A290105 -id:A290105 - cp's OEIS frontend

A290103 a(n) = LCM of the prime indices in prime factorization of n, a(1) = 1.

1, 1, 2, 1, 3, 2, 4, 1, 2, 3, 5, 2, 6, 4, 6, 1, 7, 2, 8, 3, 4, 5, 9, 2, 3, 6, 2, 4, 10, 6, 11, 1, 10, 7, 12, 2, 12, 8, 6, 3, 13, 4, 14, 5, 6, 9, 15, 2, 4, 3, 14, 6, 16, 2, 15, 4, 8, 10, 17, 6, 18, 11, 4, 1, 6, 10, 19, 7, 18, 12, 20, 2, 21, 12, 6, 8, 20, 6, 22, 3, 2, 13, 23, 4, 21, 14, 10, 5, 24, 6, 12, 9, 22, 15, 24, 2, 25, 4, 10, 3, 26, 14, 27, 6, 12

Offset: 1

Antti Karttunen, Aug 13 2017

nonn

			Here primepi (A000720) gives the index of its prime argument:
n = 14 = 2 * 7, thus a(14) = lcm(primepi(2), primepi(7)) = lcm(1,4) = 4.
n = 21 = 3 * 7, thus a(21) = lcm(primepi(3), primepi(7)) = lcm(2,4) = 4.

Cf. A000040, A000720, A003963, A007947, A156061, A290104, A290105.

Cf. also A072411, A181819.

Table[Apply[LCM, PrimePi[FactorInteger[n][[All, 1]] ]] + Boole[n == 1], {n, 105}] (* Michael De Vlieger, Aug 14 2017 *)

a(n)=if(n>1, lcm(apply(primepi, factor(n)[,1])), 1) \\ Charles R Greathouse IV, Nov 11 2021

(define (A290103 n) (if (= 1 n) n (lcm (A055396 n) (A290103 (A028234 n))))) ;; Antti Karttunen, Aug 13 2017

a(1) = 1; for n > 1, a(n) = lcm(A055396(n), a(A028234(n))).
Other identities. For all n >= 1:
a(A007947(n)) = a(n).
a(A181819(n)) = A072411(n).

A156061 a(n) = product of indices of distinct prime factors of n, where index(prime(k)) = k.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 1, 2, 3, 5, 2, 6, 4, 6, 1, 7, 2, 8, 3, 8, 5, 9, 2, 3, 6, 2, 4, 10, 6, 11, 1, 10, 7, 12, 2, 12, 8, 12, 3, 13, 8, 14, 5, 6, 9, 15, 2, 4, 3, 14, 6, 16, 2, 15, 4, 16, 10, 17, 6, 18, 11, 8, 1, 18, 10, 19, 7, 18, 12, 20, 2, 21, 12, 6, 8, 20, 12, 22, 3, 2, 13, 23, 8, 21, 14, 20, 5, 24, 6, 24, 9, 22, 15, 24, 2, 25, 4, 10, 3

Offset: 1

Ctibor O. Zizka, Feb 03 2009

a(n) = the product of the distinct parts of the partition with Heinz number n. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436. Example: a(252)= 8; indeed, the partition having Heinz number 252 = 2*2*3*3*7 is [1,1,2,2,4] and 1*2*4 = 8. - Emeric Deutsch, Jun 03 2015

Multiplicative with a(prime(k)^e) = k. Note that in contrast to A003963, this is not fully multiplicative. a(1) = 1 as an empty product. - Antti Karttunen, Aug 13 2017

			Here primepi (A000720) gives the index of its prime argument:
n = 14 = 2 * 7, thus a(14) = primepi(2)*primepi(7) = 1*4 = 4.
n = 21 = 3 * 7, thus a(21) = primepi(3)*primepi(7) = 2*4 = 8.
n = 168 = 2^3 * 3 * 7, thus a(168)= primepi(2)*primepi(3)*primepi(7) = 1*2*4 = 8.

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

Cf. A000040, A000720, A007947.
Cf. also A003963, A290105, A290106, A290107.
Differs from related A290103 for the first time at n=21.

with(numtheory): a := proc(n) options operator, arrow: product(pi(factorset(n)[j]), j = 1 .. nops(factorset(n))) end proc: seq(a(n), n = 1 .. 100);  #  Emeric Deutsch, Jun 03 2015

Table[Apply[Times, PrimePi@ FactorInteger[n][[All, 1]]] + Boole[n == 1], {n, 100}] (* Michael De Vlieger, Aug 14 2017 *)

a(n) = {my(f=factor(n)); for (k=1, #f~, f[k,1] = primepi(f[k,1]); f[k,2] = 1); factorback(f);} \\ Michel Marcus, Aug 14 2017

(define (A156061 n) (if (= 1 n) 1 (* (A055396 n) (A156061 (A028234 n))))) ;; Antti Karttunen, Aug 13 2017

From Antti Karttunen, Aug 13 2017: (Start)
a(1) = 1; for n > 1, a(n) = A055396(n) * a(A028234(n)).
a(n) = A003963(A007947(n)) = a(A007947(n)).
a(n) = A003963(n) / A290106(n) = A290103(n) * A290105(n).
a(A181819(n)) = A290107(n).
(End)

a(1) = 1 prepended by Antti Karttunen, Aug 13 2017

A290106 a(1) = 1; for n > 1, if n = Product prime(k)^e(k), then a(n) = Product (k)^(e(k)-1).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 2, 3, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1

Offset: 1

Antti Karttunen, Aug 13 2017

			For n = 21 = 3*7 = prime(2)^1 * prime(4)^1, a(n) = 2^0 * 4^0 = 1*1 = 1.
For n = 360 = 2^3 * 3^2 * 5^1 = prime(1)^3 * prime(2)^2 * prime(3)^1, a(n) = 1^2 * 2^1 * 3^0 = 1*2*1 = 2.

