A328400 -id:A328400 - cp's OEIS frontend

A328397 Lexicographically earliest infinite sequence such that a(i) = a(j) => A328400(A276087(i)) = A328400(A276087(j)) for all i, j.

1, 1, 1, 1, 1, 2, 3, 3, 3, 1, 3, 2, 4, 5, 3, 6, 3, 3, 7, 3, 8, 9, 8, 10, 11, 12, 13, 14, 15, 16, 1, 3, 5, 3, 3, 4, 3, 3, 17, 1, 17, 2, 18, 19, 17, 6, 20, 4, 7, 21, 17, 22, 23, 24, 25, 26, 27, 28, 23, 21, 5, 5, 4, 7, 17, 7, 3, 3, 17, 29, 30, 31, 18, 19, 32, 22, 33, 24, 34, 35, 36, 37, 38, 39, 15, 40, 41, 14, 42, 43, 7, 19, 44, 45, 19, 46, 21, 8, 17, 47, 48, 4, 49

Offset: 0

Antti Karttunen, Oct 15 2019

nonn

Restricted growth sequence transform of function f(n) = A328400(A276087(n)).
For all i, j:
   A328395(i) = A328395(j) => a(i) = a(j) => A328389(i) = A328389(j).

Antti Karttunen, Table of n, a(n) for n = 0..32589
Index entries for sequences related to primorial base

Cf. A276086, A276087, A328389, A328400.

Cf. also A278226, A286626, A328388, A328395, A328396.

up_to = 32589;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A007947(n) = factorback(factorint(n)[, 1]);
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A181821(n) = { my(f=factor(n),p=0,m=1); forstep(i=#f~,1,-1,while(f[i,2], f[i,2]--; m *= (p=nextprime(p+1))^primepi(f[i,1]))); (m); };
A328400(n) = A181821(A007947(A181819(n)));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A276087(n) = A276086(A276086(n));
v328397 = rgs_transform(vector(1+up_to, n, A328400(A276087(n-1))));
A328397(n) = v328397[1+n];

A328401 Lexicographically earliest infinite sequence such that a(i) = a(j) => A328400(i) = A328400(j) for all i, j.

Original entry on oeis.org

1, 2, 2, 3, 2, 2, 2, 4, 3, 2, 2, 5, 2, 2, 2, 6, 2, 5, 2, 5, 2, 2, 2, 7, 3, 2, 4, 5, 2, 2, 2, 8, 2, 2, 2, 3, 2, 2, 2, 7, 2, 2, 2, 5, 5, 2, 2, 9, 3, 5, 2, 5, 2, 7, 2, 7, 2, 2, 2, 5, 2, 2, 5, 10, 2, 2, 2, 5, 2, 2, 2, 11, 2, 2, 5, 5, 2, 2, 2, 9, 6, 2, 2, 5, 2, 2, 2, 7, 2, 5, 2, 5, 2, 2, 2, 12, 2, 5, 5, 3, 2, 2, 2, 7, 2

Offset: 1

Antti Karttunen, Oct 17 2019

nonn

Restricted growth sequence transform of A328400(n), or equally, of A007947(A181819(n)).
For all i, j:
  A101296(i) = A101296(j) => a(i) = a(j),
  a(i) = a(j) => A051903(i) = A051903(j) => A008966(i) = A008966(j),
  a(i) = a(j) => A051904(i) = A051904(j),
  a(i) = a(j) => A052409(i) = A052409(j),
  a(i) = a(j) => A072411(i) = A072411(j),
  a(i) = a(j) => A071625(i) = A071625(j),
  a(i) = a(j) => A267115(i) = A267115(j),
  a(i) = a(j) => A267116(i) = A267116(j).

