A064988 -id:A064988 - cp's OEIS frontend

A357983 Second MTF-transform of the primes (A000040). Replace prime(k) with prime(A064988(k)) in the prime factorization of n.

1, 2, 5, 4, 11, 10, 23, 8, 25, 22, 31, 20, 47, 46, 55, 16, 59, 50, 103, 44, 115, 62, 97, 40, 121, 94, 125, 92, 137, 110, 127, 32, 155, 118, 253, 100, 197, 206, 235, 88, 179, 230, 233, 124, 275, 194, 257, 80, 529, 242, 295, 188, 419, 250, 341, 184, 515, 274

Offset: 1

Gus Wiseman, Oct 24 2022

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. We define the MTF-transform as shifting a number's prime indices along a function; see the Mathematica program.

			First, we have
- 4 = prime(1) * prime(1),
- A000040(1) = 2,
- A064988(4) = prime(2) * prime(2) = 9.
Similarly, A064988(3) = 5. Next,
- 35 = prime(3) * prime(4),
- A064988(3) = 5,
- A064988(4) = 9,
- a(35) = prime(5) * prime(9) = 253.

Other multiplicative sequences: A003961, A357852, A064989, A357977, A357980.
Applying the transformation only once gives A064988.
The union is A076610 (numbers whose prime indices are themselves prime).
For partition numbers instead of primes we have A357979.
A000040 lists the primes.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000720, A003964, A063834, A215366, A296150, A299201, A299202, A357975.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mtf[f_][n_]:=Product[If[f[i]==0,1,Prime[f[i]]],{i,primeMS[n]}];
Array[mtf[mtf[Prime]],100]

A318668 a(n) = gcd(n, A064988(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 5, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 3, 1, 1, 5, 1, 1, 3, 1, 1, 1, 1, 1, 3, 11, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 9, 1, 1, 5, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 5

Offset: 1

Antti Karttunen, Sep 11 2018

nonn

a(n) > 1 if and only if the prime factorization of n contains at least two distinct primes, p and q, such that q = prime(p).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A064988, A318660.

A318668(n) = { my(f = factor(n)); for (k=1, #f~, f[k, 1] = prime(f[k, 1]); ); gcd(n,factorback(f)); }; \\ After code in A064988.

a(n) = gcd(n, A064988(n)).

A318660 Remainder when A064988(n) is divided by n.

Original entry on oeis.org

0, 1, 2, 1, 1, 3, 3, 3, 7, 3, 9, 9, 2, 9, 10, 1, 8, 3, 10, 19, 1, 5, 14, 15, 21, 19, 17, 13, 22, 15, 3, 19, 23, 7, 12, 9, 9, 11, 10, 17, 15, 3, 19, 15, 5, 19, 23, 21, 44, 13, 40, 5, 29, 51, 11, 11, 50, 37, 41, 15, 39, 9, 47, 25, 61, 3, 63, 55, 1, 1, 69, 27, 2, 27, 5, 71, 65, 69, 6, 11, 58, 45, 16, 9, 54, 57, 23, 45, 16, 15, 60, 11, 77, 69

Offset: 1

Altug Alkan and Antti Karttunen, Sep 08 2018

nonn

Inspired by A064988 and a 'minimum' version of it (A318871).

a(n) = 0 only for n = 1. Numbers n such that a(n) = 1 are 2, 4, 5, 16, 21, 69, 70, 181, 265, 370, 1043, 3760, 4531, ...

			a(6) = prime(2)*prime(3) mod 6 = 15 mod 6 = 3.

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

Cf. A064988, A076240, A318668, A318871.

Table[Mod[If[n == 1, 1, Apply[Times, FactorInteger[n] /. {p_, e_} /; p > 1 :> Prime[p]^e]], n], {n, 94}] (* Michael De Vlieger, Sep 10 2018 *)

A318660(n) = { my(f = factor(n)); for (k=1, #f~, f[k, 1] = prime(f[k, 1]); ); (factorback(f)%n); }; \\ After code in A064988.

a(n) = A064988(n) mod n.

a(A000040(n)) = A076240(n).

