A064988 -id:A064988 - cp's OEIS frontend

A357980 Replace prime(k) with prime(A000720(k)) in the prime factorization of n, assuming prime(0) = 1.

1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 6, 1, 7, 4, 7, 3, 6, 5, 7, 2, 9, 5, 8, 3, 7, 6, 11, 1, 10, 7, 9, 4, 11, 7, 10, 3, 13, 6, 13, 5, 12, 7, 13, 2, 9, 9, 14, 5, 13, 8, 15, 3, 14, 7, 17, 6, 17, 11, 12, 1, 15, 10, 19, 7, 14, 9, 19, 4, 19, 11, 18, 7, 15, 10

Offset: 1

Gus Wiseman, Oct 24 2022

In the definition, taking A000720(k) in place of prime(A000720(k)) gives A357984.

			We have 90 = prime(1) * prime(2)^2 * prime(3), so a(90) = prime(0) * prime(1)^2 * prime(2) = 12.

Other multiplicative sequences: A003961, A357852, A064988, A064989, A357980.
The version for p instead of pi is A357977, strict A357978.
The triangular version is A357984.
A000040 lists the prime numbers.
A000720 = PrimePi.
A056239 adds up prime indices, row-sums of A112798.
Cf. A063834, A066207, A076610, A215366, A296150, 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[PrimePi],100]

myprime(n) = if (n==0, 1, prime(n));
a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = myprime(primepi(primepi(f[k,1])))); factorback(f); \\ Michel Marcus, Oct 25 2022

A357852 Replace prime(k) with prime(k+2) in the prime factorization of n.

Original entry on oeis.org

1, 5, 7, 25, 11, 35, 13, 125, 49, 55, 17, 175, 19, 65, 77, 625, 23, 245, 29, 275, 91, 85, 31, 875, 121, 95, 343, 325, 37, 385, 41, 3125, 119, 115, 143, 1225, 43, 145, 133, 1375, 47, 455, 53, 425, 539, 155, 59, 4375, 169, 605, 161, 475, 61, 1715, 187, 1625, 203

Offset: 1

Gus Wiseman, Oct 28 2022

This is the same as A045966 except the first term is 1 instead of 3.

			The terms together with their prime indices begin:
    1: {}
    5: {3}
    7: {4}
   25: {3,3}
   11: {5}
   35: {3,4}
   13: {6}
  125: {3,3,3}
   49: {4,4}
   55: {3,5}
   17: {7}
  175: {3,3,4}
   19: {8}
   65: {3,6}
   77: {4,5}
  625: {3,3,3,3}

Applying the transformation only once gives A003961.
A permutation of A007310.
Other multiplicative sequences: A064988, A064989, A357977, A357980, A357983.
A000040 lists the primes.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000720, A003964, A066207, A076610, A215366, A296150, A299201, A357979.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Product[Prime[i+2],{i,primeMS[n]}],{n,30}]

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = nextprime(nextprime(f[k,1]+1)+1)); factorback(f); \\ Michel Marcus, Oct 28 2022

from math import prod
from sympy import nextprime, factorint
def A357852(n): return prod(nextprime(p,ith=2)**e for p, e in factorint(n).items()) # Chai Wah Wu, Oct 29 2022

a(n) = A003961(A003961(n)).

A357975 Divide all prime indices by 2, round down, and take the number with those prime indices, assuming prime(0) = 1.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 4, 2, 3, 2, 5, 3, 4, 1, 5, 4, 7, 2, 6, 3, 7, 2, 4, 5, 8, 3, 11, 4, 11, 1, 6, 5, 6, 4, 13, 7, 10, 2, 13, 6, 17, 3, 8, 7, 17, 2, 9, 4, 10, 5, 19, 8, 6, 3, 14, 11, 19, 4, 23, 11, 12, 1, 10, 6, 23, 5, 14, 6, 29, 4, 29, 13, 8, 7, 9, 10, 31

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.
Also the Heinz number of the part-wise half (rounded down) of the partition with Heinz number n, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Each n appears A000005(n) times at odd positions (infinitely many at even). To see this, note that our transformation does not distinguish between A066207 and A066208.

