A351944 -id:A351944 - 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

A351945 a(n) = A342001(A181819(n)).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 5, 1, 2, 2, 1, 1, 5, 1, 5, 2, 2, 1, 7, 1, 2, 1, 5, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 7, 1, 3, 1, 5, 5, 2, 1, 9, 1, 5, 2, 5, 1, 7, 2, 7, 2, 2, 1, 8, 1, 2, 5, 1, 2, 3, 1, 5, 2, 3, 1, 8, 1, 2, 5, 5, 2, 3, 1, 9, 1, 2, 1, 8, 2, 2, 2, 7, 1, 8, 2, 5, 2, 2, 2, 13, 1, 5, 5, 2, 1, 3, 1, 7, 3

Offset: 1

Antti Karttunen, Apr 02 2022

nonn

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

Cf. A003415, A003557, A181819, A342001, A351942, A351954 [= a(A276076(n))].

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A342001(n) = (A003415(n) / A003557(n));
A351945(n) = A342001(A181819(n));

a(n) = A342001(A181819(n)) = A351942(n) / A351944(n).

A351942 Arithmetic derivative of A181819(n), where A181819(n) = Product prime(e(i)) when n = Product prime(i)^e(i).

Original entry on oeis.org

0, 1, 1, 1, 1, 4, 1, 1, 1, 4, 1, 5, 1, 4, 4, 1, 1, 5, 1, 5, 4, 4, 1, 7, 1, 4, 1, 5, 1, 12, 1, 1, 4, 4, 4, 6, 1, 4, 4, 7, 1, 12, 1, 5, 5, 4, 1, 9, 1, 5, 4, 5, 1, 7, 4, 7, 4, 4, 1, 16, 1, 4, 5, 1, 4, 12, 1, 5, 4, 12, 1, 8, 1, 4, 5, 5, 4, 12, 1, 9, 1, 4, 1, 16, 4, 4, 4, 7, 1, 16, 4, 5, 4, 4, 4, 13, 1, 5, 5, 6, 1, 12

Offset: 1

Antti Karttunen, Apr 02 2022

nonn

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

Cf. A003415, A181819, A351944, A351945.

Coincides with A351943 on the positions given by A130091.

{0}~Join~Array[If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &[Apply[Times, Prime[FactorInteger[#][[All, -1]]]]] &, 101, 2] (* Michael De Vlieger, Apr 03 2022 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A351942(n) = A003415(A181819(n));

a(n) = A003415(A181819(n)).

a(n) = A351944(n) * A351945(n).

A295879 Multiplicative with a(p) = 1, a(p^e) = prime(e-1) if e > 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 5, 5, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 7, 1, 2, 2, 4, 1, 1, 1, 3, 1, 1, 1, 6, 1, 1, 1, 5, 1, 1, 1, 2, 2, 1, 1, 3, 2, 1, 1, 2, 3, 2, 1, 13

Offset: 1

Antti Karttunen, Nov 29 2017

This sequence can be used as a filter. It matches at least to the following sequences related to the counting of various non-unitary prime divisors:
For all i, j:
  a(i) = a(j) => A056170(i) = A056170(j), as A056170(n) = A001222(a(n)).
  a(i) = a(j) => A162641(i) = A162641(j).
  a(i) = a(j) => A295659(i) = A295659(j).
  a(i) = a(j) => A295662(i) = A295662(j).
  a(i) = a(j) => A295883(i) = A295883(j), as A295883(n) = A007949(a(n)).
  a(i) = a(j) => A295884(i) = A295884(j).
An encoding of the prime signature of A057521(n), the powerful part of n. - Peter Munn, Apr 06 2024

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

Cf. A003557, A008578, A057521, A064989, A181819.
Cf. also A293515, A294875, A294895, A294897, A295878, A351944.
Differs from A000688 for the first time at n=128, where a(128) = 13, while A000688(128) = 15.

Array[Apply[Times, FactorInteger[#] /. {p_, e_} /; p > 0 :> Which[p == 1, 1, e == 1, 1, True, Prime[e - 1]]] &, 128] (* Michael De Vlieger, Nov 29 2017 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == 1, 1, prime(f[i,2]-1)));} \\ Amiram Eldar, Nov 18 2022

a(1) = 1; for n>1, if n = Product prime(i)^e(i), then a(n) = Product A008578(e(i)).
a(n) = A064989(A181819(n)).
a(n) = A181819(A003557(n)). - Antti Karttunen, Apr 03 2022
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/p^2  + Sum_{k>=1} (prime(k+1)-prime(k))/p^(k+2)) = 2.208... . - Amiram Eldar, Nov 18 2022

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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A351945 a(n) = A342001(A181819(n)).

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A351942 Arithmetic derivative of A181819(n), where A181819(n) = Product prime(e(i)) when n = Product prime(i)^e(i).

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A295879 Multiplicative with a(p) = 1, a(p^e) = prime(e-1) if e > 1.

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A353524 A003557 applied to the prime shadow of primorial base exp-function: a(n) = A003557(A181819(A276086(n))).

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