A003961 Completely multiplicative with a(prime(k)) = prime(k+1).

Original entry on oeis.org

1, 3, 5, 9, 7, 15, 11, 27, 25, 21, 13, 45, 17, 33, 35, 81, 19, 75, 23, 63, 55, 39, 29, 135, 49, 51, 125, 99, 31, 105, 37, 243, 65, 57, 77, 225, 41, 69, 85, 189, 43, 165, 47, 117, 175, 87, 53, 405, 121, 147, 95, 153, 59, 375, 91, 297, 115, 93, 61, 315, 67, 111, 275, 729, 119

Offset: 1

Marc LeBrun

Meyers (see Guy reference) conjectures that for all r >= 1, the least odd number not in the set {a(i): i < prime(r)} is prime(r+1). - N. J. A. Sloane, Jan 08 2021
Meyers' conjecture would be refuted if and only if for some r there were such a large gap between prime(r) and prime(r+1) that there existed a composite c for which prime(r) < c < a(c) < prime(r+1), in which case (by Bertrand's postulate) c would necessarily be a term of A246281. - Antti Karttunen, Mar 29 2021
a(n) is odd for all n and for each odd m there exists a k with a(k) = m (see A064216). a(n) > n for n > 1: bijection between the odd and all numbers. - Reinhard Zumkeller, Sep 26 2001
a(n) and n have the same number of distinct primes with (A001222) and without multiplicity (A001221). - Michel Marcus, Jun 13 2014
From Antti Karttunen, Nov 01 2019: (Start)
More generally, a(n) has the same prime signature as n, A046523(a(n)) = A046523(n). Also A246277(a(n)) = A246277(n) and A287170(a(n)) = A287170(n).
Many permutations and other sequences that employ prime factorization of n to encode either polynomials, partitions (via Heinz numbers) or multisets in general can be easily defined by using this sequence as one of their constituent functions. See the last line in the Crossrefs section for examples.
(End)

			a(12) = a(2^2 * 3) = a(prime(1)^2 * prime(2)) = prime(2)^2 * prime(3) = 3^2 * 5 = 45.
a(A002110(n)) = A002110(n + 1) / 2.

Richard K. Guy, editor, Problems From Western Number Theory Conferences, Labor Day, 1983, Problem 367 (Proposed by Leroy F. Meyers, The Ohio State U.).

Indranil Ghosh, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Index entries for sequences computed from indices in prime factorization.
Index entries for sequences related to Heinz numbers.

See A045965 for another version.
Row 1 of table A242378 (which gives the "k-th powers" of this sequence), row 3 of A297845 and of A306697. See also arrays A066117, A246278, A255483, A308503, A329050.
Cf. A064989 (a left inverse), A064216, A000040, A002110, A000265, A027746, A046523, A048673 (= (a(n)+1)/2), A108228 (= (a(n)-1)/2), A191002 (= a(n)*n), A252748 (= a(n)-2n), A286385 (= a(n)-sigma(n)), A283980 (= a(n)*A006519(n)), A341529 (= a(n)*sigma(n)), A326042, A049084, A001221, A001222, A122111, A225546, A260443, A245606, A244319, A246269 (= A065338(a(n))), A322361 (= gcd(n, a(n))), A305293.
Cf. A191555, A252738.
Cf. A249734, A249735 (bisections).
Cf. A246261 (a(n) is of the form 4k+1), A246263 (of the form 4k+3), A246271, A246272, A246259, A246281 (n such that a(n) < 2n), A246282 (n such that a(n) > 2n), A252742.
Cf. A275717 (a(n) > a(n-1)), A275718 (a(n) < a(n-1)).
Cf. A003972 (Möbius transform), A003973 (Inverse Möbius transform), A318321.
Cf. A300841, A305421, A322991, A250469, A269379 for analogous shift-operators in other factorization and quasi-factorization systems.
Cf. also following permutations and other sequences that can be defined with the help of this sequence: A005940, A163511, A122111, A260443, A206296, A265408, A265750, A275733, A275735, A297845, A091202 & A091203, A250245 & A250246, A302023 & A302024, A302025 & A302026.
A version for partition numbers is A003964, strict A357853.
A permutation of A005408.
Applying the same transformation again gives A357852.
Other multiplicative sequences: A064988, A357977, A357978, A357980, A357983.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000720, A076610, A296150.