			Numbers 2 (= 2^1), 3 (= 3^1), 6 = (2^1 * 3^1) and 30 (2^1 * 3^1 * 5^1) all have just one distinct exponent, 1, in the multisets of exponents that occur in their prime factorization, thus they all have the same value a(2) = a(3) = a(6) = a(30) = 2 in this sequence.
Number 4 (2^2), 9 (3^2) and 36 (2^2 * 3^2) all have just one distinct exponent, 2, in the multisets of exponents that occur in their prime factorization, thus they all have the same value a(4) = a(9) = a(36) = 3 in this sequence.
Numbers 12 = 2^2 * 3^1, 18 = 2^1 * 3^2, 60 = 2^2 * 3^1 * 5^1 and 300 = 2^2 * 3^1 * 5^2 all have both 1 and 2 and none other values occurring in the multisets of exponents in their prime factorization, thus they all have the value of a(12) = 5 that was allotted to 12 by the restricted growth sequence transform, as 12 is the smallest number with prime signature (1, 2).

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

Cf. A007947, A181819, A328400.
Cf. A005117 (gives indices of terms <= 2), A062503 (after its initial 1, gives indices of 3's).
Cf. A008966, A051903, A051904, A052409, A072411, A071625, A267115, A267116.

up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A007947(n) = factorback(factorint(n)[, 1]);
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
v328401 = rgs_transform(vector(up_to, n, A007947(A181819(n)))); \\ Faster than with A328400(n).
A328401(n) = v328401[n];

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

A108951 Primorial inflation of n: Fully multiplicative with a(p) = p# for prime p, where x# is the primorial A034386(x).

Original entry on oeis.org

1, 2, 6, 4, 30, 12, 210, 8, 36, 60, 2310, 24, 30030, 420, 180, 16, 510510, 72, 9699690, 120, 1260, 4620, 223092870, 48, 900, 60060, 216, 840, 6469693230, 360, 200560490130, 32, 13860, 1021020, 6300, 144, 7420738134810, 19399380, 180180, 240, 304250263527210, 2520

Offset: 1

Paul Boddington, Jul 21 2005

This sequence is a permutation of A025487.
And thus also a permutation of A181812, see the formula section. - Antti Karttunen, Jul 21 2014
A previous description of this sequence was: "Multiplicative with a(p^e) equal to the product of the e-th powers of all primes at most p" (see extensions), Giuseppe Coppoletta, Feb 28 2015

			a(12) = a(2^2) * a(3) = (2#)^2 * (3#) = 2^2 * 6 = 24
a(45) = (3#)^2 * (5#) = (2*3)^2 * (2*3*5) = 1080 (as 45 = 3^2 * 5).

Amiram Eldar, Table of n, a(n) for n = 1..2370 (terms 1..256 from Antti Karttunen)
Index to divisibility sequences.
Index entries for sequences computed from indices in prime factorization.
Index entries for sequences related to primorial numbers.

Cf. A319626, A329900 (left inverses).

Cf. A034386, A002110, A025487, A048673, A064216, A064989, A085082, A122111, A124859, A161360, A181811, A181812, A181814, A181815, A181817, A181819, A181822, A238690, A283477, A283478, A307035, A324886, A324887, A324888, A324896, A325226, A329040, A329046, A329047, A329344, A329348, A329349, A329378, A329382, A329600, A329602, A329605, A329607, A329615, A329616, A329617, A329619, A329622, A319627, A329647, A331292, A337474, A346108, A346109, A344698, A344699.

