A056169 -id:A056169 - cp's OEIS frontend

A275812 Sum of exponents larger than one in the prime factorization of n: A001222(n) - A056169(n).

0, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 3, 2, 0, 3, 2, 0, 0, 0, 5, 0, 0, 0, 4, 0, 0, 0, 3, 0, 0, 0, 2, 2, 0, 0, 4, 2, 2, 0, 2, 0, 3, 0, 3, 0, 0, 0, 2, 0, 0, 2, 6, 0, 0, 0, 2, 0, 0, 0, 5, 0, 0, 2, 2, 0, 0, 0, 4, 4, 0, 0, 2, 0, 0, 0, 3, 0, 2, 0, 2, 0, 0, 0, 5, 0, 2, 2, 4, 0, 0, 0, 3, 0, 0, 0, 5, 0, 0, 0, 4, 0, 0, 0, 2, 2, 0, 0, 3

Offset: 1

Antti Karttunen, Aug 11 2016

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

Cf. A001222, A001694, A005117, A028234, A046660, A056169, A056170, A057521, A067029, A247180.
Differs from A212172 for the first time at n=36, where a(36)=4, while A212172(36)=2.
Cf. A085548, A136141.

Table[Total@ Map[Last, Select[FactorInteger@ n, Last@ # > 1 &] /. {} -> {{0, 0}}], {n, 120}] (* Michael De Vlieger, Aug 11 2016 *)

a(n) = my(f = factor(n)); sum(k=1, #f~, if (f[k,2] > 1, f[k,2])); \\ Michel Marcus, Jul 19 2017

sub a275812 { vecsum( grep {$> 1} map {$->[1]} factor_exp(shift) ); } # Dana Jacobsen, Aug 15 2016

from sympy import factorint, primefactors
def a001222(n):
    return 0 if n==1 else a001222(n//primefactors(n)[0]) + 1
def a056169(n):
    f=factorint(n)
    return 0 if n==1 else sum(1 for i in f if f[i]==1)
def a(n):
    return a001222(n) - a056169(n)
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 19 2017

a(1) = 0, and for n > 1, if A067029(n)=1 [when n is one of the terms of A247180], a(n) = a(A028234(n)), otherwise a(n) = A067029(n)+a(A028234(n)).
a(n) = A001222(n) - A056169(n).
a(n) = A001222(A057521(n)). - Antti Karttunen, Jul 19 2017
From Amiram Eldar, Sep 28 2023: (Start)
Additive with a(p) = 0, and a(p^e) = e for e >= 2.
a(n) >= 0, with equality if and only if n is squarefree (A005117).
a(n) <= A001222(n), with equality if and only if n is powerful (A001694).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} (1/p^2 + 1/(p*(p-1))) = A085548 + A136141 = 1.22540408909086062637... . (End)
a(n) = A046660(n) + A056170(n). - Amiram Eldar, Jan 09 2024

A275946 Number of nonzero digits that are the sole occupants of their slope in factorial base representation: a(n) = A056169(A275734(n)). (See comments for more exact definition.)

Original entry on oeis.org

0, 1, 1, 2, 1, 0, 1, 2, 2, 3, 2, 1, 1, 2, 0, 1, 2, 1, 1, 0, 2, 1, 0, 0, 1, 2, 2, 3, 2, 1, 2, 3, 3, 4, 3, 2, 2, 3, 1, 2, 3, 2, 2, 1, 3, 2, 1, 1, 1, 2, 2, 3, 2, 1, 0, 1, 1, 2, 1, 0, 2, 3, 1, 2, 3, 2, 2, 1, 3, 2, 1, 1, 1, 2, 0, 1, 2, 1, 2, 3, 1, 2, 3, 2, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 1, 1, 1, 0, 2, 1, 0, 0, 2, 1, 3, 2, 1, 1, 2, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1

Offset: 0

Antti Karttunen, Aug 15 2016

Total number of such nonzero digits d_x in the factorial base representation (A007623) of n for which it holds that for all other nonzero digits d_y present (i_x - d_x) <> (i_y - d_y), where i_x and i_y are the indices of the digits d_x and d_y respectively.

Equally: Number of digit slopes occupied by just one nonzero digit in the factorial base representation of n. In other words, a(n) is the number of elements with multiplicity one in multiset [(i_x - d_x) | where d_x ranges over each nonzero digit present and i_x is its position from the right].

			For n=2, in factorial base "10", there is only one slope occupied by a single nonzero digit (1 is on the sub-maximal slope as 2-1 = 1), thus a(2) = 1.
For n=3, in factorial base "11", there are two occupied slopes, each having just one digit present, thus a(3) = 2.
For n=5, in factorial base "21", there is just one distinct occupied slope, but it contains two nonzero digits (2 and 1 both occupy the maximal slope as 2-2 = 1-1 = 0), thus there are no slopes with just one nonzero digit and a(5) = 0.
For n=525, in factorial base "41311", there are three occupied slopes. The maximal slope contains the nonzero digits "3.1", the sub-maximal digits "4..1.", and the sub-sub-sub-maximal just "1..." (the 1 in the position 4 from right is the sole occupier of its own slope). Thus only one of the slopes is occupied by a lonely occupant and a(525) = 1.