a003961 1 = 1
a003961 n = product $ map (a000040 . (+ 1) . a049084) $ a027746_row n
-- Reinhard Zumkeller, Apr 09 2012, Oct 09 2011
(MIT/GNU Scheme, with Aubrey Jaffer's SLIB Scheme library)
(require 'factor)
(define (A003961 n) (apply * (map A000040 (map 1+ (map A049084 (factor n))))))
;; Antti Karttunen, May 20 2014

a:= n-> mul(nextprime(i[1])^i[2], i=ifactors(n)[2]):
seq(a(n), n=1..80);  # Alois P. Heinz, Sep 13 2017

a[p_?PrimeQ] := a[p] = Prime[ PrimePi[p] + 1]; a[1] = 1; a[n_] := a[n] = Times @@ (a[#1]^#2& @@@ FactorInteger[n]); Table[a[n], {n, 1, 65}] (* Jean-François Alcover, Dec 01 2011, updated Sep 20 2019 *)
Table[Times @@ Map[#1^#2 & @@ # &, FactorInteger[n] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[n == 1], {n, 65}] (* Michael De Vlieger, Mar 24 2017 *)

a(n)=local(f); if(n<1,0,f=factor(n); prod(k=1,matsize(f)[1],nextprime(1+f[k,1])^f[k,2]))

a(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Michel Marcus, May 17 2014

use ntheory ":all";  sub a003961 { vecprod(map { next_prime($) } factor(shift)); }  # _Dana Jacobsen, Mar 06 2016

from sympy import factorint, prime, primepi, prod
def a(n):
    f=factorint(n)
    return 1 if n==1 else prod(prime(primepi(i) + 1)**f[i] for i in f)
[a(n) for n in range(1, 11)] # Indranil Ghosh, May 13 2017

If n = Product p(k)^e(k) then a(n) = Product p(k+1)^e(k).
Multiplicative with a(p^e) = A000040(A000720(p)+1)^e. - David W. Wilson, Aug 01 2001
a(n) = Product_{k=1..A001221(n)} A000040(A049084(A027748(n,k))+1)^A124010(n,k). - Reinhard Zumkeller, Oct 09 2011 [Corrected by Peter Munn, Nov 11 2019]
A064989(a(n)) = n and a(A064989(n)) = A000265(n). - Antti Karttunen, May 20 2014 & Nov 01 2019
A001221(a(n)) = A001221(n) and A001222(a(n)) = A001222(n). - Michel Marcus, Jun 13 2014
From Peter Munn, Oct 31 2019: (Start)
a(n) = A225546((A225546(n))^2).
a(A225546(n)) = A225546(n^2).
(End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((p^2-p)/(p^2-nextprime(p))) = 2.06399637... . - Amiram Eldar, Nov 18 2022

A076610 Numbers having only prime factors of form prime(prime); a(1)=1.

Original entry on oeis.org

1, 3, 5, 9, 11, 15, 17, 25, 27, 31, 33, 41, 45, 51, 55, 59, 67, 75, 81, 83, 85, 93, 99, 109, 121, 123, 125, 127, 135, 153, 155, 157, 165, 177, 179, 187, 191, 201, 205, 211, 225, 241, 243, 249, 255, 275, 277, 279, 283, 289, 295, 297, 327, 331, 335, 341, 353, 363

Offset: 1

Reinhard Zumkeller, Oct 21 2002

nonn

Numbers n such that the partition B(n) has only prime parts. For n>=2, B(n) is defined as the partition obtained by taking the prime decomposition of n and replacing each prime factor p by its index i (i.e. i-th prime = p); also B(1) = the empty partition. For example, B(350) = B(2*5^2*7) = [1,3,3,4]. B is a bijection between the positive integers and the set of all partitions. In the Maple program the command B(n) yields B(n). - Emeric Deutsch, May 09 2015
Multiplicative closure of A006450.
Sequence A064988 sorted into ascending order. - Antti Karttunen, Aug 08 2017
From David A. Corneth, Sep 28 2020: (Start)
Product_{p in A006450} p/(p-1) where primepi(p) <= 10^k for k = 3..10 respectively is
2.7609365004752546...
2.8489587563778631...
2.9038201166664191...
2.9413699333962213...
2.9687172228411300...
2.9895324403761206...
3.0059192857697702...
3.0191633206253085... (End)

			99 = 11*3*3 = A000040(A000040(3))*A000040(A000040(1))^2, therefore 99 is a term.

