A064988 -id:A064988 - cp's OEIS frontend

A304203 If n = Product (p_j^k_j) then a(n) = Product (p_j^prime(k_j)).

1, 4, 9, 8, 25, 36, 49, 32, 27, 100, 121, 72, 169, 196, 225, 128, 289, 108, 361, 200, 441, 484, 529, 288, 125, 676, 243, 392, 841, 900, 961, 2048, 1089, 1156, 1225, 216, 1369, 1444, 1521, 800, 1681, 1764, 1849, 968, 675, 2116, 2209, 1152, 343, 500, 2601, 1352, 2809, 972, 3025

Offset: 1

Ilya Gutkovskiy, May 09 2018

			a(12) = a(2^2*3^1) = 2^prime(2)*3^prime(1) = 2^3*3^2 = 72.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Ilya Gutkovskiy, Scatter plot of a(n) up to n=50000.
Index entries for sequences computed from exponents in factorization of n.

Cf. A000040, A001248, A002110, A003963, A008477, A030078, A048767, A050997, A056166 (records), A061742, A181819.

Cf. A064988 (apply prime to p), A321874 (apply prime to both p & e).

a:= n-> mul(i[1]^ithprime(i[2]), i=ifactors(n)[2]):
seq(a(n), n=1..55);  # Alois P. Heinz, Jan 20 2021

a[n_] := Times @@ (#[[1]]^Prime[#[[2]]] & /@ FactorInteger[n]); Table[a[n], {n, 55}]

a(n) = my(f=factor(n)); prod(k=1, #f~, f[k,1]^prime(f[k,2])); \\ Michel Marcus, May 09 2018

apply( A304203(n)=factorback((n=factor(n))[,1],apply(prime,n[,2])), [1..50]) \\ M. F. Hasler, Nov 20 2018

a(prime(i)^k) = prime(i)^prime(k).
a(A000040(k)) = A001248(k).
a(A001248(k)) = A030078(k).
a(A030078(k)) = A050997(k).
a(A002110(k)) = A061742(k).
Multiplicative with a(p^e) = p^prime(e). - M. F. Hasler, Nov 20 2018
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + Sum_{k>=1} 1/p^prime(k)) = 1.80728269690724154161... . - Amiram Eldar, Jan 20 2024

A320105 If A001222(n) <= 2, a(n) = 1, otherwise, a(n) = Sum_{p|n} Sum_{q|n, q>=(p+[p^2 does not divide n])} a(prime(primepi(p)primepi(q)) (n/(p*q))), where p and q range over distinct primes dividing n. (See formula section for exact details.)

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 6, 1, 1, 1, 4, 1, 3, 1, 2, 2, 1, 1, 8, 1, 2, 1, 2, 1, 4, 1, 4, 1, 1, 1, 11, 1, 1, 2, 1, 1, 3, 1, 2, 1, 3, 1, 16, 1, 1, 2, 2, 1, 3, 1, 8, 2, 1, 1, 11, 1, 1, 1, 4, 1, 10, 1, 2, 1, 1, 1, 16, 1, 2, 2, 6, 1, 3, 1, 4, 3

Offset: 1

Antti Karttunen, Oct 08 2018

nonn

This is an auxiliary function for computing A317145 with help of A064988. Note the similarity of the formula to that of A300385, with only difference being in the value of a(1) and that here we have multiplication (*) instead of addition (+) between primepi(p) and primepi(q).
From Gus Wiseman, Oct 09 2018: (Start)
Combinatorial interpretation is: In the poset of multiset partitions ordered by refinement, number of maximal chains from the n-th multiset multisystem (A302242) to the maximal multiset partition of the same multiset, assuming n is odd. For example, the a(315) = 10 maximal chains are
  {{1},{1},{2},{1,1}} < {{1},{1},{1,1,2}} < {{1},{1,1,1,2}} < {{1,1,1,1,2}}
  {{1},{1},{2},{1,1}} < {{1},{1},{1,1,2}} < {{1,1},{1,1,2}} < {{1,1,1,1,2}}
  {{1},{1},{2},{1,1}} < {{1},{2},{1,1,1}} < {{1},{1,1,1,2}} < {{1,1,1,1,2}}
  {{1},{1},{2},{1,1}} < {{1},{2},{1,1,1}} < {{2},{1,1,1,1}} < {{1,1,1,1,2}}
  {{1},{1},{2},{1,1}} < {{1},{2},{1,1,1}} < {{1,2},{1,1,1}} < {{1,1,1,1,2}}
  {{1},{1},{2},{1,1}} < {{1},{1,1},{1,2}} < {{1},{1,1,1,2}} < {{1,1,1,1,2}}
  {{1},{1},{2},{1,1}} < {{1},{1,1},{1,2}} < {{1,1},{1,1,2}} < {{1,1,1,1,2}}
  {{1},{1},{2},{1,1}} < {{1},{1,1},{1,2}} < {{1,2},{1,1,1}} < {{1,1,1,1,2}}
  {{1},{1},{2},{1,1}} < {{2},{1,1},{1,1}} < {{2},{1,1,1,1}} < {{1,1,1,1,2}}
  {{1},{1},{2},{1,1}} < {{2},{1,1},{1,1}} < {{1,1},{1,1,2}} < {{1,1,1,1,2}}.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..12960
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A064988, A300385, A302242, A317145.