Antti Karttunen, Table of n, a(n) for n = 0..40320
Indranil Ghosh, Python program for generating this sequence
Index entries for sequences related to factorial base representation

Cf. A056169, A275734.

Cf. also A060502, A060130, A225901, A275947, A275948, A275962.

(define (A275946 n) (A056169 (A275734 n)))

a(n) = A056169(A275734(n)).
Other identities. For all n >= 0.
a(n) = A275948(A225901(n)).
A060502(n) = a(n) + A275947(n).
A060130(n) = a(n) + A275962(n).

A275948 Number of nonzero digits that occur only once in factorial base representation of n: a(n) = A056169(A275735(n)).

Original entry on oeis.org

0, 1, 1, 0, 1, 2, 1, 0, 0, 0, 2, 1, 1, 2, 2, 1, 0, 1, 1, 2, 2, 1, 2, 3, 1, 0, 0, 0, 2, 1, 0, 0, 0, 0, 1, 1, 2, 1, 1, 1, 1, 0, 2, 1, 1, 1, 3, 2, 1, 2, 2, 1, 0, 1, 2, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 3, 3, 2, 1, 2, 1, 2, 2, 1, 2, 3, 2, 1, 1, 1, 3, 2, 2, 3, 3, 2, 1, 2, 0, 1, 1, 0, 1, 2, 1, 2, 2, 1, 2, 3, 2, 1, 1, 1, 3, 2, 2, 3, 3, 2, 1, 2, 2, 3, 3, 2, 3, 4, 1

Offset: 0

Antti Karttunen, Aug 15 2016

			For n=0, with factorial base representation (A007623) also 0, there are no nonzero digits, thus a(0) = 0.
For n=2, with factorial base representation "10", there is one nonzero digit, thus a(2) = 1.
For n=3 (= "11") there is no nonzero digit which would occur just once, thus a(3) = 0.
For n=23 (= "321") there are three nonzero digits and each of those digits occurs just once, thus a(23) = 3.
For n=44 (= "1310") there are two distinct nonzero digits ("1" and "3"), but only the other (3) occurs just once, thus a(44) = 1.

Antti Karttunen, Table of n, a(n) for n = 0..40320
Index entries for sequences related to factorial base representation.

Cf. A056169, A275735.

Cf. also A060130, A225901, A275806, A275946, A275949, A275964.

a[n_] := Module[{k = n, m = 2, r, s = {}}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, AppendTo[s, r]; m++]; Count[Tally[Select[s, # > 0 &]][[;;,2]], 1]]; Array[a, 100, 0] (* Amiram Eldar, Feb 07 2024 *)

from sympy import prime, factorint
from operator import mul
from functools import reduce
import collections
def a056169(n):
    f=factorint(n)
    return 0 if n==1 else sum([1 for i in f if f[i]==1])
def a007623(n, p=2): return n if nIndranil Ghosh, Jun 20 2017

(define (A275948 n) (A056169 (A275735 n)))

a(n) = A056169(A275735(n)).
Other identities. For all n >= 0.
a(n) = A275946(A225901(n)).
A275806(n) = a(n) + A275949(n).
A060130(n) = a(n) + A275964(n).

A119251 Positive integers each with exactly 1 unitary prime divisor (i.e., n is included if and only if A056169(n) = 1).

Original entry on oeis.org

2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 20, 23, 24, 28, 29, 31, 37, 40, 41, 43, 44, 45, 47, 48, 50, 52, 53, 54, 56, 59, 61, 63, 67, 68, 71, 73, 75, 76, 79, 80, 83, 88, 89, 92, 96, 97, 98, 99, 101, 103, 104, 107, 109, 112, 113, 116, 117, 124, 127, 131, 135, 136, 137, 139, 147

Offset: 1

Leroy Quet, Jul 23 2006

nonn

Also, numbers expressible as the product of a prime and a powerful number not divisible by that prime. - Franklin T. Adams-Watters, Jul 25 2006

			28 has the prime factorization of 2^2 * 7^1. 28 is therefore included in this sequence because there is exactly one prime raised to an exponent of 1 in 28's prime factorization.

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

Cf. A001694, A056169.

Select[Range@147, Count[FactorInteger@#, 1, 2] == 1 &] (* Robert G. Wilson v, Jul 25 2006 *)

More terms from Robert G. Wilson v and Franklin T. Adams-Watters, Jul 25 2006

A284267 Number of terms with coefficient 1 in the Stern polynomial B(2n+1,x): a(n) = A056169(A277324(n)).