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

Cf. A000040, A006450, A064988.

with(numtheory): B := proc (n) local pf: pf := op(2, ifactors(n)): [seq(seq(pi(op(1, op(i, pf))), j = 1 .. op(2, op(i, pf))), i = 1 .. nops(pf))] end proc: S := {}: for r to 400 do s := 0: for t to nops(B(r)) do if isprime(B(r)[t]) = false then s := s+1 else end if end do: if s = 0 then S := `union`(S, {r}) else end if end do: S; # Emeric Deutsch, May 09 2015

{1}~Join~Select[Range@ 400, AllTrue[PrimePi@ First@ Transpose@ FactorInteger@ #, PrimeQ] &] (* Michael De Vlieger, May 09 2015, Version 10 *)

isok(k) = my(f = factor(k)[,1]); sum(i=1, #f, isprime(primepi(f[i]))) == #f; \\ Michel Marcus, Sep 16 2022

Sum_{n>=1} 1/a(n) = Product_{p in A006450} p/(p-1) converges since the sum of the reciprocals of A006450 converges. - Amiram Eldar, Sep 27 2020

A048767 If n = Product (p_j^k_j) then a(n) = Product ( prime(k_j)^pi(p_j) ) where pi is A000720.

Original entry on oeis.org

1, 2, 4, 3, 8, 8, 16, 5, 9, 16, 32, 12, 64, 32, 32, 7, 128, 18, 256, 24, 64, 64, 512, 20, 27, 128, 25, 48, 1024, 64, 2048, 11, 128, 256, 128, 27, 4096, 512, 256, 40, 8192, 128, 16384, 96, 72, 1024, 32768, 28, 81, 54, 512, 192, 65536, 50, 256, 80, 1024, 2048

Offset: 1

Naohiro Nomoto

If the prime power factors p^e of n are replaced by prime(e)^pi(p), then the prime terms q in the sequence pertain to 2^m with m > 1, since pi(2) = 1. - Michael De Vlieger, Apr 25 2017

Also the Heinz number of the integer partition obtained by applying the map described in A217605 (which interchanges the parts with their multiplicities) to the integer partition with Heinz number n, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The image of this map (which is the union of this sequence) is A130091. - Gus Wiseman, May 04 2019

			For n=6, 6 = (2^1)*(3^1), a(6) = ([first prime]^pi(2))*([first prime]^pi(3)) = (2^1)*(2^2) = 8.
From _Gus Wiseman_, May 04 2019: (Start)
For n = 1..20, the prime indices of n together with the prime indices of a(n) are the following:
   1: {} {}
   2: {1} {1}
   3: {2} {1,1}
   4: {1,1} {2}
   5: {3} {1,1,1}
   6: {1,2} {1,1,1}
   7: {4} {1,1,1,1}
   8: {1,1,1} {3}
   9: {2,2} {2,2}
  10: {1,3} {1,1,1,1}
  11: {5} {1,1,1,1,1}
  12: {1,1,2} {1,1,2}
  13: {6} {1,1,1,1,1,1}
  14: {1,4} {1,1,1,1,1}
  15: {2,3} {1,1,1,1,1}
  16: {1,1,1,1} {4}
  17: {7} {1,1,1,1,1,1,1}
  18: {1,2,2} {1,2,2}
  19: {8} {1,1,1,1,1,1,1,1}
  20: {1,1,3} {1,1,1,2}
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Stephan Wagner, The Number of Fixed Points of Wilf's Partition Involution, The Electronic Journal of Combinatorics, 20(4) (2013), #P13.

