Hasler
A246378(n) = if(1==n, 1, if(!(n%2), A002808(A246378(n/2)), prime(A246378((n-1)/2))));
for(n=1, 4096, write("b246378.txt", n, " ", A246378(n)));
(Scheme, with memoizing definec-macro)
(definec (A246378 n) (cond ((< n 2) n) ((even? n) (A002808 (A246378 (/ n 2)))) (else (A000040 (A246378 (/ (- n 1) 2))))))

a(1) = 1, a(2n) = nthcomposite(a(n)), a(2n+1) = nthprime(a(n)), where nthcomposite = A002808, nthprime = A000040.
As a composition of related permutations:
a(n) = A227413(A054429(n)).
a(n) = A236854(A227413(n)).
a(n) = A246380(A246375(n)).
a(n) = A246682(A163511(n)). [For n >= 1].
Other identities. For all n > 1 the following holds:
A010051(a(n)) = A000035(n). [Maps odd numbers larger than one to primes, and even numbers to composites, in some order. Permutations A246380 & A246682 have the same property].

A330991 Positive integers whose number of factorizations into factors > 1 (A001055) is a prime number (A000040).

Original entry on oeis.org

4, 6, 8, 9, 10, 14, 15, 16, 21, 22, 24, 25, 26, 27, 30, 32, 33, 34, 35, 38, 39, 40, 42, 46, 49, 51, 54, 55, 56, 57, 58, 60, 62, 64, 65, 66, 69, 70, 74, 77, 78, 81, 82, 84, 85, 86, 87, 88, 90, 91, 93, 94, 95, 96, 102, 104, 105, 106, 110, 111, 114, 115, 118, 119

Offset: 1

Gus Wiseman, Jan 07 2020

nonn

In short, A001055(a(n)) belongs to A000040.

			Factorizations of selected terms:
  (4)    (8)      (16)       (24)       (60)       (96)
  (2*2)  (2*4)    (2*8)      (3*8)      (2*30)     (2*48)
         (2*2*2)  (4*4)      (4*6)      (3*20)     (3*32)
                  (2*2*4)    (2*12)     (4*15)     (4*24)
                  (2*2*2*2)  (2*2*6)    (5*12)     (6*16)
                             (2*3*4)    (6*10)     (8*12)
                             (2*2*2*3)  (2*5*6)    (2*6*8)
                                        (3*4*5)    (3*4*8)
                                        (2*2*15)   (4*4*6)
                                        (2*3*10)   (2*2*24)
                                        (2*2*3*5)  (2*3*16)
                                                   (2*4*12)
                                                   (2*2*3*8)
                                                   (2*2*4*6)
                                                   (2*3*4*4)
                                                   (2*2*2*12)
                                                   (2*2*2*2*6)
                                                   (2*2*2*3*4)
                                                   (2*2*2*2*2*3)

Amiram Eldar, Table of n, a(n) for n = 1..10000
R. E. Canfield, P. Erdős and C. Pomerance, On a Problem of Oppenheim concerning "Factorisatio Numerorum", J. Number Theory 17 (1983), 1-28.

Factorizations are A001055, with image A045782, with complement A330976.
Numbers whose number of strict integer partitions is prime are A035359.
Numbers whose number of integer partitions is prime are A046063.
Numbers whose number of set partitions is prime are A051130.
Numbers whose number of factorizations is a power of 2 are A330977.
The least number with prime(n) factorizations is A330992(n).
Cf. A001222, A033833, A045783, A181819, A181821, A326622, A330972, A330973, A330993.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],PrimeQ[Length[facs[#]]]&]

A025129 a(n) = p(1)p(n) + p(2)p(n-1) + ... + p(k)p(n-k+1), where k = [ n/2 ], p = A000040, the primes.

