A156061 -id:A156061 - cp's OEIS frontend

A333235 a(n) is the product of indices of unitary prime power divisors of n.

1, 1, 2, 3, 4, 2, 5, 6, 7, 4, 8, 6, 9, 5, 8, 10, 11, 7, 12, 12, 10, 8, 13, 12, 14, 9, 15, 15, 16, 8, 17, 18, 16, 11, 20, 21, 19, 12, 18, 24, 20, 10, 21, 24, 28, 13, 22, 20, 23, 14, 22, 27, 24, 15, 32, 30, 24, 16, 25, 24, 26, 17, 35, 27, 36, 16, 28, 33, 26, 20

Offset: 1

Ilya Gutkovskiy, Mar 12 2020

Equivalently: replace each prime power p^e in the prime factorization of n by its index in A246655. - M. F. Hasler, Jun 16 2021

			a(600) = a(2^3 * 3 * 5^2) = a(A246655(6) * A246655(2) * A246655(14)) = 6 * 2 * 14 = 168.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A003963, A025528, A027883, A141128, A141809, A156061, A182908, A246655.

Cf. A322981 (the index of n = p^e in A246655).

N:= 1000: # for a(1)..a(N)
R:= NULL: p:= 2:
while p < N do
  R:= R,  seq(p^k,k=1..ilog[p](N));
  p:= nextprime(p);
od:
L:= sort([R]):
f:= proc(n) local F, t;
  F:= ifactors(n)[2];
  mul(ListTools:-BinarySearch(L,t[1]^t[2]),t=F)
end proc:
map(f, [$1..N]); # Robert Israel, Feb 11 2021

PrimePowerPi[n_] := Sum[Boole[PrimePowerQ[k]], {k, 1, n}]; a[1] = 1; a[n_] := Times @@ (PrimePowerPi[#[[1]]^#[[2]]] & /@ FactorInteger[n]); Table[a[n], {n, 1, 70}]

apply( {A333235(n)=vecprod([A322981(f[1]^f[2])|f<-factor(n)~])}, [1..99]) \\ M. F. Hasler, Jun 16 2021

If n = Product (p_j^k_j) then a(n) = Product (A025528(p_j^k_j)).
a(prime(n)) = A027883(n).
a(2^n) = A182908(n).
a(A246655(n)) = 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

A328879 If n = Product (p_j^k_j) then a(n) = Product (pi(p_j) + 1), where pi = A000720.

Original entry on oeis.org

1, 2, 3, 2, 4, 6, 5, 2, 3, 8, 6, 6, 7, 10, 12, 2, 8, 6, 9, 8, 15, 12, 10, 6, 4, 14, 3, 10, 11, 24, 12, 2, 18, 16, 20, 6, 13, 18, 21, 8, 14, 30, 15, 12, 12, 20, 16, 6, 5, 8, 24, 14, 17, 6, 24, 10, 27, 22, 18, 24, 19, 24, 15, 2, 28, 36, 20, 16, 30, 40, 21, 6, 22, 26, 12

Offset: 1

Ilya Gutkovskiy, Oct 29 2019

a(n) is the product of indices of distinct prime factors of n if 1 is considered as a prime (see A008578).

			a(36) = 6 because 36 = 2^2 * 3^2 = prime(1)^2 * prime(2)^2 and (1 + 1) * (2 + 1) = 6.

Index entries for sequences computed from indices in prime factorization

Cf. A000720, A007947, A008578, A036234, A048250, A064553, A156061.

a[n_] := Times @@ ((PrimePi[#[[1]]] + 1) & /@ FactorInteger[n]); Table[a[n], {n, 1, 75}]

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

A333352 a(n) is the product of indices of the smallest and greatest prime factors of n.

Original entry on oeis.org

1, 1, 4, 1, 9, 2, 16, 1, 4, 3, 25, 2, 36, 4, 6, 1, 49, 2, 64, 3, 8, 5, 81, 2, 9, 6, 4, 4, 100, 3, 121, 1, 10, 7, 12, 2, 144, 8, 12, 3, 169, 4, 196, 5, 6, 9, 225, 2, 16, 3, 14, 6, 256, 2, 15, 4, 16, 10, 289, 3, 324, 11, 8, 1, 18, 5, 361, 7, 18, 4, 400, 2, 441, 12, 6, 8, 20, 6, 484, 3

Offset: 1

Ilya Gutkovskiy, Mar 15 2020

nonn

			a(315) = a(3^2 * 5 * 7) = a(prime(2)^2 * prime(3) * prime(4)) = 2 * 4 = 8.