Cf. A008477, A048768, A048769, A064988.

Cf. A056239, A098859, A109298, A112798, A118914, A130091, A217605, A320348, A325326, A325337.

A048767 := proc(n)
    local a,p,e,f;
    a := 1 ;
    for f in ifactors(n)[2] do
        p := op(1,f) ;
        e := op(2,f) ;
        a := a*ithprime(e)^numtheory[pi](p) ;
    end do:
    a ;
end proc: # R. J. Mathar, Nov 08 2012

Table[{p, k} = Transpose@ FactorInteger[n]; Times @@ (Prime[k]^PrimePi[p]), {n, 58}] (* Ivan Neretin, Jun 02 2016 *)
Array[Apply[Times, FactorInteger[#] /. {p_, e_} /; e >= 0 :> Prime[e]^PrimePi[p]] &, 65] (* Michael De Vlieger, Apr 25 2017 *)

a(1)=1 prepended by Alois P. Heinz, Jul 26 2015

A357982 Replace prime(k) with A000009(k) in the prime factorization of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 4, 2, 2, 1, 5, 1, 6, 2, 2, 3, 8, 1, 4, 4, 1, 2, 10, 2, 12, 1, 3, 5, 4, 1, 15, 6, 4, 2, 18, 2, 22, 3, 2, 8, 27, 1, 4, 4, 5, 4, 32, 1, 6, 2, 6, 10, 38, 2, 46, 12, 2, 1, 8, 3, 54, 5, 8, 4, 64, 1, 76, 15, 4, 6, 6, 4, 89, 2, 1

Offset: 1

Gus Wiseman, Oct 25 2022

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. This sequence gives the number of ways to choose a strict partition of each prime index of n.

The indices i, where a(i) = 1, form A003586, and the indices j, where a(j) > 1, form A059485. - Ivan N. Ianakiev, Oct 27 2022

			The a(121) = 9 twice-partitions are: (5)(5), (5)(41), (5)(32), (41)(5), (41)(41), (41)(32), (32)(5), (32)(41), (32)(32).

Other multiplicative sequences: A003961, A357852, A064988, A064989, A357980.
The non-strict version is A299200.
A horizontal version is A357978, non-strict A357977.
A000040 lists the primes.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000041, A000720, A003964, A063834, A076610, A215366, A273873, A296150, A299201-A299203, A357975, A357979, A357983.
Cf. A003586, A059485.

Table[Times@@Cases[FactorInteger[n],{p_,k_}:>PartitionsQ[PrimePi[p]]^k],{n,100}]

f9(n) = polcoeff( prod( k=1, n, 1 + x^k, 1 + x * O(x^n)), n); \\ A000009
a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = f9(primepi(f[k,1]))); factorback(f); \\ Michel Marcus, Oct 26 2022

A317145 Number of maximal chains of factorizations of n into factors > 1, ordered by refinement.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 3, 1, 4, 1, 1, 1, 7, 1, 1, 1, 5, 1, 3, 1, 2, 2, 1, 1, 15, 1, 2, 1, 2, 1, 5, 1, 5, 1, 1, 1, 11, 1, 1, 2, 11, 1, 3, 1, 2, 1, 3, 1, 26, 1, 1, 2, 2, 1, 3, 1, 15, 2, 1, 1, 11, 1, 1, 1, 5, 1, 11, 1, 2, 1, 1, 1, 52, 1, 2, 2, 7, 1, 3, 1, 5, 3

Offset: 1

Gus Wiseman, Jul 22 2018

nonn

If x and y are factorizations of the same integer and it is possible to produce x by further factoring the factors of y, flattening, and sorting, then x <= y.

a(n) depends only on prime signature of n (cf. A025487). - Antti Karttunen, Oct 08 2018

			The a(36) = 7 maximal chains:
  (2*2*3*3) < (2*2*9) < (2*18) < (36)
  (2*2*3*3) < (2*2*9) < (4*9)  < (36)
  (2*2*3*3) < (2*3*6) < (2*18) < (36)
  (2*2*3*3) < (2*3*6) < (3*12) < (36)
  (2*2*3*3) < (2*3*6) < (6*6)  < (36)
  (2*2*3*3) < (3*3*4) < (3*12) < (36)
  (2*2*3*3) < (3*3*4) < (4*9)  < (36)

Antti Karttunen, Table of n, a(n) for n = 1..11520
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Cf. A001055, A002846, A007716, A045778, A064988, A162247, A213427, A275024, A281113, A299202, A300385, A317144, A317146, A320105.