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

Cf. A003557, A003963, A007947, A156061, A290105.

Differs from A290104 for the first time at n=21.

Table[If[n == 1, 1, Apply[Times, Map[PrimePi[#1]^#2 & @@ # &, #]] / Apply[Times, PrimePi[#[[All, 1]] ]]] &@ FactorInteger@ n, {n, 120}] (* Michael De Vlieger, Aug 14 2017 *)

(define (A290106 n) (/ (A003963 n) (A156061 n)))
(define (A290106 n) (if (= 1 n) 1 (* (expt (A055396 n) (- (A067029 n) 1)) (A290106 (A028234 n)))))

Multiplicative with a(prime(k)^e) = k^(e-1).
a(n) = A003963(n) / A156061(n).
a(n) = A003963(A003557(n)) = A003963(n/A007947(n)).

A290104 a(n) = A003963(n) / A290103(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 3, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 1, 4, 1, 1, 2, 1, 1, 1, 1, 1, 4, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 8, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 4, 2, 3, 1, 1, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 3, 1, 4, 1, 1, 1

Offset: 1

Antti Karttunen, Aug 13 2017

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k). Then a(n) is the product divided by the LCM of the integer partition with Heinz number n. - Gus Wiseman, Aug 01 2018

			n = 21 = 3 * 7 = prime(2) * prime(4), thus A003963(21) = 2*4 = 8, while A290103(21) = lcm(2,4) = 4, so a(21) = 8/4 = 2.

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

Differs from A290106 for the first time at n=21.

Cf. A003963, A056239, A074761, A289509, A290103, A290105, A296150, A316429, A316431.

Table[If[n == 1, 1, Apply[Times, Map[PrimePi[#1]^#2 & @@ # &, #]] / Apply[LCM, PrimePi[#[[All, 1]] ]]] &@ FactorInteger@ n, {n, 120}] (* Michael De Vlieger, Aug 14 2017 *)

(define (A290104 n) (/ (A003963 n) (A290103 n)))

a(n) = A003963(n) / A290103(n).
Other identities. For all n >= 1:
a(A181819(n)) = A005361(n)/A072411(n).

A323082 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = -(n mod 2) if n is a prime, and f(n) = A300840(n) for any other number.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 4, 6, 7, 3, 8, 3, 9, 10, 11, 3, 6, 3, 12, 13, 14, 3, 8, 15, 16, 17, 18, 3, 10, 3, 11, 19, 20, 21, 22, 3, 23, 24, 12, 3, 13, 3, 25, 26, 27, 3, 28, 29, 15, 30, 31, 3, 17, 32, 18, 33, 34, 3, 35, 3, 36, 37, 38, 39, 19, 3, 40, 41, 21, 3, 22, 3, 42, 43, 44, 45, 24, 3, 46, 47, 48, 3, 49, 50, 51, 52, 25, 3, 26, 53, 54, 55, 56, 57, 28, 3, 29, 58, 59, 3, 30, 3, 31

Offset: 1

Antti Karttunen, Jan 04 2019

nonn

For all i, j: A323074(i) = A323074(j) => a(i) = a(j).
Like the related A322822 also this filter sequence satisfies the following two implications, for all i, j >= 1:
  a(i) = a(j) => A322356(i) = A322356(j),
  a(i) = a(j) => A290105(i) = A290105(j).

Antti Karttunen, Table of n, a(n) for n = 1..65539

Cf. A052330, A300840, A305801, A322822, A323074.

Cf. also A290105, A322356.

up_to = 65537;
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; };
ispow2(n) = (n && !bitand(n,n-1));
A302777(n) = ispow2(isprimepower(n));
A050376list(up_to) = { my(v=vector(up_to), i=0); for(n=1,oo,if(A302777(n), i++; v[i] = n); if(i == up_to,return(v))); };
v050376 = A050376list(up_to);
A050376(n) = v050376[n];
A052330(n) = { my(p=1,i=1); while(n>0, if(n%2, p *= A050376(i)); i++; n >>= 1); (p); };
A052331(n) = { my(s=0,e); while(n > 1, fordiv(n, d, if(((n/d)>1)&&A302777(n/d), e = vecsearch(v050376, n/d); if(!e, print("v050376 too short!"); return(1/0)); s += 2^(e-1); n = d; break))); (s); };
A300840(n) = A052330(A052331(n)>>1);
A323082aux(n) = if(isprime(n),-(n%2),A300840(n));
v323082 = rgs_transform(vector(up_to,n,A323082aux(n)));
A323082(n) = v323082[n];

cp's OEIS Frontend

A290103 a(n) = LCM of the prime indices in prime factorization of n, a(1) = 1.

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A156061 a(n) = product of indices of distinct prime factors of n, where index(prime(k)) = k.

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A290106 a(1) = 1; for n > 1, if n = Product prime(k)^e(k), then a(n) = Product (k)^(e(k)-1).

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A290104 a(n) = A003963(n) / A290103(n).

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A323082 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = -(n mod 2) if n is a prime, and f(n) = A300840(n) for any other number.

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