Original entry on oeis.org

0, 6, 10, 29, 43, 94, 128, 231, 279, 484, 584, 903, 1051, 1552, 1796, 2489, 2823, 3784, 4172, 5515, 6091, 7758, 8404, 10575, 11395, 14076, 15174, 18339, 19667, 23414, 24906, 29437, 31089, 36500, 38614, 44731, 47071, 54198, 56914, 65051, 68371, 77402, 81052, 91341

Offset: 1

Clark Kimberling

nonn

This is the sum of distinct squarefree semiprimes with prime indices summing to n + 1. A squarefree semiprime is a product of any two distinct prime numbers. A prime index of n is a number m such that the m-th prime number divides n. The multiset of prime indices of n is row n of A112798. - Gus Wiseman, Dec 05 2020

			From _Gus Wiseman_, Dec 05 2020: (Start)
The sequence of sums begins (n > 1):
    6 =  6
   10 = 10
   29 = 14 + 15
   43 = 22 + 21
   94 = 26 + 33 + 35
  128 = 34 + 39 + 55
  231 = 38 + 51 + 65 + 77
  279 = 46 + 57 + 85 + 91
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Gus Wiseman, Sum of prime(i) * prime(j) for i + j = n, i != j.

Cf. A000040, A258323.
The nonsquarefree version is A024697 (shifted right).
Row sums of A338905 (shifted right).
A332765 is the greatest among these squarefree semiprimes.
A001358 lists semiprimes.
A006881 lists squarefree semiprimes.
A014342 is the self-convolution of the primes.
A056239 is the sum of prime indices of n.
A338899/A270650/A270652 give the prime indices of squarefree semiprimes.
A339194 sums squarefree semiprimes grouped by greater prime factor.
Cf. A001221, A005117, A062198, A098350, A168472, A320656, A338900, A338901, A338904, A339114, A339116.

a025129 n = a025129_list !! (n-1)
a025129_list= f (tail a000040_list) [head a000040_list] 1 where
   f (p:ps) qs k = sum (take (div k 2) $ zipWith (*) qs $ reverse qs) :
                   f ps (p : qs) (k + 1)
-- Reinhard Zumkeller, Apr 07 2014

f[n_] := Block[{primeList = Prime@ Range@ n}, Total[ Take[ primeList, Floor[n/2]]*Reverse@ Take[ primeList, {Floor[(n + 3)/2], n}]]]; Array[f, 44] (* Robert G. Wilson v, Apr 07 2014 *)

A025129=n->sum(k=1,n\2,prime(k)*prime(n-k+1)) \\ M. F. Hasler, Apr 06 2014

a(n) = A024697(n) for even n. - M. F. Hasler, Apr 06 2014

Following suggestions by Robert Israel and N. J. A. Sloane, initial 0=a(1) added by M. F. Hasler, Apr 06 2014

A105560 a(1) = 1, and for n >= 2, a(n) = prime(bigomega(n)), where prime(n) = A000040(n) and bigomega(n) = A001222(n).

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 5, 3, 3, 2, 5, 2, 3, 3, 7, 2, 5, 2, 5, 3, 3, 2, 7, 3, 3, 5, 5, 2, 5, 2, 11, 3, 3, 3, 7, 2, 3, 3, 7, 2, 5, 2, 5, 5, 3, 2, 11, 3, 5, 3, 5, 2, 7, 3, 7, 3, 3, 2, 7, 2, 3, 5, 13, 3, 5, 2, 5, 3, 5, 2, 11, 2, 3, 5, 5, 3, 5, 2, 11, 7, 3, 2, 7, 3, 3, 3, 7, 2, 7, 3, 5, 3, 3, 3, 13, 2, 5, 5, 7

Offset: 1

Cino Hilliard, May 03 2005

From Antti Karttunen, Jul 21 2014: (Start)
a(n) divides A122111(n), A242424(n), A243072(n), A243073(n) because a(n) divides all the terms in column n of A243070.
a(2n-1) divides A243505(n) and a(2n-1)^2 divides A122111(2n-1).
(End)

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

Cf. A000040, A001222, A006530, A008578, A243070, A242424, A243072, A243073, A122111, A243505.

Table[Prime[Sum[FactorInteger[n][[i,2]],{i,1,Length[FactorInteger[n]]}]],{n,2,40}] (* Stefan Steinerberger, May 16 2007 *)

d(n) = for(x=2,n,print1(prime(bigomega(x))","))

from sympy import prime, primefactors
def a001222(n): return 0 if n==1 else a001222(n/primefactors(n)[0]) + 1
def a(n): return 1 if n==1 else prime(a001222(n)) # Indranil Ghosh, Jun 15 2017

a(1) = 1, and for n >= 2, a(n) = A000040(A001222(n)).
From Antti Karttunen, Jul 21 2014: (Start)
a(n) = A008578(1 + A001222(n)).
a(n) = A006530(A122111(n)).
a(n) = A122111(n) / A122111(A064989(n)).
a(2n-1) = A122111(2n-1) / A243505(n).
a(n) = A242424(n) / A064989(n).
(End)

a(1) = 1 prepended by Antti Karttunen, Jul 21 2014

A329605 Number of divisors of A108951(n), where A108951 is fully multiplicative with a(prime(i)) = prime(i)# = Product_{i=1..i} A000040(i).