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 1, 4, 3, 1, 1, 2, 2, 2, 1, 5, 4, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 6, 5, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 7, 6, 1, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2

Offset: 0

Antti Karttunen, Mar 25 2017

nonn

Number of 1's on row 2n+1 of table A125184.

Antti Karttunen, Table of n, a(n) for n = 0..8192

Cf. A007306, A056169, A125184, A260443, A277324, A284268.

Odd bisection of A284271.

A003961[p_?PrimeQ] := A003961[p] = Prime[ PrimePi[p] + 1]; A003961[1] = 1; A003961[n_]:= A003961[n] = Times @@ (A003961[First[#]] ^ Last[#] & ) /@ FactorInteger[n] (* after Jean-François Alcover, Dec 01 2011 *); A260443[n_]:= If[n<2, n + 1, If[EvenQ[n], A003961[A260443[n/2]], A260443[(n - 1)/2] * A260443[(n + 1)/2]]]; a[n_]:= If[n<2, 0, Count[Transpose[FactorInteger[n]][[2]], 1]]; A277324[n_]:=A260443[2n + 1]; Table[a[A277324[n]], {n, 0, 150}] (* Indranil Ghosh, Mar 28 2017 *)

A284267(n) = A284271(n+n+1); \\ Other code as in A284271.

(define (A284267 n) (A284271 (+ n n 1)))
(define (A284267 n) (A056169 (A277324 n)))

a(n) = A284271((2*n)+1).
a(n) = A056169(A277324(n)).
Other identities. For all n >= 0:
A007306(1+n) = a(n) + A284268(n).

A295663 a(n) = A295664(n) - A056169(n); 2-adic valuation of tau(n) minus the number of unitary prime divisors of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2

Offset: 1

Antti Karttunen, Nov 28 2017

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

Cf. A000005, A007814, A056169, A295662, A295664.

Cf. A295661 (positions of nonzero terms).

Table[IntegerExponent[DivisorSigma[0, n], 2] - DivisorSum[n, 1 &, And[PrimeQ@ #, CoprimeQ[#, n/#]] &], {n, 105}] (* Michael De Vlieger, Nov 28 2017 *)

a(n) = vecsum(apply(x -> if(x == 1, 0, valuation(x+1, 2)), factor(n)[, 2])); \\ Amiram Eldar, Sep 28 2023

Additive with a(p) = 0, a(p^e) = A007814(1+e) if e > 1.
a(1) = 0; and for n > 1, if A067029(n) = 1, a(n) = a(A028234(n)), otherwise A007814(1+A067029(n)) + a(A028234(n)).
a(n) = A295664(n) - A056169(n).
a(n) = 0 iff A295662(n) = 0, and when A295662(n) > 0, a(n) >= A295662(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} f(1/p) = 0.22852676306472099280..., where f(x) = -1 + (1-x)*(-x + Sum_{k>=0} x^(2^k-1)/(1-x^(2^k))). - Amiram Eldar, Sep 28 2023

A328830 The second prime shadow of n: a(1) = 1; for n > 1, a(n) = a(A003557(n)) * prime(A056169(n)) when A056169(n) > 0, otherwise a(n) = a(A003557(n)).

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 4, 2, 3, 3, 2, 2, 4, 2, 4, 3, 3, 2, 4, 2, 3, 2, 4, 2, 5, 2, 2, 3, 3, 3, 3, 2, 3, 3, 4, 2, 5, 2, 4, 4, 3, 2, 4, 2, 4, 3, 4, 2, 4, 3, 4, 3, 3, 2, 6, 2, 3, 4, 2, 3, 5, 2, 4, 3, 5, 2, 4, 2, 3, 4, 4, 3, 5, 2, 4, 2, 3, 2, 6, 3, 3, 3, 4, 2, 6, 3, 4, 3, 3, 3, 4, 2, 4, 4, 3, 2, 5, 2, 4, 5

Offset: 1

Antti Karttunen, Oct 29 2019

nonn

a(n) depends only on prime signature of n (cf. A025487).

			For n = 30 = 2 * 3 * 5, there are three unitary prime factors, while A003557(30) = 1, which terminates the recursion, thus a(30) = prime(3) = 5.
For n = 60060 = 2^2 * 3 * 5 * 7 * 11 * 13, there are 5 unitary prime factors, while in A003557(60060) = 2 there is only one, thus a(60060) = prime(5) * prime(1) = 11 * 2 = 22.
The number 1260 = 2^2*3^2*5*7 has prime exponents (2,2,1,1) so its prime shadow is prime(2)*prime(2)*prime(1)*prime(1) = 36.  Next, 36 = 2^2*3^2 has prime exponents (2,2) so its prime shadow is prime(2)*prime(2) = 9. In fact, the term a(1260) = 9 is the first appearance of 9 in the sequence. - _Gus Wiseman_, Apr 28 2022

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Column 2 of A353510.
Cf. A001221, A001222, A003557, A008578, A056169, A071625, A323022.
Differs from A182860 for the first time at a(30) = 5, while A182860(30) = 4.
Cf. A182863 for the first appearances.
A005361 gives product of prime exponents.
A112798 gives prime indices, sum A056239.
A124010 gives prime signature, sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A325131 lists numbers relatively prime to their prime shadow.
A325755 lists numbers divisible by their prime shadow.
Cf. A109297, A182850, A353393, A353507.