A320105(n) = if(bigomega(n)<=2,1,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]))))); (s));

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))); \\ Memoized version.

If A001222(n) <= 2 [when n is one, a prime or semiprime], a(n) = 1, otherwise, a(n) = Sum_{p|n} Sum_{q|n, q>=(p+[p^2 does not divide n])} a(prime(primepi(p)*primepi(q)) * (n/(p*q))), where p ranges over all distinct primes dividing n, and q also ranges over primes dividing n, but with condition that q > p if p is a unitary prime factor of n, otherwise q >= p. Here primepi = A000720.

a(A064988(n)) = A317145(n).

A328878 If n = Product (p_j^k_j) then a(n) = Product (prime(p_j)).

Original entry on oeis.org

1, 3, 5, 3, 11, 15, 17, 3, 5, 33, 31, 15, 41, 51, 55, 3, 59, 15, 67, 33, 85, 93, 83, 15, 11, 123, 5, 51, 109, 165, 127, 3, 155, 177, 187, 15, 157, 201, 205, 33, 179, 255, 191, 93, 55, 249, 211, 15, 17, 33, 295, 123, 241, 15, 341, 51, 335, 327, 277, 165, 283, 381, 85, 3, 451

Offset: 1

Ilya Gutkovskiy, Oct 29 2019

			a(54) = 15 because 54 = 2 * 3^3 = prime(1) * prime(2)^3 and prime(prime(1)) * prime(prime(2)) = 3 * 5 = 15.

Cf. A006450, A007947, A064988, A156061, A181819, A321874.

a:= n-> mul(ithprime(i[1]), i=ifactors(n)[2]):
seq(a(n), n=1..65);  # Alois P. Heinz, Nov 26 2024

a[n_] := Times @@ (Prime[#[[1]]] & /@ FactorInteger[n]); Table[a[n], {n, 1, 65}]

a(n)={my(f=factor(n)[,1]); prod(i=1, #f, prime(f[i]))} \\ Andrew Howroyd, Oct 29 2019

A301315 Multiplicative with a(p^e) = prime(p) ^ a(e) (where prime(k) denotes the k-th prime number).

Original entry on oeis.org

1, 3, 5, 27, 11, 15, 17, 243, 125, 33, 31, 135, 41, 51, 55, 7625597484987, 59, 375, 67, 297, 85, 93, 83, 1215, 1331, 123, 3125, 459, 109, 165, 127, 177147, 155, 177, 187, 3375, 157, 201, 205, 2673, 179, 255, 191, 837, 1375, 249, 211, 38127987424935, 4913, 3993

Offset: 1

Rémy Sigrist, Mar 18 2018

This sequence is a recursive version of A064988.

This sequence is injective (all terms are distinct).

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

Cf. A064988, A225395, A279690.

Fold[Function[{a, n}, Append[a, Times @@ Map[Prime[#1]^a[[#2]] & @@ # &, FactorInteger@ n]]], {1}, Range[2, 50]] (* Michael De Vlieger, Mar 19 2018 *)

a(n) = my (f=factor(n)); prod (i=1, #f~, prime(f[i,1])^a(f[i,2]))

A225395(a(n)) = n.

A279690(a(n)) = A279690(n).

A357853 Fully multiplicative with a(prime(k)) = A000009(k+1).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 4, 2, 4, 2, 5, 3, 4, 1, 6, 4, 8, 2, 6, 4, 10, 2, 4, 5, 8, 3, 12, 4, 15, 1, 8, 6, 6, 4, 18, 8, 10, 2, 22, 6, 27, 4, 8, 10, 32, 2, 9, 4, 12, 5, 38, 8, 8, 3, 16, 12, 46, 4, 54, 15, 12, 1, 10, 8, 64, 6, 20, 6, 76, 4, 89, 18, 8, 8, 12, 10

Offset: 1

Gus Wiseman, Oct 28 2022

			We have 525 = prime(2)*prime(3)*prime(3)*prime(4) so a(525) = Q(3)*Q(4)*Q(4)*Q(5) = 2*2*2*3 = 24, where Q = A000009.

Other multiplicative sequences: A003961, A064988, A064989, A357852, A357980.
The non-strict version is A003964.
The unshifted horizontal version is A357978, non-strict A357977.
The unshifted version is A357982.
A000040 lists the primes.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000009, A000720, A076610, A273873, A296150, A357979.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ptf[f_][n_]:=Product[f[i],{i,primeMS[n]}];
Array[ptf[PartitionsQ[#+1]&],100]

A357984 Replace prime(k) with A000720(k) in the prime factorization of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Oct 25 2022

			We have 91 = prime(4) * prime(6), so a(91) = pi(4) * pi(6) = 6.