			The prime indices of n = 1501500 are {1,1,2,3,3,3,4,5,6}, so the prime indices of a(n) are {1,1,1,1,2,2,3}; hence we have a(1501500) = 720.
The 6 odd positions of 2124 are: 63, 99, 105, 165, 175, 275, with prime indices:
   63: {2,2,4}
   99: {2,2,5}
  105: {2,3,4}
  165: {2,3,5}
  175: {3,3,4}
  275: {3,3,5}

SiYang Hu, Table of n, a(n) for n = 1..10000

Positions of 1's are A000079.
Positions of 2's are 3 and A164095.
Positions of first appearances are A297002, sorted A066207.
A004526 is floor(n/2), with an extra first zero.
A056239 adds up prime indices, row-sums of A112798.
A109763 lists primes of index floor(n/2).
Cf. A003961, A064988, A064989, A076610, A215366, A248601, A357980.

Table[Times@@(If[#1<=2,1,Prime[Floor[PrimePi[#1]/2]]^#2]&@@@FactorInteger[n]),{n,100}]

Completely multiplicative with a(prime(2k)) = prime(k) and a(prime(2k+1)) = prime(k). Cf. A297002.

a(prime(n)) = A109763(n-1).

A357978 Replace prime(k) with prime(A000009(k)) in the prime factorization of n.

Original entry on oeis.org

1, 2, 2, 4, 3, 4, 3, 8, 4, 6, 5, 8, 7, 6, 6, 16, 11, 8, 13, 12, 6, 10, 19, 16, 9, 14, 8, 12, 29, 12, 37, 32, 10, 22, 9, 16, 47, 26, 14, 24, 61, 12, 79, 20, 12, 38, 103, 32, 9, 18, 22, 28, 131, 16, 15, 24, 26, 58, 163, 24, 199, 74, 12, 64, 21, 20, 251, 44, 38

Offset: 1

Gus Wiseman, Oct 24 2022

In the definition, taking A000009(k) instead of prime(A000009(k)) gives A357982.

			We have 90 = prime(1) * prime(2)^2 * prime(3), so a(90) = prime(1) * prime(1)^2 * prime(2) = 24.

The non-strict version is A357977.
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, A357979, 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[PartitionsQ],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] = prime(f9(primepi(f[k,1])))); factorback(f); \\ Michel Marcus, Oct 25 2022

A357979 Second MTF-transform of A000041. Replace prime(k) with prime(A357977(k)) in the prime factorization of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 31, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 59, 32, 33, 62, 35, 36, 37, 38, 39, 40, 127, 42, 79, 44, 45, 46, 47, 48, 49, 50, 93, 52, 53, 54, 55, 56, 57, 58, 211, 60, 61, 118, 63, 64, 65, 66

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 applying a function horizontally along a number's prime indices; see the Mathematica program.

			We have:
- 51 = prime(2) * prime(7),
- A357977(2) = 2,
- A357977(7) = 11,
- a(51) = prime(2) * prime(11) = 93.

Other multiplicative sequences: A003961, A357852, A064988, A064989, A357980.
Applying the transformation only once gives A357977, strict A357978.
For primes instead of partition numbers we have A357983.
A000040 lists the primes.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000041, A000720, A003964, A063834, A076610, 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[PartitionsP]],100]

A321874 If n = Product (p_j^k_j) then a(n) = Product (prime(p_j)^prime(k_j)).

Original entry on oeis.org

1, 9, 25, 27, 121, 225, 289, 243, 125, 1089, 961, 675, 1681, 2601, 3025, 2187, 3481, 1125, 4489, 3267, 7225, 8649, 6889, 6075, 1331, 15129, 3125, 7803, 11881, 27225, 16129, 177147, 24025, 31329, 34969, 3375, 24649, 40401, 42025, 29403, 32041, 65025, 36481, 25947, 15125

Offset: 1

Ilya Gutkovskiy, Nov 20 2018

			a(12) = a(2^2 * 3^1) = prime(2)^prime(2) * prime(3)^prime(1) = 3^3 * 5^2 = 675.

Index entries for sequences computed from indices in prime factorization.

Cf. A000040, A064988, A304203.

a[n_] := Times @@ (Prime[#[[1]]]^Prime[#[[2]]] & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 45}]

a(n) = my(f=factor(n)); prod(k=1, #f~, prime(f[k,1])^prime(f[k,2])); \\ Michel Marcus, Nov 20 2018

apply( A321874(n)=factorback(apply(prime,factor(n))), [1..49]) \\ M. F. Hasler, Nov 20 2018

Multiplicative with a(p^e) = prime(p)^prime(e). - M. F. Hasler, Nov 20 2018

Sum_{n>=1} 1/a(n) = Product_{m>=1} (1 + Sum_{k>=1} 1/prime(m)^prime(k)) = 1.22718741... . - Amiram Eldar, Jan 20 2024

A340323 Multiplicative with a(p^e) = (p + 1) * (p - 1)^(e - 1).