A064988(n) = { my(f = factor(n)); for (k=1, #f~, f[k, 1] = prime(f[k, 1]); ); factorback(f); }; \\ From A064988
memoA320105 = Map();
A320105(n) = if(bigomega(n)<=2,1,if(mapisdefined(memoA320105,n), mapget(memoA320105,n), my(f=factor(n), u = #f~, s = 0); for(i=1,u,for(j=i+(1==f[i,2]),u, s += A320105(prime(primepi(f[i,1])*primepi(f[j,1]))*(n/(f[i,1]*f[j,1]))))); mapput(memoA320105,n,s); (s)));
A317145(n) = A320105(A064988(n)); \\ Antti Karttunen, Oct 08 2018

a(prime^n) = A002846(n).

a(n) = A320105(A064988(n)). - Antti Karttunen, Oct 08 2018

Data section extended to 105 terms by Antti Karttunen, Oct 08 2018

A318871 Minimum Heinz number of a factorization of n into factors > 1.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 35, 41, 43, 47, 49, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 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

Offset: 1

Gus Wiseman, Sep 05 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			a(1) = 1 = the empty product.
a(12) = 35 = 5 * 7 = prime(3) * prime(4).
a(16) = 49 = 7^2 = prime(4)^2.
a(23) = 83 = prime(23).

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000040, A001055, A007716, A056239, A064988, A162247, A215366, A246868.

a:= proc(n) option remember; `if`(n=1, 1, min(seq(a(d)*
      ithprime(n/d), d=numtheory[divisors](n) minus {n})))
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Sep 05 2018

facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
Table[Min[Times@@Prime/@#&/@facs[n]],{n,100}]

A357977 Replace prime(k) with prime(A000041(k)) in the prime factorization of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 11, 8, 9, 10, 17, 12, 31, 22, 15, 16, 47, 18, 79, 20, 33, 34, 113, 24, 25, 62, 27, 44, 181, 30, 263, 32, 51, 94, 55, 36, 389, 158, 93, 40, 547, 66, 761, 68, 45, 226, 1049, 48, 121, 50, 141, 124, 1453, 54, 85, 88, 237, 362, 1951, 60, 2659, 526

Offset: 1

Gus Wiseman, Oct 23 2022

In the definition, taking A000041(k) instead of prime(A000041(k)) gives A299200.

			We have 35 = prime(3) * prime(4), so a(35) = prime(A000041(3)) * prime(A000041(4)) = prime(3) * prime(5) = 55.

Applying the same transformation again gives A357979.
The strict version is A357978.
Other multiplicative sequences: A003961, A357852, A064988, A064989, A357980.
A000040 lists the primes.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000041, A000720, A003964, A063834, A076610, A215366, A296150, A299201, A299202, A357975, A357983.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mtf[f_][n_]:=Product[If[f[i]==0,1,Prime[f[i]]],{i,primeMS[n]}];
Array[mtf[PartitionsP],100]

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = prime(numbpart(primepi(f[k,1])))); factorback(f); \\ Michel Marcus, Oct 25 2022

cp's OEIS Frontend

A357983 Second MTF-transform of the primes (A000040). Replace prime(k) with prime(A064988(k)) in the prime factorization of n.

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A318668 a(n) = gcd(n, A064988(n)).

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A318660 Remainder when A064988(n) is divided by n.

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A003961 Completely multiplicative with a(prime(k)) = prime(k+1).

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A076610 Numbers having only prime factors of form prime(prime); a(1)=1.

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A048767 If n = Product (p_j^k_j) then a(n) = Product ( prime(k_j)^pi(p_j) ) where pi is A000720.

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A357982 Replace prime(k) with A000009(k) in the prime factorization of n.

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