A182860 -id:A182860 - cp's OEIS frontend

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)).

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

A238748 Numbers k such that each integer that appears in the prime signature of k appears an even number of times.

Original entry on oeis.org

1, 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194

Offset: 1

Matthew Vandermast, May 08 2014

nonn

Values of n for which all numbers in row A238747(n) are even. Also, numbers n such that A000005(n^m) is a perfect square for all nonnegative integers m; numbers n such that A181819(n) is a perfect square; numbers n such that A182860(n) is odd.

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 3, 33, 314, 3119, 31436, 315888, 3162042, 31626518, 316284320, 3162915907, ... . Apparently, the asymptotic density of this sequence exists and equals 0.3162... . - Amiram Eldar, Nov 28 2023

			The prime signature of 36 = 2^2 * 3^2 is {2,2}. One distinct integer (namely, 2) appears in the prime signature, and it appears an even number of times (2 times). Hence, 36 appears in the sequence.
The prime factorization of 1260 = 2^2 * 3^2 * 5^1 * 7^1. Exponent 2 occurs twice (an even number of times), as well as exponent 1, thus 1260 is included. It is also the first term k > 1 in this sequence for which A182850(k) = 4, not 3. - _Antti Karttunen_, Feb 06 2016

Antti Karttunen, Table of n, a(n) for n = 1..10000

Subsequences include A006881, A030229, A046386, A077448, A162142, A179690, A190108, A190465, A367590.
Subsequence of A000379, A028260, A063774 and A268390.
Cf. A000005, A181819, A182850, A182860, A238747.

q[n_] := n == 1 || AllTrue[Tally[FactorInteger[n][[;; , 2]]][[;; , 2]], EvenQ]; Select[Range[200], q] (* Amiram Eldar, Nov 28 2023 *)

is(n) = {my(e = factor(n)[, 2], m = #e); if(m%2, return(0)); e = vecsort(e); forstep(i = 1, m, 2, if(e[i] != e[i+1], return(0))); 1;} \\ Amiram Eldar, Nov 28 2023

(define A238748 (MATCHING-POS 1 1 (lambda (n) (square? (A181819 n)))))
(define (square? n) (not (zero? (A010052 n))))
;; Requires also MATCHING-POS macro from my IntSeq-library - Antti Karttunen, Feb 06 2016

A182862 Numbers k that set a record for the number of distinct prime signatures represented among their unitary divisors.

Original entry on oeis.org

1, 2, 6, 12, 60, 360, 1260, 2520, 27720, 138600, 360360, 831600, 10810800, 75675600, 183783600, 1286485200, 24443218800, 38594556000, 424540116000, 733296564000, 8066262204000, 185524030692000, 1693915062840000, 5380196890068000, 38960046445320000, 166786103592108000

Offset: 1

Matthew Vandermast, Jan 14 2011

nonn

In other words, the sequence includes k iff A182860(k) > A182860(m) for all m < k.

The records for the number of distinct prime signatures are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 20, 24, 32, 36, 40, 48, 60, 64, 72, 80, 96, ... (see the link for more values). - Amiram Eldar, Jul 07 2019

			60 has 8 unitary divisors (1, 3, 4, 5, 12, 15, 20 and 60). Primes 3 and 5 have the same prime signature, as do 12 (2^2*3) and 20 (2^2*5); each of the other four numbers listed is the only unitary divisor of 60 with its particular prime signature.  This makes a total of 6 distinct prime signatures that appear among the unitary divisors of 60.  Since no positive integer smaller than 60 has more than 4 distinct prime signatures appearing among its unitary divisors, 60 belongs to this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..60
Amiram Eldar, Table of n, a(n), A182860(a(n)) for n = 1..60
Eric Weisstein's World of Mathematics, Unitary Divisor

Subsequence of A025487, A129912, A181826, A182863. See also A034444, A085082, A182860, A182861.

f[1] = 1; f[n_] := Times @@ (Values[Counts[FactorInteger[n][[;; , 2]]]] + 1); fm = 0; s={}; Do[f1 = f[n]; If[f1 > fm, fm = f1; AppendTo[s, n]], {n, 1, 10^6}]; s (* Amiram Eldar, Jan 19 2019 *)

a(14)-a(26) from Amiram Eldar, Jan 19 2019

A212180 Number of distinct second signatures (cf. A212172) represented among divisors of n.

Original entry on oeis.org

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

Offset: 1

Matthew Vandermast, Jun 04 2012

nonn

Completely determined by the exponents >=2 in the prime factorization of n (cf. A212172, A212173).

The fraction of the divisors of n which have a given second signature {S} is also a function of n's second signature. For example, if n has second signature {3,2}, it follows that 1/3 of n's divisors are squarefree. Squarefree numbers are represented with 0's in A212172, in accord with the usual OEIS custom of using 0 for nonexistent elements; in comments, their second signature is represented as { }.