A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A056169(n) = { my(f=factor(n)[, 2]); sum(i=1, #f, f[i]==1); }; \\ From A056169
A328830(n) = if(1==n,n, my(u=A056169(n)); if(0==u,1,prime(u)) * A328830(A003557(n)));

a(1) = 1; for n > 1, a(n) = A008578(1+A056169(n)) * a(A003557(n)).
A001221(a(n)) = A323022(n).
A001222(a(n)) = A071625(n).
a(n) = A181819(A181819(n)). - Gus Wiseman, Apr 27 2022

Added Gus Wiseman's new name to the front of the definition. - Antti Karttunen, Apr 27 2022

A284271 Number of terms with coefficient 1 in the Stern polynomial B(n,x): a(n) = A056169(A260443(n)).

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 2, 1, 3, 4, 1, 3, 2, 1, 1, 1, 2, 2, 2, 2, 1, 2, 3, 1, 4, 5, 1, 4, 3, 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 3, 2, 1, 2, 4, 1, 5, 6, 1, 5, 4, 1, 3, 2, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 3, 2, 2, 1, 1, 1, 2, 2, 4

Offset: 0

Antti Karttunen, Mar 25 2017

nonn

Number of 1's on row n of table A125184.

Antti Karttunen, Table of n, a(n) for n = 0..8192

Cf. A002487, A056169, A125184, A260443, A277700, A284272, A284267 (odd bisection).

A003961[p_?PrimeQ] := A003961[p] = Prime[ PrimePi[p] + 1]; A003961[1] = 1; A003961[n_] := A003961[n] = Times @@ ( A003961[First[#]] ^ Last[#] & ) /@ FactorInteger[n](* Jean-François Alcover, Dec 01 2011 *); A260443[n_]:= If[n<2, n + 1, If[EvenQ[n], A003961[A260443[n/2]], A260443[(n - 1)/2] * A260443[(n + 1)/2]]]; a[n_]:= If[n<2, 0, Count[Transpose[FactorInteger[n]][[2]], 1]]; Table[a[A260443[n]], {n, 0, 150}] (* Indranil Ghosh, Mar 28 2017 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From Michel Marcus
A260443(n) = if(n<2, n+1, if(n%2, A260443(n\2)*A260443(n\2+1), A003961(A260443(n\2)))); \\ Cf. Charles R Greathouse IV's code for "ps" in A186891 and A277013.
A056169(n) = { my(f=factor(n)[, 2]); sum(i=1, #f, f[i]==1); }; \\ This function from Charles R Greathouse IV, Apr 29 2015
A284271(n) = A056169(A260443(n));

(define (A284271 n) (A056169 (A260443 n)))

a(n) = A056169(A260443(n)).
Other identities and observations. For all n >= 0:
A002487(n) = a(n) + A284272(n).
a(n) <= A277700(n).

A276086 Primorial base exp-function: digits in primorial base representation of n become the exponents of successive prime factors whose product a(n) is.

Original entry on oeis.org

1, 2, 3, 6, 9, 18, 5, 10, 15, 30, 45, 90, 25, 50, 75, 150, 225, 450, 125, 250, 375, 750, 1125, 2250, 625, 1250, 1875, 3750, 5625, 11250, 7, 14, 21, 42, 63, 126, 35, 70, 105, 210, 315, 630, 175, 350, 525, 1050, 1575, 3150, 875, 1750, 2625, 5250, 7875, 15750, 4375, 8750, 13125, 26250, 39375, 78750, 49, 98, 147, 294, 441, 882, 245, 490, 735, 1470, 2205, 4410, 1225, 2450