Cf. A000079 (positions of 1's), A000720, A002110, A006530, A020639, A033845 (positions of 2's), A055396, A061395, A066048, A156061, A243055.

a[1] = 1; a[n_] := PrimePi[FactorInteger[n] [[1, 1]]] PrimePi[ FactorInteger[ n] [[-1, 1]]]; Table[a[n], {n, 1, 80}]

a(n) = if (n==1, 1, my(f=factor(n)[,1]); primepi(vecmin(f))*primepi(vecmax(f))); \\ Michel Marcus, Mar 16 2020

If n = Product (p_j^k_j) then a(n) = min{pi(p_j)} * max{pi(p_j)}, where pi = A000720.
a(n) = A055396(n) * A061395(n) for n > 1.
a(2*n) = A061395(n) for n > 1.
a(n^k) = a(n) for k > 0
a(2*prime(n)^k) = n for k > 0.
a(prime(n)^k) = n^2 for k > 0.
a(n!) = pi(n) for n > 1.
a(A002110(n)) = n.

A381077 Sorted positions of first appearances in A380986 (product of prime indices minus product of distinct prime indices).

Original entry on oeis.org

1, 9, 25, 49, 63, 81, 99, 121, 125, 135, 169, 171, 245, 279, 289, 343, 361, 363, 369, 375, 387, 477, 529, 531, 575, 603, 625, 675, 711, 729, 747, 833, 841, 847, 873, 875, 891, 909, 961, 981, 1029, 1083, 1125, 1127, 1179, 1225, 1251, 1377, 1413, 1445, 1467

Offset: 1

Gus Wiseman, Feb 20 2025

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. A position of first appearance in a sequence q is an index k such that q(k) is different from q(j) for all j < k.

All terms are odd.

			The terms together with their prime indices begin:
     1: {}
     9: {2,2}
    25: {3,3}
    49: {4,4}
    63: {2,2,4}
    81: {2,2,2,2}
    99: {2,2,5}
   121: {5,5}
   125: {3,3,3}
   135: {2,2,2,3}
   169: {6,6}
   171: {2,2,8}
   245: {3,4,4}
   279: {2,2,11}

For length instead of product we have A151821, firsts of A046660.
For factors instead of indices we have A381076, sorted firsts of A066503.
For sum of factors instead of product of indices we have A381075 (unsorted A280286), A280292.
For quotient instead of difference we have A380988 (unsorted A380987), firsts of A290106.
For quotient and factors we have A001694 (unsorted A064549), firsts of A003557.
For sum instead of product we have A380957 (unsorted A380956), firsts of A380955.
Sorted firsts of A380986, which has nonzero terms at positions A038838.
A000040 lists the primes, differences A001223.
A003963 gives product of prime indices, distinct A156061.
A005117 lists the squarefree numbers, complement A013929.
A007947 gives squarefree kernel.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, length A001222.
A304038 lists distinct prime indices, sum A066328, length A001221.
Cf. A000079, A000720, A001222, A081770, A075255, A116861, A178503, A374248, A379681, A380958.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
q=Table[Times@@prix[n]-Times@@Union[prix[n]],{n,10000}];
Select[Range[Length[q]],FreeQ[Take[q,#-1],q[[#]]]&]

A304194 Numbers k such that k = Product (p_j^e_j) = Product (pi(p_j)*p_j), where pi() = A000720.

Original entry on oeis.org

1, 2, 12, 56, 180, 304, 336, 936, 1696, 1824, 2484, 5040, 5328, 6664, 8384, 8512, 9900, 10176, 13176, 14040, 25632, 26208, 27360, 33372, 33712, 37260, 39808, 39984, 47488, 50304, 51072, 52200, 65232, 69552, 79920, 126900, 128448, 142272, 149184, 152640, 162648, 167776, 184064, 193752, 197640

Offset: 1

Ilya Gutkovskiy, May 07 2018

nonn

Numbers k such that A007947(k)*A156061(k) = k or A156061(k) = A003557(k).

			9900 is a term because 9900 = 2^2 * 3^2 * 5^2 * 11 = prime(1)^2 * prime(2)^2 * prime(3)^2 * prime(5) = 1*prime(1) * 2*prime(2) * 3*prime(3) * 5*prime(5).

Eric Weisstein's World of Mathematics, Distinct Prime Factors
Index entries for sequences computed from indices in prime factorization

Cf. A000720, A003557, A005117, A007947, A008478, A033286, A046022, A048768, A078779, A109298, A156061.

a[n_] := Times @@ (PrimePi[#[[1]]] #[[1]] & /@ FactorInteger[n]); a[1] = 1; Select[Range[200000], a[#] == # &]

isok(n) = {my(f=factor(n)); prod(k=1, #f~, primepi(f[k,1])*f[k,1]) == n;} \\ Michel Marcus, May 08 2018

A333373 a(n) = n + 1 for n <= 1; thereafter a(n) = a(n-1) * prime(gpf(a(n-1))).