a[n_] := a[n] = Module[{f = FactorInteger[n], p, e}, If[Length[f]>1, Times @@ a /@ Power @@@ f, {{p, e}} = f; Times @@ (Prime[Range[PrimePi[p]]]^e)]]; a[1] = 1; Table[a[n], {n, 1, 42}] (* Jean-François Alcover, Feb 24 2015 *)
Table[Times @@ Map[#1^#2 & @@ # &, FactorInteger[n] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}], {n, 42}] (* Michael De Vlieger, Mar 18 2017 *)

primorial(n)=prod(i=1,primepi(n),prime(i))
a(n)=my(f=factor(n)); prod(i=1,#f~, primorial(f[i,1])^f[i,2]) \\ Charles R Greathouse IV, Jun 28 2015

from sympy import primerange, factorint
from operator import mul
def P(n): return reduce(mul, [i for i in primerange(2, n + 1)])
def a(n):
    f = factorint(n)
    return 1 if n==1 else reduce(mul, [P(i)**f[i] for i in f])
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, May 14 2017

def sharp_primorial(n): return sloane.A002110(prime_pi(n))
def p(f):
    return sharp_primorial(f[0])^f[1]
[prod(p(f) for f in factor(n)) for n in range (1,51)]
# Giuseppe Coppoletta, Feb 07 2015

Dirichlet g.f.: 1/(1-2*2^(-s))/(1-6*3^(-s))/(1-30*5^(-s))...
Completely multiplicative with a(p_i) = A002110(i) = prime(i)#. [Franklin T. Adams-Watters, Jun 24 2009; typos corrected by Antti Karttunen, Jul 21 2014]
From Antti Karttunen, Jul 21 2014: (Start)
a(1) = 1, and for n > 1, a(n) = n * a(A064989(n)).
a(n) = n * A181811(n).
a(n) = A002110(A061395(n)) * A331188(n). - [added Jan 14 2020]
a(n) = A181812(A048673(n)).
Other identities:
A006530(a(n)) = A006530(n). [Preserves the largest prime factor of n.]
A071178(a(n)) = A071178(n). [And also its exponent.]
a(2^n) = 2^n. [Fixes the powers of two.]
A067029(a(n)) = A007814(a(n)) = A001222(n). [The exponent of the least prime of a(n), that prime always being 2 for n>1, is equal to the total number of prime factors in n.]
(End)
From Antti Karttunen, Nov 19 2019: (Start)
Further identities:
a(A307035(n)) = A000142(n).
a(A003418(n)) = A181814(n).
a(A025487(n)) = A181817(n).
a(A181820(n)) = A181822(n).
a(A019565(n)) = A283477(n).
A001221(a(n)) = A061395(n).
A001222(a(n)) = A056239(n).
A181819(a(n)) = A122111(n).
A124859(a(n)) = A181821(n).
A085082(a(n)) = A238690(n).
A328400(a(n)) = A329600(n). (smallest number with the same set of distinct prime exponents)
A000188(a(n)) = A329602(n). (square root of the greatest square divisor)
A072411(a(n)) = A329378(n). (LCM of exponents of prime factors)
A005361(a(n)) = A329382(n). (product of exponents of prime factors)
A290107(a(n)) = A329617(n). (product of distinct exponents of prime factors)
A000005(a(n)) = A329605(n). (number of divisors)
A071187(a(n)) = A329614(n). (smallest prime factor of number of divisors)
A267115(a(n)) = A329615(n). (bitwise-AND of exponents of prime factors)
A267116(a(n)) = A329616(n). (bitwise-OR of exponents of prime factors)
A268387(a(n)) = A329647(n). (bitwise-XOR of exponents of prime factors)
A276086(a(n)) = A324886(n). (prime product form of primorial base expansion)
A324580(a(n)) = A324887(n).
A276150(a(n)) = A324888(n). (digit sum in primorial base)
A267263(a(n)) = A329040(n). (number of distinct nonzero digits in primorial base)
A243055(a(n)) = A329343(n).
A276088(a(n)) = A329348(n). (least significant nonzero digit in primorial base)
A276153(a(n)) = A329349(n). (most significant nonzero digit in primorial base)
A328114(a(n)) = A329344(n). (maximal digit in primorial base)
A062977(a(n)) = A325226(n).
A097248(a(n)) = A283478(n).
A324895(a(n)) = A324896(n).
A324655(a(n)) = A329046(n).
A327860(a(n)) = A329047(n).
A329601(a(n)) = A329607(n).
(End)
a(A181815(n)) = A025487(n), and A319626(a(n)) = A329900(a(n)) = n. - Antti Karttunen, Dec 29 2019
From Antti Karttunen, Jul 09 2021: (Start)
a(n) = A346092(n) + A346093(n).
a(n) = A346108(n) - A346109(n).
a(A342012(n)) = A004490(n).
a(A337478(n)) = A336389(n).
A336835(a(n)) = A337474(n).
A342002(a(n)) = A342920(n).
A328571(a(n)) = A346091(n).
A328572(a(n)) = A344592(n).
(End)
Sum_{n>=1} 1/a(n) = A161360. - Amiram Eldar, Aug 04 2022