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 16, 4, 9, 12, 32, 8, 64, 24, 18, 5, 128, 12, 256, 16, 36, 48, 512, 10, 27, 96, 16, 32, 1024, 24, 2048, 6, 72, 192, 54, 15, 4096, 384, 144, 20, 8192, 48, 16384, 64, 32, 768, 32768, 12, 81, 36, 288, 128, 65536, 20, 108, 40, 576, 1536, 131072, 30, 262144, 3072, 64, 7, 216, 96, 524288, 256, 1152, 72, 1048576, 18, 2097152, 6144, 48

Offset: 1

Antti Karttunen, Nov 18 2019

nonn

Cf. A000005, A000040, A000142, A002110, A010052, A034386, A052126, A056239, A108951, A329902, A329378, A329382, A329614, A329617, A331283, A331286 (odd part).
Cf. A329606 (rgs-transform), A329608, A331284 (ordinal transform).
Cf. A331285 (the position where for the first time some term has occurred n times in this sequence).

Block[{a}, a[n_] := a[n] = Module[{f = FactorInteger[n], p, e}, If[Length[f] > 1, Times @@ a /@ Power @@@ f, {{p, e}} = f; Times @@ (Prime[Range[PrimePi[p]]]^e)]]; a[1] = 1; Array[DivisorSigma[0, a@ #] &, 75]] (* Michael De Vlieger, Jan 24 2020, after Jean-François Alcover at A108951 *)

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A329605(n) = numdiv(A108951(n));

A329605(n) = if(1==n,1,my(f=factor(n),e=1,m=1); forstep(i=#f~,1,-1, e += f[i,2]; m *= e^(primepi(f[i,1])-if(1==i,0,primepi(f[i-1,1])))); (m)); \\ Antti Karttunen, Jan 14 2020

A329605(n) = if(1==n,1,my(f=factor(n),e=0,d); forstep(i=#f~,1,-1, e += f[i,2]; d = (primepi(f[i,1])-if(1==i,0,primepi(f[i-1,1]))); f[i,1] = (e+1); f[i,2] = d); factorback(f)); \\ Antti Karttunen, Jan 14 2020

a(n) = A000005(A108951(n)).
a(n) >= A329382(n) >= A329617(n) >= A329378(n).
A020639(a(n)) = A329614(n).
From Antti Karttunen, Jan 14 2020: (Start)
a(A052126(n)) = A329382(n).
a(A002110(n)) = A000142(1+n), for all n >= 0.
a(n) > A056239(n).
a(A329902(n)) = A002183(n).
A000265(a(n)) = A331286(n).
gcd(n,a(n)) = A331283(n).
If n = p(k1)^e(k1) * p(k2)^e(k2) * p(k3)^e(k3) * ... * p(kx)^e(kx), with p(n) = A000040(n) and k1 > k2 > ... > kx, then a(n) = (1+e(k1))^(k1-k2) * (1+e(k1)+e(k2))^(k2-k3) * ... * (1+e(k1)+e(k2)+...+e(kx))^kx.
A000035(a(n)) = A000035(A000005(n)) = A010052(n).
(End)

A078782 Nonprimes (A018252) with prime (A000040) subscripts.

Original entry on oeis.org

4, 6, 9, 12, 18, 21, 26, 28, 34, 42, 45, 52, 57, 60, 65, 74, 81, 84, 91, 95, 98, 106, 112, 119, 128, 133, 135, 141, 143, 147, 165, 170, 177, 180, 192, 195, 203, 209, 214, 220, 228, 231, 244, 246, 250, 253, 267, 284, 288, 290, 295, 301, 303, 316, 323, 329, 336

Offset: 1

Joseph L. Pe, Jan 09 2003

a(n) = A018252(A000040(n)). Subsequence of A175250 (nonprimes (A018252) with noncomposite (A008578) subscripts), a(n) = A175250(n+1). a(n) U A102615(n) = A018252(n). [From Jaroslav Krizek, Mar 13 2010]

			a(4) = nonprime(prime(4)) = nonprime(7) = 12.