Offset: 0

Antti Karttunen, Aug 21 2016

Prime product form of primorial base expansion of n.
Sequence is a permutation of A048103. It maps the smallest prime not dividing n to the smallest prime dividing n, that is, A020639(a(n)) = A053669(n) holds for all n >= 1.
The sequence satisfies the exponential function identity, a(x + y) = a(x) * a(y), whenever A329041(x,y) = 1, that is, when adding x and y together will not generate any carries in the primorial base. Examples of such pairs of x and y are A328841(n) & A328842(n), and also A328770(n) (when added with itself). - Antti Karttunen, Oct 31 2019
From Antti Karttunen, Feb 18 2022: (Start)
The conjecture given in A327969 asks whether applying this function together with the arithmetic derivative (A003415) in some combination or another can eventually transform every positive integer into zero.
Another related open question asks whether there are any other numbers than n=6 such that when starting from that n and by iterating with A003415, one eventually reaches a(n). See comments in A351088.
This sequence is used in A351255 to list the terms of A099308 in a different order, by the increasing exponents of the successive primes in their prime factorization. (End)
From Bill McEachen, Oct 15 2022: (Start)
From inspection, the least significant decimal digits of a(n) terms form continuous chains of 30 as follows. For n == i (mod 30), i=0..5, there are 6 ordered elements of these 8 {1,2,3,6,9,8,7,4}. Then for n == i (mod 30), i=6..29, there are 12 repeated pairs = {5,0}.
Moreover, when the individual elements of any of the possible groups of 6 are transformed via (7*digit) (mod 10), the result matches one of the other 7 groupings (not all 7 may be seen). As example, {1,2,3,6,9,8} transforms to {7,4,1,2,3,6}. (End)
The least significant digit of a(n) in base 4 is given by A353486, and in base 6 by A358840. - Antti Karttunen, Oct 25 2022, Feb 17 2024

			For n = 24, which has primorial base representation (see A049345) "400" as 24 = 4*A002110(2) + 0*A002110(1) + 0*A002110(0) = 4*6 + 0*2 + 0*1, thus a(24) = prime(3)^4 * prime(2)^0 * prime(1)^0 = 5^4 = 625.
For n = 35 = "1021" as 35 = 1*A002110(3) + 0*A002110(2) + 2*A002110(1) + 1*A002110(0) = 1*30 + 0*6 + 2*2 + 1*1, thus a(35) = prime(4)^1 * prime(2)^2 * prime(1) = 7 * 3*3 * 2 = 126.

Antti Karttunen, Table of n, a(n) for n = 0..2310
Antti Karttunen, Program in LODA-assembly
Antti Karttunen, Program in LODA-assembly [Cached copy]
Index entries for sequences related to primorial base

Cf. A276085 (a left inverse) and also A276087, A328403.
Cf. A000040, A001221, A001222, A002110, A020639, A049345, A053669, A055396, A057588, A071178, A143293, A257993, A267263, A276084, A276088, A276092, A276093, A276147, A276150, A276151, A276153, A276156, A283477, A324198 (= gcd(n, a(n))), A328584 (= lcm(n, a(n))), A324646, A324289, A328386, A328403, A328475, A328571, A328572, A328578, A328612, A328613, A328620, A328624, A328627, A328763, A328766, A328828, A328835, A328841, A328842, A328843, A328844, A329041, A324580 [= n*a(n)], A324895 (largest proper divisor of a(n)), A351252, A353486 (reduced modulo 4), A358840 (modulo 6), A353489, A353516.
Cf. A048103 (terms sorted into ascending order), A100716 (natural numbers not present in this sequence).
Cf. A278226 (associated filter-sequence), A286626 (and its rgs-version), A328477.
Cf. A328316 (iterates started from zero).
Cf. A327858, A327859, A327860, A327963, A328097, A328098, A328099, A328110, A328112, A328382 for various combinations with arithmetic derivative (A003415).
Cf. also A327167, A329037.
Cf. A019565 and A054842 for base-2 and base-10 analogs and A276076 for the analogous "factorial base exp-function", from which this differs for the first time at n=24, where a(24)=625 while A276076(24)=7.
Cf. A327969, A351088, A351458 for sequences with conjectures involving this sequence.

b = MixedRadix[Reverse@ Prime@ Range@ 12]; Table[Function[k, Times @@ Power @@@ # &@ Transpose@ {Prime@ Range@ Length@ k, Reverse@ k}]@ IntegerDigits[n, b], {n, 0, 51}] (* Michael De Vlieger, Aug 23 2016, Version 10.2 *)
f[n_] := Block[{a = {{0, n}}}, Do[AppendTo[a, {First@ #, Last@ #} &@ QuotientRemainder[a[[-1, -1]], Times @@ Prime@ Range[# - i]]], {i, 0, #}] &@ NestWhile[# + 1 &, 0, Times @@ Prime@ Range[# + 1] <= n &]; Rest[a][[All, 1]]]; Table[Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &, Reverse@ f@ n], {n, 0, 73}] (* Michael De Vlieger, Aug 30 2016, Pre-Version 10 *)
a[n0_] := Module[{m = 1, i = 1, n = n0, p}, While[n > 0, p = Prime[i]; m *= p^Mod[n, p]; n = Quotient[n, p]; i++]; m];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Dec 01 2021, after Antti Karttunen's Sage code *)

A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; }; \\ Antti Karttunen, May 12 2017