More terms computed by Antti Karttunen, Jul 21 2014
The name of the sequence was changed for more clarity, in accordance with the above remark of Franklin T. Adams-Watters (dated Jun 24 2009). It is implicitly understood that a(n) is then uniquely defined by completely multiplicative extension. - Giuseppe Coppoletta, Feb 28 2015
Name "Primorial inflation" (coined by Matthew Vandermast in A181815) prefixed to the name by Antti Karttunen, Jan 14 2020

A329600 Smallest number with the same set of distinct prime exponents as A108951(n).

Original entry on oeis.org

1, 2, 2, 4, 2, 12, 2, 8, 4, 12, 2, 24, 2, 12, 12, 16, 2, 72, 2, 24, 12, 12, 2, 48, 4, 12, 8, 24, 2, 360, 2, 32, 12, 12, 12, 144, 2, 12, 12, 48, 2, 360, 2, 24, 24, 12, 2, 96, 4, 72, 12, 24, 2, 432, 12, 48, 12, 12, 2, 720, 2, 12, 24, 64, 12, 360, 2, 24, 12, 360, 2, 288, 2, 12, 72, 24, 12, 360, 2, 96, 16, 12, 2, 720, 12, 12, 12, 48, 2, 2160, 12, 24, 12, 12, 12

Offset: 1

Antti Karttunen, Nov 17 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Cf. A007947, A034386, A071364, A108951, A122111, A181819, A181821, A328400, A329607.

Cf. A077462 (rgs-transform, from its term a(1)=1 onward).

Array[Times @@ MapIndexed[Prime[#2[[1]]]^#1 &, Reverse[Flatten[Cases[FactorInteger[#], {p_, k_} :> Table[PrimePi[p], {k}]]]]] &[Times @@ FactorInteger[#][[All, 1]]] &@ If[# == 1, 1, Times @@ Prime@ FactorInteger[#][[All, -1]]] &[Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}]] &, 105] (* Michael De Vlieger, Nov 18 2019, after Gus Wiseman at A181821 *)

A007947(n) = factorback(factorint(n)[, 1]);
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A181821(n) = { my(f=factor(n),p=0,m=1); forstep(i=#f~,1,-1,while(f[i,2], f[i,2]--; m *= (p=nextprime(p+1))^primepi(f[i,1]))); (m); };
A328400(n) = A181821(A007947(A181819(n)));
A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A329600(n) = A328400(A108951(n));

a(n) = A328400(A108951(n)) = A181821(A007947(A122111(n)))

a(n) = A181821(A329607(n)) = A181821(A122111(A071364(n))).