Let A = primes A000040, B = nonprimes A018252. The 2-level compounds are AA = A006450, AB = A007821, BA = A078782, BB = A102615. The 3-level compounds AAA, AAB, ..., BBB are A038580, A049078, A270792, A102617, A270794, A270796, A102216.

from sympy import prime, composite
def A078782(n): return composite(prime(n)-1) # Chai Wah Wu, Nov 13 2024

Corrected by Jaroslav Krizek, Mar 13 2010

A138837 Non-Mersenne primes: A000040 \ A000668.

Original entry on oeis.org

2, 5, 11, 13, 17, 19, 23, 29, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283

Offset: 1

Omar E. Pol, Apr 03 2008

Primes that are not Mersenne primes A000668.
Pandigital primes in base 2. (Pandigital interpreted as including all digits, not necessarily only once each.) - Franklin T. Adams-Watters, May 11 2011
Primes whose sum of divisors is not a power of 2. - Omar E. Pol, Dec 19 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Microsoft Ltd., The Prime Challenge, Dec. 2013
OEIS wiki, Pandigital primes.

Primes in A133398.

Cf. A000040, A000203, A000668, A155151.

max = 300; Complement[Prime[Range[PrimePi[max]]], 2^Range[Ceiling[Log[2, max]]] - 1] (* Alonso del Arte, Dec 30 2013 *)

is_A138837(n)={isprime(n)&&1<M. F. Hasler, Feb 05 2014

A138837 = A000040 \ A000668. - M. F. Hasler, Feb 09 2014

New name from Omar E. Pol, Jan 02 2014

A229125 Numbers of the form p * m^2, where p is prime and m > 0: union of A228056 and A000040.

Original entry on oeis.org

2, 3, 5, 7, 8, 11, 12, 13, 17, 18, 19, 20, 23, 27, 28, 29, 31, 32, 37, 41, 43, 44, 45, 47, 48, 50, 52, 53, 59, 61, 63, 67, 68, 71, 72, 73, 75, 76, 79, 80, 83, 89, 92, 97, 98, 99, 101, 103, 107, 108, 109, 112, 113, 116, 117, 124, 125, 127, 128, 131, 137, 139, 147, 148, 149

Offset: 1

Chris Boyd, Sep 14 2013

nonn

No term is the product of two other terms.
Squares of terms and pairwise products of distinct terms form a subsequence of A028260.
Numbers n such that A162642(n) = 1. - Jason Kimberley, Oct 10 2016
Numbers k such that A007913(k) is a prime number. - Amiram Eldar, Jul 27 2020

Chris Boyd, Table of n, a(n) for n = 1..10000
Eckford Cohen, Arithmetical notes, IX. On the set of integers representable as a product of a prime and square, Acta Arithmetica, Vol. 7 (1962), pp. 417-420.

Subsequence of A026424.
Cf. A028260, A162642, A229153.
Cf. A007913, A013661.
Subsequences: A000040, A030078, A050997, A054753, A092759, A179643, A179665, A246551.

With[{nn=70},Take[Union[Flatten[Table[p*m^2,{p,Prime[Range[nn]]},{m,nn}]]], nn]] (* Harvey P. Dale, Dec 02 2014 *)

test(n)=isprime(core(n))
for(n=1,200,if(test(n), print1(n",")))

from math import isqrt
from sympy import primepi
def A229125(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(x//y**2) for y in range(1,isqrt(x)+1))
    return bisection(f,n,n) # Chai Wah Wu, Jan 30 2025

The number of terms not exceeding x is (Pi^2/6) * x/log(x) + O(x/(log(x))^2) (Cohen, 1962). - Amiram Eldar, Jul 27 2020

