A328830 -id:A328830 - 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

A182860 Number of distinct prime signatures represented among the unitary divisors of 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, 4, 2, 2, 3, 3, 3, 3, 2, 3, 3, 4, 2, 4, 2, 4, 4, 3, 2, 4, 2, 4, 3, 4, 2, 4, 3, 4, 3, 3, 2, 6, 2, 3, 4, 2, 3, 4, 2, 4, 3, 4, 2, 4, 2, 3, 4, 4, 3, 4, 2, 4, 2, 3, 2, 6, 3, 3, 3, 4, 2, 6, 3, 4, 3, 3, 3, 4, 2, 4, 4, 3, 2, 4, 2, 4, 4

Offset: 1

Matthew Vandermast, Jan 14 2011

nonn

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

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

			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, a(60) = 6.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Unitary Divisor
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A034444, A064553, A085082, A146289, A146290, A182861, A182862, A212180, A328830.

Table[Length@ Union@ Map[Sort[FactorInteger[#] /. {p_, e_} /; p > 0 :> If[p == 1, 0, e]] &, Select[Divisors@ n, CoprimeQ[#, n/#] &]], {n, 105}] (* Michael De Vlieger, Jul 19 2017 *)

A181819(n) = {my(f=factor(n)); prod(k=1, #f~, prime(f[k, 2])); }; \\ From A181819
A182860(n) = numdiv(A181819(n)); \\ Antti Karttunen, Jul 19 2017

a(n) = A000005(A181819(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.)
a(n) = A064553(A328830(n)). - Antti Karttunen, Apr 29 2022

A353510 Square array A(n,k), n >= 1, k >= 0, with A(n,0) = n, and for k > 0, A(n,k) = A181819(A(n,k-1)), read by descending antidiagonals.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen and Gus Wiseman, Apr 27 2022

The row indexing of this array starts from 1, and the column indexing starts from 0, thus it is read by descending antidiagonals as A(1,0), A(1,1), A(2,0), A(1,2), A(2,1), A(3,0), etc.

A(n, k) gives the k-th prime shadow (the k-fold iterate of A181819) of n.

			The top left {0..6} x {1..16} corner of the array:
   1, 1, 1, 1, 1, 1, 1,
   2, 2, 2, 2, 2, 2, 2,
   3, 2, 2, 2, 2, 2, 2,
   4, 3, 2, 2, 2, 2, 2,
   5, 2, 2, 2, 2, 2, 2,
   6, 4, 3, 2, 2, 2, 2,
   7, 2, 2, 2, 2, 2, 2,
   8, 5, 2, 2, 2, 2, 2,
   9, 3, 2, 2, 2, 2, 2,
  10, 4, 3, 2, 2, 2, 2,
  11, 2, 2, 2, 2, 2, 2,
  12, 6, 4, 3, 2, 2, 2,
  13, 2, 2, 2, 2, 2, 2,
  14, 4, 3, 2, 2, 2, 2,
  15, 4, 3, 2, 2, 2, 2,
  16, 7, 2, 2, 2, 2, 2,

Michael De Vlieger, Table of n, a(n) for n = 1..11476 (rows n = 1..150, flattened)

This is a full square array version of irregular triangle A325239, which after 1, lists the terms on each row only up to the first 2.
Columns 0..2: A000027, A181819, A328830.
Rows 1..2: A000012, A007395.

f[n_] := If[n == 1, 1, Times @@ Prime[FactorInteger[n][[All, -1]]]]; Table[Function[m, Which[m == 1, a[1, k] = 1, k == 0, a[m, 0] = m, True, Set[a[m, k], f[a[m, k - 1]]]]][n - k + 1], {n, 0, 13}, {k, n, 0, -1}] // Flatten (* Michael De Vlieger, Apr 28 2022 *)

up_to = 105;
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A353510sq(n, k) = { while(k, n = A181819(n); k--); (n); };
A353510list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, forstep(col=a-1,0,-1, i++; if(i > up_to, return(v)); v[i] = A353510sq(a-col,col))); (v); };
v353510 = A353510list(up_to);
A353510(n) = v353510[n];

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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A182860 Number of distinct prime signatures represented among the unitary divisors of n.

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A353510 Square array A(n,k), n >= 1, k >= 0, with A(n,0) = n, and for k > 0, A(n,k) = A181819(A(n,k-1)), read by descending antidiagonals.

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