A297404 A binary representation of the positive exponents that appear in the prime factorization of a number, shown in decimal.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 4, 2, 1, 1, 3, 1, 1, 1, 8, 1, 3, 1, 3, 1, 1, 1, 5, 2, 1, 4, 3, 1, 1, 1, 16, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 1, 3, 3, 1, 1, 9, 2, 3, 1, 3, 1, 5, 1, 5, 1, 1, 1, 3, 1, 1, 3, 32, 1, 1, 1, 3, 1, 1, 1, 6, 1, 1, 3, 3, 1, 1, 1, 9, 8, 1, 1, 3, 1, 1

Offset: 1

Rémy Sigrist, Dec 29 2017

This sequence is similar to A087207; here we encode the exponents, there the prime numbers appearing in the prime factorization of a number.
The binary representation of a(n) shows which exponents appear in the prime factorization of n, but without multiplicities:
- for any prime number p and k > 0, if p^k divides n but p^(k+1) does not divide n, then a(n) AND 2^(k-1) = 2^(k-1) (where AND denotes the bitwise AND operator),
- conversely, if a(n) AND 2^(k-1) = 2^(k-1) for some k > 0, then there is prime number p such that p^k divides n but p^(k+1) does not divide n.

			For n = 90:
- 90 = 5^1 * 3^2 * 2^1,
- the exponents appearing in the prime factorization of 90 are 1 and 2,
- hence a(90) = 2^(1-1) + 2^(2-1) = 3.

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

Cf. A000120, A000695, A001694, A004709, A005117, A038109, A052485, A059404, A087207, A328400.

Array[Total@ Map[2^(# - 1) &, Union[FactorInteger[#][[All, -1]] ]] - Boole[# == 1] &, 86] (* Michael De Vlieger, Dec 29 2017 *)

a(n) = my (x=Set(factor(n)[,2]~)); sum(i=1, #x, 2^(x[i]))/2

a(p^k) = 2^(k-1) for any prime number p and k > 0.
a(n^2) = A000695(2 * a(n)) / 2 for any n > 0.
a(n) <= 1 iff n is squarefree (A005117).
a(n) <= 3 iff n is cubefree (A004709).
a(n) is odd iff n belongs to A052485 (weak numbers).
a(n) is even iff n belongs to A001694 (powerful numbers).
a(n) AND 2 = 2 iff n belongs to A038109 (where AND denotes the bitwise AND operator).
A000120(a(n)) <= 1 iff n belongs to A072774 (powers of squarefree numbers).
A000120(a(n)) > 1 iff n belongs to A059404.
If gcd(m, n) = 1, then a(m * n) = a(m) OR a(n) (where OR denotes the bitwise OR operator).
a(n) = a(A328400(n)). - Peter Munn, Oct 02 2023

A328569 Exponent of least prime factor in A276086(A276086(n)), where A276086 converts the primorial base expansion of n into its prime product form.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Oct 20 2019

nonn

Equally, the least significant nonzero digit in primorial base expansion of A276086(n).

Antti Karttunen, Table of n, a(n) for n = 0..32768
Index entries for sequences related to primorial base

Cf. A003557, A051903, A067029, A276086, A276087, A276088, A328389, A328400, A328476, A328570, A328579.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A276088(n) = { my(e=0, p=2); while(n && !(e=(n%p)), n = n/p; p = nextprime(1+p)); (e); };
A328569(n) = A276088(A276086(n));

a(n) = A276088(A276086(n)) = A067029(A276087(n)).
max(a(n),1+A051903(A328400(A003557(A276086(A328476(n)))))) = A328389(n). [A328400 is optional in the formula]
For all even n, a(n) < A328579(n).

cp's OEIS Frontend

A328397 Lexicographically earliest infinite sequence such that a(i) = a(j) => A328400(A276087(i)) = A328400(A276087(j)) for all i, j.

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A328401 Lexicographically earliest infinite sequence such that a(i) = a(j) => A328400(i) = A328400(j) for all i, j.

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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)).

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A108951 Primorial inflation of n: Fully multiplicative with a(p) = p# for prime p, where x# is the primorial A034386(x).

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A329600 Smallest number with the same set of distinct prime exponents as A108951(n).

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A297404 A binary representation of the positive exponents that appear in the prime factorization of a number, shown in decimal.

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A328569 Exponent of least prime factor in A276086(A276086(n)), where A276086 converts the primorial base expansion of n into its prime product form.

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