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); }; \\ (Better than above one, avoids unnecessary construction of primorials). - Antti Karttunen, Oct 14 2019

from sympy import prime
def a(n):
    i=0
    m=pr=1
    while n>0:
        i+=1
        N=prime(i)*pr
        if n%N!=0:
            m*=(prime(i)**((n%N)/pr))
            n-=n%N
        pr=N
    return m # Indranil Ghosh, May 12 2017, after Antti Karttunen's PARI code

from sympy import nextprime
def a(n):
    m, p = 1, 2
    while n > 0:
        n, r = divmod(n, p)
        m *= p**r
        p = nextprime(p)
    return m
print([a(n) for n in range(74)])  # Peter Luschny, Apr 20 2024

def A276086(n):
    m=1
    i=1
    while n>0:
        p = sloane.A000040(i)
        m *= (p**(n%p))
        n = floor(n/p)
        i += 1
    return (m)
# Antti Karttunen, Oct 14 2019, after Indranil Ghosh's Python code above, and my own leaner PARI code from Oct 14 2019. This avoids unnecessary construction of primorials.

(define (A276086 n) (let loop ((n n) (t 1) (i 1)) (if (zero? n) t (let* ((p (A000040 i)) (d (modulo n p))) (loop (/ (- n d) p) (* t (expt p d)) (+ 1 i))))))

(definec (A276086 n) (if (zero? n) 1 (* (expt (A053669 n) (A276088 n)) (A276086 (A276093 n))))) ;; Needs macro definec from http://oeis.org/wiki/Memoization#Scheme

(definec (A276086 n) (if (zero? n) 1 (* (A053669 n) (A276086 (- n (A002110 (A276084 n))))))) ;; Needs macro definec from http://oeis.org/wiki/Memoization#Scheme

a(0) = 1; for n >= 1, a(n) = A053669(n) * a(A276151(n)) = A053669(n) * a(n-A002110(A276084(n))).
a(0) = 1; for n >= 1, a(n) = A053669(n)^A276088(n) * a(A276093(n)).
a(n) = A328841(a(n)) + A328842(a(n)) = A328843(n) + A328844(n).
a(n) = a(A328841(n)) * a(A328842(n)) = A328571(n) * A328572(n).
a(n) = A328475(n) * A328580(n) = A328476(n) + A328580(n).
a(A002110(n)) = A000040(n+1). [Maps primorials to primes]
a(A143293(n)) = A002110(n+1). [Maps partial sums of primorials to primorials]
a(A057588(n)) = A276092(n).
a(A276156(n)) = A019565(n).
a(A283477(n)) = A324289(n).
a(A003415(n)) = A327859(n).
Here the text in brackets shows how the right hand side sequence is a function of the primorial base expansion of n:
A001221(a(n)) = A267263(n). [Number of nonzero digits]
A001222(a(n)) = A276150(n). [Sum of digits]
A067029(a(n)) = A276088(n). [The least significant nonzero digit]
A071178(a(n)) = A276153(n). [The most significant digit]
A061395(a(n)) = A235224(n). [Number of significant digits]
A051903(a(n)) = A328114(n). [Largest digit]
A055396(a(n)) = A257993(n). [Number of trailing zeros + 1]
A257993(a(n)) = A328570(n). [Index of the least significant zero digit]
A079067(a(n)) = A328620(n). [Number of nonleading zeros]
A056169(a(n)) = A328614(n). [Number of 1-digits]
A056170(a(n)) = A328615(n). [Number of digits larger than 1]
A277885(a(n)) = A328828(n). [Index of the least significant digit > 1]
A134193(a(n)) = A329028(n). [The least missing nonzero digit]
A005361(a(n)) = A328581(n). [Product of nonzero digits]
A072411(a(n)) = A328582(n). [LCM of nonzero digits]
A001055(a(n)) = A317836(n). [Number of carry-free partitions of n in primorial base]
Various number theoretical functions applied:
A000005(a(n)) = A324655(n). [Number of divisors of a(n)]
A000203(a(n)) = A324653(n). [Sum of divisors of a(n)]
A000010(a(n)) = A324650(n). [Euler phi applied to a(n)]
A023900(a(n)) = A328583(n). [Dirichlet inverse of Euler phi applied to a(n)]
A069359(a(n)) = A329029(n). [Sum a(n)/p over primes p dividing a(n)]
A003415(a(n)) = A327860(n). [Arithmetic derivative of a(n)]
Other identities:
A276085(a(n)) = n.          [A276085 is a left inverse]
A020639(a(n)) = A053669(n). [The smallest prime not dividing n -> the smallest prime dividing n]
A046523(a(n)) = A278226(n). [Least number with the same prime signature as a(n)]
A246277(a(n)) = A329038(n).
A181819(a(n)) = A328835(n).
A053669(a(n)) = A326810(n), A326810(a(n)) = A328579(n).
A257993(a(n)) = A328570(n), A328570(a(n)) = A328578(n).
A328613(a(n)) = A328763(n), A328620(a(n)) = A328766(n).
A328828(a(n)) = A328829(n).
A053589(a(n)) = A328580(n). [Greatest primorial number which divides a(n)]
A276151(a(n)) = A328476(n). [... and that primorial subtracted from a(n)]
A111701(a(n)) = A328475(n).
A328114(a(n)) = A328389(n). [Greatest digit of primorial base expansion of a(n)]
A328389(a(n)) = A328394(n), A328394(a(n)) = A328398(n).
A235224(a(n)) = A328404(n), A328405(a(n)) = A328406(n).
a(A328625(n)) = A328624(n), a(A328626(n)) = A328627(n). ["Twisted" variants]
a(A108951(n)) = A324886(n).
a(n) mod n = A328386(n).
a(a(n)) = A276087(n), a(a(a(n))) = A328403(n). [2- and 3-fold applications]
a(2n+1) = 2 * a(2n). - Antti Karttunen, Feb 17 2022