A329382 Product of exponents of prime factors of A108951(n), where A108951 is fully multiplicative with a(prime(i)) = prime(i)# = Product_{i=1..i} A000040(i).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 4, 2, 1, 3, 1, 2, 4, 4, 1, 6, 1, 3, 4, 2, 1, 4, 8, 2, 9, 3, 1, 6, 1, 5, 4, 2, 8, 8, 1, 2, 4, 4, 1, 6, 1, 3, 9, 2, 1, 5, 16, 12, 4, 3, 1, 12, 8, 4, 4, 2, 1, 8, 1, 2, 9, 6, 8, 6, 1, 3, 4, 12, 1, 10, 1, 2, 18, 3, 16, 6, 1, 5, 16, 2, 1, 8, 8, 2, 4, 4, 1, 12, 16, 3, 4, 2, 8, 6, 1, 24, 9, 16, 1, 6, 1, 4, 18

Offset: 1

Antti Karttunen, Nov 17 2019

nonn

Also the product of parts of the conjugate of the integer partition with Heinz number n, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). For example, the partition (3,2) with Heinz number 15 has conjugate (2,2,1) with product a(15) = 4. - Gus Wiseman, Mar 27 2022

Antti Karttunen, Table of n, a(n) for n = 1..10201
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to primorial numbers

Cf. A000005, A005361, A019565, A034386, A108951, A284001, A331188, A329378, A329605, A329617.
This is the conjugate version of A003963 (product of prime indices).
The solutions to a(n) = A003963(n) are A325040, counted by A325039.
The Heinz number of the conjugate partition is given by A122111.
These are the row products of A321649 and of A321650.
A000700 counts self-conj partitions, ranked by A088902, complement A330644.
A008480 counts permutations of prime indices, conjugate A321648.
A056239 adds up prime indices, row sums of A112798 and of A296150.
A124010 gives prime signature, sorted A118914, sum A001222.
A238744 gives the conjugate of prime signature, rank A238745.
Cf. A000701, A000720, A001221, A046682, A175508, A290822, A303975, A316524, A324850, A352486-A352491, A353570.

Table[Times @@ FactorInteger[Times @@ Map[#1^#2 & @@ # &, FactorInteger[n] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}]][[All, -1]], {n, 105}] (* Michael De Vlieger, Jan 21 2020 *)

A005361(n) = factorback(factor(n)[, 2]); \\ from A005361
A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A329382(n) = A005361(A108951(n));

A329382(n) = if(1==n,1,my(f=factor(n),e=0,m=1); forstep(i=#f~,1,-1, e += f[i,2]; m *= e^(primepi(f[i,1])-if(1==i,0,primepi(f[i-1,1])))); (m)); \\ Antti Karttunen, Jan 14 2020

a(n) = A005361(A108951(n)).
A329605(n) >= a(n) >= A329617(n) >= A329378(n).
a(A019565(n)) = A284001(n).
From Antti Karttunen, Jan 14 2020: (Start)
If n = p(k1)^e(k1) * p(k2)^e(k2) * p(k3)^e(k3) * ... * p(kx)^e(kx), with p(n) = A000040(n) and k1 > k2 > k3 > ... > kx, then a(n) = e(k1)^(k1-k2) * (e(k1)+e(k2))^(k2-k3) * (e(k1)+e(k2)+e(k3))^(k3-k4) * ... * (e(k1)+e(k2)+...+e(kx))^kx.
a(n) = A000005(A331188(n)) = A329605(A052126(n)).
(End)
a(n) = A003963(A122111(n)). - Gus Wiseman, Mar 27 2022

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A246378 Permutation of natural numbers: a(1) = 1, a(2n) = nthcomposite(a(n)), a(2n+1) = nthprime(a(n)), where nthcomposite = A002808, nthprime = A000040.

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A330991 Positive integers whose number of factorizations into factors > 1 (A001055) is a prime number (A000040).

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A025129 a(n) = p(1)p(n) + p(2)p(n-1) + ... + p(k)p(n-k+1), where k = [ n/2 ], p = A000040, the primes.

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A105560 a(1) = 1, and for n >= 2, a(n) = prime(bigomega(n)), where prime(n) = A000040(n) and bigomega(n) = A001222(n).

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A329605 Number of divisors of A108951(n), where A108951 is fully multiplicative with a(prime(i)) = prime(i)# = Product_{i=1..i} A000040(i).

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A078782 Nonprimes (A018252) with prime (A000040) subscripts.

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