			The divisors of 72 represent a total of 5 distinct second signatures (cf. A212172), as can be seen from the exponents >= 2, if any, in the canonical prime factorization of each divisor:
{ }: 1, 2 (prime), 3 (prime), 6 (2*3)
{2}: 4 (2^2), 9 (3^2), 12 (2^2*3), 18 (2*3^2)
{3}: 8 (2^3), 24 (2^3*3)
{2,2}: 36 (2^2*3^2)
{3,2}: 72 (2^3*3^2)
Hence, a(72) = 5.

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

Cf. A212172, A085082, A088873, A181796, A182860, A212173, A212642, A212643, A212644.

Array[Length@ Union@ Map[Sort@ Select[FactorInteger[#][[All, -1]], # >= 2 &] &, Divisors@ #] &, 88] (* Michael De Vlieger, Jul 19 2017 *)

A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ This function from Charles R Greathouse IV, Aug 17 2011
A057521(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); } \\ This function from Charles R Greathouse IV, Aug 13 2013
A212173(n) = A046523(A057521(n));
A212180(n) = { my(vals = Set()); fordiv(n, d, vals = Set(concat(vals, A212173(d)))); length(vals); }; \\ Antti Karttunen, Jul 19 2017

from sympy import factorint, divisors, prod
def P(n): return sorted(factorint(n).values())
def a046523(n):
    x=1
    while True:
        if P(n)==P(x): return x
        else: x+=1
def a057521(n): return 1 if n==1 else prod(p**e for p, e in factorint(n).items() if e != 1)
def a212173(n): return a046523(a057521(n))
def a(n):
    l=[]
    for d in divisors(n):
        x=a212173(d)
        if not x in l:l+=[x, ]
    return len(l)
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jul 19 2017

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

A182861 Number of distinct prime signatures represented among the unitary divisors of A025487(n).

Original entry on oeis.org

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

Offset: 1

Matthew Vandermast, Jan 14 2011

nonn

a(n) = number of members m of A025487 such that d(m^k) divides d(A025487(n)^k) for all values of k. (Here d(n) represents the number of divisors of n, or A000005(n).)

			60 has 8 unitary divisors (1, 3, 4, 5, 12, 15, 20 and 60). Primes 3 and 5 have the same prime signature, as do 12 (2^2*3) and 20 (2^2*5); each of the other four numbers listed is the only unitary divisor of 60 with its particular prime signature.  Since a total of 6 distinct prime signatures appear among the unitary divisors of 60, and since 60 = A025487(13), a(13) = 6.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Unitary Divisor

Cf. A000005, A025487, A034444, A085082, A146289, A146290, A181820, A182860, A182862.

a(n) = A000005(A181820(n)).

If the canonical factorization of n into prime powers is Product p^e(p), then the formula for d(n^k) is Product_p (ek + 1). (See also A146289, A146290.)

More terms from Amiram Eldar, Jun 20 2019

A306491 Numbers that set a record for the number of distinct prime signatures represented among their divisors (A085082).

Original entry on oeis.org

1, 2, 4, 8, 12, 24, 48, 96, 144, 192, 288, 480, 576, 720, 1152, 1440, 2880, 4320, 5760, 8640, 11520, 17280, 34560, 51840, 69120, 103680, 120960, 138240, 207360, 241920, 345600, 362880, 414720, 483840, 604800, 725760, 829440, 967680, 1209600, 1451520, 1814400

Offset: 1

Amiram Eldar, Feb 19 2019

nonn

The records are 1, 2, 3, 4, 5, 7, 9, 11, 12, 13, 15, 16, 18, 19, 21, 24, 29, 30, 34, 37, 39, 44, 51, 52, 58, 61, 62, 65, 70, ... (see the link for more values).

Numbers that set a record for the number of divisors of least prime signature (A322584). - Hal M. Switkay, Aug 20 2024

Amiram Eldar, Table of n, a(n) for n = 1..533
Amiram Eldar, Table of n, a(n), A085082(a(n)) for n = 1..533

Subsequence of A025487.

Cf. A085082, A182860, A182862 (unitary version), A322584.

ps[1] = {}; ps[n_] := FactorInteger[n][[All, 2]] // Sort; a[n_] := ps /@ Divisors[n] // Union // Length; s={}; am=0; Do[a1=a[n]; If[a1>am, am=a1; AppendTo[s, n]], {n, 1, 2*10^6}]; s (* after Jean-François Alcover at A085082 *)

cp's OEIS Frontend

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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A238748 Numbers k such that each integer that appears in the prime signature of k appears an even number of times.

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A182862 Numbers k that set a record for the number of distinct prime signatures represented among their unitary divisors.

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A212180 Number of distinct second signatures (cf. A212172) represented among 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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A182861 Number of distinct prime signatures represented among the unitary divisors of A025487(n).

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A306491 Numbers that set a record for the number of distinct prime signatures represented among their divisors (A085082).

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