Original entry on oeis.org

1, 3, 4, 3, 6, 12, 8, 3, 8, 18, 12, 12, 14, 24, 24, 3, 18, 24, 20, 18, 32, 36, 24, 12, 24, 42, 16, 24, 30, 72, 32, 3, 48, 54, 48, 24, 38, 60, 56, 18, 42, 96, 44, 36, 48, 72, 48, 12, 48, 72, 72, 42, 54, 48, 72, 24, 80, 90, 60, 72, 62, 96, 64, 3, 84, 144, 68, 54

Offset: 1

José María Grau Ribas, Jan 04 2021

Starting with any integer and repeatedly applying the map x -> a(x) reaches the fixed point 12 or the loop {3, 4}.

			a(2^s) = 3 for all s>0.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A003958, A003963, A048250, A068468, A064989, A167344, A327938, A293442, A064988, A108548, A124859, A279513, A324106, A322360, A326297, A340324, A340325, A340368.

f:= proc(n) local  t;
  mul((t[1]+1)*(t[1]-1)^(t[2]-1),t=ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, Jan 07 2021

fa[n_]:=fa[n]=FactorInteger[n];
phi[1]=1; phi[p_, s_]:= (p + 1)*( p - 1)^(s - 1)
phi[n_]:=Product[phi[fa[n][[i, 1]], fa[n][[i, 2]]], {i,Length[fa[n]]}];
Array[phi, 245]

A340323(n) = if(1==n,n,my(f=factor(n)); prod(i=1,#f~,(f[i,1]+1)*((f[i,1]-1)^(f[i,2]-1)))); \\ Antti Karttunen, Jan 06 2021

a(n) = A167344(n) / A340368(n) = A048250(n) * A326297(n). - Antti Karttunen, Jan 06 2021

Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(6)/(2*zeta(2)*zeta(3))) * Product_{p prime} (1 + 2/p^2) = 0.56361239505... . - Amiram Eldar, Nov 12 2022

A378175 Triangle T(n,k) read by rows in which n-th row lists in increasing order all multiplicative partitions mu of n (with factors > 1) encoded as Product_{j in mu} prime(j); n>=1, 1<=k<=A001055(n).

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 27, 23, 25, 29, 33, 31, 35, 37, 39, 45, 41, 43, 51, 47, 55, 49, 53, 57, 63, 81, 59, 61, 65, 69, 75, 67, 71, 77, 87, 99, 73, 85, 79, 93, 83, 89, 91, 95, 105, 111, 117, 135, 97, 121, 101, 123, 103, 115, 125, 107, 119, 129, 153

Offset: 1

Alois P. Heinz, Nov 18 2024

			The multiplicative partitions of n=8 are {[8], [4,2], [2,2,2]}, encodings give {prime(8), prime(4)*prime(2), prime(2)^3} = {19, 7*3, 3^3} => row 8 = [19, 21, 27].
For n=1 the empty partition [] gives the empty product 1.
Triangle T(n,k) begins:
   1 ;
   3 ;
   5 ;
   7,  9 ;
  11 ;
  13, 15 ;
  17 ;
  19, 21, 27 ;
  23, 25 ;
  29, 33 ;
  31 ;
  35, 37, 39, 45 ;
  41 ;
  43, 51 ;
  47, 55 ;
  49, 53, 57, 63, 81 ;
  59 ;
  ...

Alois P. Heinz, Rows n = 1..2047, flattened

Row sums give A378176.
Row lengths give A001055.
Column k=1 gives A318871.
Rightmost elements of rows give A064988.
Sorted terms give A005408.
Cf. A000040, A006450, A215366, A377852.

b:= proc(n) option remember; `if`(n=1, {1}, {seq(map(x-> x*
      ithprime(d), b(n/d))[], d=numtheory[divisors](n) minus {1})})
    end:
T:= n-> sort([b(n)[]])[]:
seq(T(n), n=1..28);

T(prime(n),1) = T(A000040(n),1) = A006450(n).

A290641 Multiplicative with a(p^e) = prime(p-1)^e.