Name edited and new link-formulas added by Antti Karttunen, Oct 29 2019

Name changed again by Antti Karttunen, Feb 05 2022

A181819 Prime shadow of n: a(1) = 1; for n>1, if n = Product prime(i)^e(i), then a(n) = Product prime(e(i)).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 7, 2, 6, 2, 6, 4, 4, 2, 10, 3, 4, 5, 6, 2, 8, 2, 11, 4, 4, 4, 9, 2, 4, 4, 10, 2, 8, 2, 6, 6, 4, 2, 14, 3, 6, 4, 6, 2, 10, 4, 10, 4, 4, 2, 12, 2, 4, 6, 13, 4, 8, 2, 6, 4, 8, 2, 15, 2, 4, 6, 6, 4, 8, 2, 14, 7, 4, 2, 12, 4, 4, 4, 10, 2, 12, 4, 6, 4, 4, 4, 22, 2, 6, 6, 9, 2, 8, 2, 10, 8

Offset: 1

Matthew Vandermast, Dec 07 2010

a(n) depends only on prime signature of n (cf. A025487). a(m) = a(n) iff m and n have the same prime signature, i.e., iff A046523(m) = A046523(n).

Because A046523 (the smallest representative of prime signature of n) and this sequence are functions of each other as A046523(n) = A181821(a(n)) and a(n) = a(A046523(n)), it implies that for all i, j: a(i) = a(j) <=> A046523(i) = A046523(j) <=> A101296(i) = A101296(j), i.e., that equivalence-class-wise this is equal to A101296, and furthermore, applying any function f on this sequence gives us a sequence b(n) = f(a(n)) whose equivalence class partitioning is equal to or coarser than that of A101296, i.e., b is then a sequence that depends only on the prime signature of n (the multiset of exponents of its prime factors), although not necessarily in a very intuitive way. - Antti Karttunen, Apr 28 2022

			20 = 2^2*5 has the exponents (2,1) in its prime factorization. Accordingly, a(20) = prime(2)*prime(1) = A000040(2)*A000040(1) = 3*2 = 6.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n

Column 1 of A353510 (see also A325239 and A325277).
Cf. A000040, A000265, A001511, A001222, A003963, A005361, A007814, A008578, A028234, A046523, A056239, A064553, A064989, A067029, A101296 (restricted growth sequence transform), A108951, A122111, A124010, A124859, A156552, A181820, A181821, A182850, A182855, A182857 (also A323014), A115621, A101296, A238690, A238745, A238747, A238748, A246029, A304465, A304647, A305732, A305733, A320118, A323022, A325501, A325502, A325507, A325508, A325755 (A353566), A325756, A328830 [= a(a(n))], A328835, A351564 (characteristic function of A130091), A351944, A351946, A353379.
Left inverse of A304660.

a181819 = product . map a000040 . a124010_row
-- Reinhard Zumkeller, Mar 26 2012

A181819 := proc(n)
    local a;
    a := 1;
    for pf in ifactors(n)[2] do
        a := a*ithprime(pf[2]) ;
    end do:
    a ;
end proc:
seq(A181819(n),n=1..80) ; # R. J. Mathar, Jan 09 2019
# second Maple program:
a:= n-> mul(ithprime(i[2]), i=ifactors(n)[2]):
seq(a(n), n=1..105);  # Alois P. Heinz, Nov 26 2024

{1}~Join~Table[Times @@ Prime@ Map[Last, FactorInteger@ n], {n, 2, 120}] (* Michael De Vlieger, Feb 07 2016 *)

a(n) = {my(f=factor(n)); prod(k=1, #f~, prime(f[k,2]));} \\ Michel Marcus, Nov 16 2015