Original entry on oeis.org

1, 2, 6, 30, 330, 10230, 1299210, 921139890, 4956653748090, 261270175715571990, 169405230502395438168090, 1649555141343581679123602303970, 287748466487735183193170029972221262770, 1052440045241486547790272887133572295775622539770

Offset: 0

Ilya Gutkovskiy, Mar 17 2020

nonn

For n > 1, a(n) belongs to A083207 (see "product" formula below and my Jan 16 2020 comment in A083207). - Ivan N. Ianakiev, Mar 17 2020

			a(7) = a(6) * prime(gpf(a(6))) = prime(1) * prime(2) * prime(3) * prime(5) * prime(11) * prime(31) * prime(127) = 2 * 3 * 5 * 11 * 31 * 127 * 709 = 921139890.

Cf. A006450, A006530, A007097, A052126, A076146, A156061.

a[0] = 1; a[n_] := a[n] = a[n - 1] Prime[FactorInteger[a[n - 1]][[-1, 1]]]; Table[a[n], {n, 0, 13}]
Table[Times @@ NestList[Prime@# &, 1, n], {n, 0, 13}]

a(n) = Product_{k=0..n} A007097(k).
A156061(a(n+1)) = a(n).
A052126(a(n+1)) = a(n).

A341041 If n = Product (p_j^k_j) then a(n) = 1 + Product (a(pi(p_j))), where pi = A000720, with a(1) = 1.

Original entry on oeis.org

1, 2, 3, 2, 4, 3, 3, 2, 3, 4, 5, 3, 4, 3, 7, 2, 4, 3, 3, 4, 5, 5, 4, 3, 4, 4, 3, 3, 5, 7, 6, 2, 9, 4, 7, 3, 4, 3, 7, 4, 5, 5, 4, 5, 7, 4, 8, 3, 3, 4, 7, 4, 3, 3, 13, 3, 5, 5, 5, 7, 4, 6, 5, 2, 10, 9, 4, 4, 7, 7, 5, 3, 6, 4, 7, 3, 9, 7, 6, 4, 3, 5, 5, 5, 10, 4, 9, 5, 4, 7, 7, 4, 11, 8, 7, 3, 5, 3, 9, 4

Offset: 1

Ilya Gutkovskiy, Feb 03 2021

nonn

			a(45) = a(3^2 * 5) = a(prime(2)^2 * prime(3)) = 1 + a(2) * a(3) = 1 + 2 * 3 = 7.

Index entries for sequences computed from indices in prime factorization

Cf. A000720, A093320, A156061, A214567, A328880.

a:= n-> `if`(n=1, 1, 1+mul(a(numtheory[pi](i[1])), i=ifactors(n)[2])):
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 03 2021

a[1] = 1; a[n_] := a[n] = 1 + Times @@ (a[PrimePi[#[[1]]]] & /@ FactorInteger[n]); Table[a[n], {n, 100}]

a(n) = a(prime(n)^k) - 1 for k > 0.

a(2*n) = a(n) for n > 1.

A367103 Composite numbers c such that pi(c) is the product of pi applied to the distinct prime factors of c.

Original entry on oeis.org

15, 21, 39, 57, 65, 91, 95, 133

Offset: 1

Tanya Khovanova, Nov 04 2023

Prime numbers trivially satisfy this property and so are excluded.

			15 is a term since its distinct prime factors are 3,5 and pi(15) = pi(3)*pi(5).

Cf. A000720, A002808, A027748, A156061, A235623.

Select[Range[2,10000], ! PrimeQ[#] && Times @@ PrimePi[Transpose[FactorInteger[#]][[1]]] == PrimePi[#] &]

cp's OEIS Frontend

A333235 a(n) is the product of indices of unitary prime power divisors of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A328879 If n = Product (p_j^k_j) then a(n) = Product (pi(p_j) + 1), where pi = A000720.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A333352 a(n) is the product of indices of the smallest and greatest prime factors of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A381077 Sorted positions of first appearances in A380986 (product of prime indices minus product of distinct prime indices).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A304194 Numbers k such that k = Product (p_j^e_j) = Product (pi(p_j)*p_j), where pi() = A000720.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A333373 a(n) = n + 1 for n <= 1; thereafter a(n) = a(n-1) * prime(gpf(a(n-1))).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A341041 If n = Product (p_j^k_j) then a(n) = 1 + Product (a(pi(p_j))), where pi = A000720, with a(1) = 1.

Views

Author