Original entry on oeis.org

1, 2, 3, 4, 7, 6, 13, 8, 9, 14, 29, 12, 37, 26, 21, 16, 53, 18, 61, 28, 39, 58, 79, 24, 49, 74, 27, 52, 107, 42, 113, 32, 87, 106, 91, 36, 151, 122, 111, 56, 173, 78, 181, 116, 63, 158, 199, 48, 169, 98, 159, 148, 239, 54, 203, 104, 183, 214, 271, 84, 281, 226, 117, 64, 259

Offset: 1

Michel Marcus, Aug 08 2017

a(n) = A064554(n) for 1 <= n < 91, but a(91) = 481 differs from A064554(91) = 463. - Georg Fischer, Oct 23 2018

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from indices in prime factorization

Cf. A046523, A055003, A064554, A064988, A064989.

Array[If[# == 1, 1, Times @@ Map[Prime[#1 - 1]^#2 & @@ # &, FactorInteger[#]]] &, 65] (* Michael De Vlieger, Apr 22 2021 *)

a(n) = {my(f = factor(n)); for (k=1, #f~, f[k, 1] = prime(f[k, 1]-1);); factorback(f);}

from sympy import factorint, prime
from operator import mul
from functools import reduce
def a(n):
    return 1 if n==1 else reduce(mul, [prime(p - 1)**e for p, e in factorint(n).items()])
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Aug 08 2017

(define (A290641 n) (if (= 1 n) n (* (A000040 (+ -1 (A020639 n))) (A290641 (A032742 n))))) ;; Antti Karttunen, Aug 08 2017

From Antti Karttunen, Aug 08 2017: (Start)
a(n) = A064989(A064988(n)).
A046523(a(n)) = A046523(n). [Preserves the prime signature of n].
(End)

A304251 If n = Product (p_j^k_j) then a(n) = Sum (prime(p_j)^k_j).

Original entry on oeis.org

0, 3, 5, 9, 11, 8, 17, 27, 25, 14, 31, 14, 41, 20, 16, 81, 59, 28, 67, 20, 22, 34, 83, 32, 121, 44, 125, 26, 109, 19, 127, 243, 36, 62, 28, 34, 157, 70, 46, 38, 179, 25, 191, 40, 36, 86, 211, 86, 289, 124, 64, 50, 241, 128, 42, 44, 72, 112, 277, 25, 283, 130, 42, 729, 52, 39, 331, 68, 88, 31

Offset: 1

Ilya Gutkovskiy, May 09 2018

nonn

			a(12) = 14 because 12 = 2^2*3 and prime(2)^2 + prime(3) = 3^2 + 5 = 14.

Robert Israel, Table of n, a(n) for n = 1..10000
Ilya Gutkovskiy, Scatter plot of a(n) up to n=100000
Index entries for sequences computed from indices in prime factorization

Cf. A000040, A002110, A006450, A008475, A038580, A064988, A083186, A222416, A304037, A304253 (fixed points).

f:= proc(n) local t;
   add(ithprime(t[1])^t[2],t=ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, Apr 25 2024

a[n_] := Plus @@ (Prime[#[[1]]]^#[[2]] & /@ FactorInteger[n]); a[1] = 0; Table[a[n], {n, 70}]

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

a(prime(i)^k) = prime(prime(i))^k.
a(A000040(k)) = A006450(k).
a(A006450(k)) = A038580(k).
a(A002110(k)) = A083186(k).

cp's OEIS Frontend

A357980 Replace prime(k) with prime(A000720(k)) in the prime factorization of n, assuming prime(0) = 1.

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A357852 Replace prime(k) with prime(k+2) in the prime factorization of n.

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A357975 Divide all prime indices by 2, round down, and take the number with those prime indices, assuming prime(0) = 1.

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A357978 Replace prime(k) with prime(A000009(k)) in the prime factorization of n.

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A357979 Second MTF-transform of A000041. Replace prime(k) with prime(A357977(k)) in the prime factorization of n.

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A321874 If n = Product (p_j^k_j) then a(n) = Product (prime(p_j)^prime(k_j)).

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A340323 Multiplicative with a(p^e) = (p + 1) * (p - 1)^(e - 1).

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A378175 Triangle T(n,k) read by rows in which n-th row lists in increasing order all multiplicative partitions mu of n (with factors > 1) encoded as Product_{j in mu} prime(j); n>=1, 1<=k<=A001055(n).

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