Other multiplicative sequences: A003961, A357852, A064988, A064989, A357983.
The version for p instead of pi is A299200, horz A357977, strict A357982.
The version for nu is A355741.
The version for bigomega is A355742.
The horizontal version is A357980.
A000040 lists the prime numbers.
A000720 is PrimePi.
A056239 adds up prime indices, row-sums of A112798.
Cf. A033879, A033880, A063834, A066207, A076610, A296150, A357975.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@PrimePi/@primeMS[n],{n,100}]

A335403 If n = Product_{i=1..k} p_i^e_i then a(n) = Sum_{i=1..k} e_i * prime(p_i).

Original entry on oeis.org

0, 3, 5, 6, 11, 8, 17, 9, 10, 14, 31, 11, 41, 20, 16, 12, 59, 13, 67, 17, 22, 34, 83, 14, 22, 44, 15, 23, 109, 19, 127, 15, 36, 62, 28, 16, 157, 70, 46, 20, 179, 25, 191, 37, 21, 86, 211, 17, 34, 25, 64, 47, 241, 18, 42, 26, 72, 112, 277, 22, 283, 130, 27, 18

Offset: 1

Gus Wiseman, Jun 06 2020

Totally additive with a(p) = prime(p) for p prime.

			The prime factors of 18 are 2 * 3 * 3, so a(18) = prime(2) + prime(3) + prime(3) = 13.

Wikipedia, Additive function

Partitions into prime parts are A000607.
Sum of prime factors is A001414.
Primes of prime index are A006450.
Sum of prime indices is A056239.
The multiplicative version is A064988.
Products of primes of prime index are A076610.
Numbers whose prime indices are not all prime are A330945.
Cf. A000040, A000720, A001222, A003961, A003963, A007097, A112798, A178503, A257994, A302242, A324851, A331915.

Table[Total[Cases[FactorInteger[n],{p_,k_}:>k*Prime[p]]],{n,30}]

a(n) = my(f=factor(n)); sum(k=1, #f~, prime(f[k,1])*f[k,2]); \\ Michel Marcus, Jun 07 2020

Edited by N. J. A. Sloane, Jun 20 2020 following a suggestion from Bernard Schott.

A357981 Numbers whose prime indices have only prime numbers as their own prime indices.

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 11, 16, 20, 22, 23, 25, 31, 32, 40, 44, 46, 47, 50, 55, 59, 62, 64, 80, 88, 92, 94, 97, 100, 103, 110, 115, 118, 121, 124, 125, 127, 128, 137, 155, 160, 176, 179, 184, 188, 194, 197, 200, 206, 220, 230, 233, 235, 236, 242, 248, 250, 253, 254

Offset: 1

Gus Wiseman, Oct 23 2022

nonn

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.

Numbers whose prime indices are prime numbers are listed by A076610.

			The terms together with their prime indices begin:
     1: {}
     2: {1}
     4: {1,1}
     5: {3}
     8: {1,1,1}
    10: {1,3}
    11: {5}
    16: {1,1,1,1}
    20: {1,1,3}
    22: {1,5}
    23: {9}
    25: {3,3}
    31: {11}
    32: {1,1,1,1,1}

Contains all elements of A000079.
Contains all primes indexed by elements of A076610.
A000040 lists the prime numbers.
A056239 adds up prime indices, row-sums of A112798.
Cf. A003961, A045966, A064988, A066207, A215366, A357977, A357980, A357983.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],And@@PrimeQ/@Join@@primeMS/@primeMS[#]&]

cp's OEIS Frontend

A304203 If n = Product (p_j^k_j) then a(n) = Product (p_j^prime(k_j)).

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A320105 If A001222(n) <= 2, a(n) = 1, otherwise, a(n) = Sum_{p|n} Sum_{q|n, q>=(p+[p^2 does not divide n])} a(prime(primepi(p)*primepi(q)) * (n/(p*q))), where p and q range over distinct primes dividing n. (See formula section for exact details.)

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A328878 If n = Product (p_j^k_j) then a(n) = Product (prime(p_j)).

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A301315 Multiplicative with a(p^e) = prime(p) ^ a(e) (where prime(k) denotes the k-th prime number).

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A357853 Fully multiplicative with a(prime(k)) = A000009(k+1).

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A357984 Replace prime(k) with A000720(k) in the prime factorization of n.

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A335403 If n = Product_{i=1..k} p_i^e_i then a(n) = Sum_{i=1..k} e_i * prime(p_i).

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A357981 Numbers whose prime indices have only prime numbers as their own prime indices.

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A320105 If A001222(n) <= 2, a(n) = 1, otherwise, a(n) = Sum_{p|n} Sum_{q|n, q>=(p+[p^2 does not divide n])} a(prime(primepi(p)primepi(q)) (n/(p*q))), where p and q range over distinct primes dividing n. (See formula section for exact details.)