;; With memoization-macro definec.
(definec (A181819 n) (cond ((= 1 n) 1) (else (* (A000040 (A067029 n)) (A181819 (A028234 n))))))
;; Antti Karttunen, Feb 05 2016

;; With memoization-macro definec.
(definec (A181819 n) (cond ((= 1 n) 1) ((even? n) (* (A000040 (A007814 n)) (A181819 (A000265 n)))) (else (A181819 (A064989 n)))))
;; Antti Karttunen, Feb 07 2016

From Antti Karttunen, Feb 07 2016: (Start)
a(1) = 1; for n > 1, a(n) = A000040(A067029(n)) * a(A028234(n)).
a(1) = 1; for n > 1, a(n) = A008578(A001511(n)) * a(A064989(n)).
Other identities. For all n >= 1:
a(A124859(n)) = A122111(a(n)) = A238745(n). - from Matthew Vandermast's formulas for the latter sequence.
(End)
a(n) = A246029(A156552(n)). - Antti Karttunen, Oct 15 2016
From Antti Karttunen, Apr 28 & Apr 30 2022: (Start)
A181821(a(n)) = A046523(n) and a(A046523(n)) = a(n). [See comments]
a(n) = A329900(A124859(n)) = A319626(A124859(n)).
a(n) = A246029(A156552(n)).
a(a(n)) = A328830(n).
a(A304660(n)) = n.
a(A108951(n)) = A122111(n).
a(A185633(n)) = A322312(n).
a(A025487(n)) = A181820(n).
a(A276076(n)) = A275735(n) and a(A276086(n)) = A328835(n).
As the sequence converts prime exponents to prime indices, it effects the following mappings:
A001221(a(n)) = A071625(n). [Number of distinct indices --> Number of distinct exponents]
A001222(a(n)) = A001221(n). [Number of indices (i.e., the number of prime factors with multiplicity) --> Number of exponents (i.e., the number of distinct prime factors)]
A056239(a(n)) = A001222(n). [Sum of indices --> Sum of exponents]
A066328(a(n)) = A136565(n). [Sum of distinct indices --> Sum of distinct exponents]
A003963(a(n)) = A005361(n). [Product of indices --> Product of exponents]
A290103(a(n)) = A072411(n). [LCM of indices --> LCM of exponents]
A156061(a(n)) = A290107(n). [Product of distinct indices --> Product of distinct exponents]
A257993(a(n)) = A134193(n). [Index of the least prime not dividing n --> The least number not among the exponents]
A055396(a(n)) = A051904(n). [Index of the least prime dividing n --> Minimal exponent]
A061395(a(n)) = A051903(n). [Index of the greatest prime dividing n --> Maximal exponent]
A008966(a(n)) = A351564(n). [All indices are distinct (i.e., n is squarefree) --> All exponents are distinct]
A007814(a(n)) = A056169(n). [Number of occurrences of index 1 (i.e., the 2-adic valuation of n) --> Number of occurrences of exponent 1]
A056169(a(n)) = A136567(n). [Number of unitary prime divisors --> Number of exponents occurring only once]
A064989(a(n)) = a(A003557(n)) = A295879(n). [Indices decremented after <--> Exponents decremented before]
Other mappings:
A007947(a(n)) = a(A328400(n)) = A329601(n).
A181821(A007947(a(n))) = A328400(n).
A064553(a(n)) = A000005(n) and A000005(a(n)) = A182860(n).
A051903(a(n)) = A351946(n).
A003557(a(n)) = A351944(n).
A258851(a(n)) = A353379(n).
A008480(a(n)) = A309004(n).
a(A325501(n)) = A325507(n) and a(A325502(n)) = A038754(n+1).
a(n!) = A325508(n).
(End)

Name "Prime shadow" (coined by Gus Wiseman in A325755) prefixed to the definition by Antti Karttunen, Apr 27 2022

cp's OEIS Frontend

A275812 Sum of exponents larger than one in the prime factorization of n: A001222(n) - A056169(n).

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A275946 Number of nonzero digits that are the sole occupants of their slope in factorial base representation: a(n) = A056169(A275734(n)). (See comments for more exact definition.)

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A275948 Number of nonzero digits that occur only once in factorial base representation of n: a(n) = A056169(A275735(n)).

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A119251 Positive integers each with exactly 1 unitary prime divisor (i.e., n is included if and only if A056169(n) = 1).

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A284267 Number of terms with coefficient 1 in the Stern polynomial B(2n+1,x): a(n) = A056169(A277324(n)).

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A295663 a(n) = A295664(n) - A056169(n); 2-adic valuation of tau(n) minus the number of unitary prime divisors of n.

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A328830 The second prime shadow of n: a(1) = 1; for n > 1, a(n) = a(A003557(n)) * prime(A056169(n)) when A056169(n) > 0, otherwise a(n) = a(A003557(n)).

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