A290104 -id:A290104 - 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).

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)).

A290105 a(n) = A156061(n) / A290103(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 2, 1, 1, 1

Offset: 1

Antti Karttunen, Aug 13 2017

nonn

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

Cf. A156061, A290103, A290104, A290106.

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

(define (A290105 n) (/ (A156061 n) (A290103 n)))

a(n) = A156061(n) / A290103(n).

A316438 Heinz numbers of integer partitions whose product is strictly greater than the LCM of the parts.

Original entry on oeis.org

9, 18, 21, 25, 27, 36, 39, 42, 45, 49, 50, 54, 57, 63, 65, 72, 75, 78, 81, 84, 87, 90, 91, 98, 99, 100, 105, 108, 111, 114, 115, 117, 121, 125, 126, 129, 130, 133, 135, 144, 147, 150, 153, 156, 159, 162, 168, 169, 171, 174, 175, 180, 182, 183, 185, 189, 195

Offset: 1

Gus Wiseman, Jul 03 2018

nonn

Also numbers n > 1 such that A290104(n) > 1.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			Sequence of partitions whose product is greater than their LCM begins: (22), (221), (42), (33), (222), (2211), (62), (421), (322), (44), (331), (2221), (82), (422), (63), (22111), (332), (621), (2222), (4211).

Cf. A056239, A074761, A143773, A285572, A289508, A290103, A296150, A316429, A316431, A316433, A316437.

Select[Range[2,300],With[{pms=Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]},Times@@pms/LCM@@pms>1]&]

A330225 Position of first appearance of n in A290103 = LCM of prime indices.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 35, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271

Offset: 1

Gus Wiseman, Mar 26 2020

nonn

Appears to be the prime numbers (A000040) with 2 replaced by 1 and 37 replaced by 35.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

The version for product instead of lcm is A318871
The version for standard compositions is A333225.
The version for binary indices is A333492.
Let q(k) be the prime indices of k:
- The product of q(k) is A003963(k).
- The sum of q(k) is A056239(k).
- The terms of q(k) are row k of A112798.
- The GCD of q(k) is A289508(k).
- The LCM of q(k) is A290103(k).
- The LCM of q(k) + 1 is A328219(k).
Cf. A000837, A074761, A074971, A076078, A285572, A289509, A290104, A319333, A324837, A328451, A331579, A333226.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
q=Table[If[n==1,1,LCM@@primeMS[n]],{n,100}];
Table[Position[q,i][[1,1]],{i,First[Split[Union[q],#1+1==#2&]]}]

A316436 Sum divided by GCD of the integer partition with Heinz number n > 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 3, 2, 4, 1, 4, 1, 5, 5, 4, 1, 5, 1, 5, 3, 6, 1, 5, 2, 7, 3, 6, 1, 6, 1, 5, 7, 8, 7, 6, 1, 9, 4, 6, 1, 7, 1, 7, 7, 10, 1, 6, 2, 7, 9, 8, 1, 7, 8, 7, 5, 11, 1, 7, 1, 12, 4, 6, 3, 8, 1, 9, 11, 8, 1, 7, 1, 13, 8, 10, 9, 9, 1, 7, 4, 14, 1, 8, 10, 15, 6, 8, 1, 8, 5, 11, 13, 16, 11, 7, 1, 9, 9, 8, 1, 10, 1, 9, 9

Offset: 2

Gus Wiseman, Jul 03 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Cf. A056239, A289508, A289509, A290103, A290104, A296150, A316430, A316431, A316432, A316437.

a:= n-> (l-> add(i, i=l)/igcd(l[]))(map(i->
      numtheory[pi](i[1])$i[2], ifactors(n)[2])):
seq(a(n), n=2..100);  # Alois P. Heinz, Jul 03 2018

Table[With[{pms=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]},Total[pms]/GCD@@pms],{n,2,100}]

A316436(n) = { my(f = factor(n), pis = apply(p -> primepi(p), f[, 1]~), es = f[, 2]~, g = gcd(pis)); sum(i=1, #f~, pis[i]*es[i])/g; }; \\ Antti Karttunen, Sep 10 2018

More terms from Antti Karttunen, Sep 10 2018

A319329 Heinz numbers of integer partitions, whose length is equal to the GCD of the parts and whose sum is equal to the LCM of the parts, in increasing order.

Original entry on oeis.org

2, 1495, 179417, 231133, 727531, 1378583, 1787387, 3744103, 4556993, 7566167, 18977519, 29629391, 30870587, 34174939, 39973571, 53508983, 70946617, 110779141, 138820187, 139681069, 170583017, 225817751, 409219217, 441317981, 493580417, 539462099, 544392433, 712797613, 802903541

Offset: 1

Gus Wiseman, Sep 17 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The corresponding sequence of partitions, whose length is equal to their GCD and whose sum is equal to their LCM: (1), (9,6,3), (20,8,8,4), (24,16,4,4), (16,16,12,4).

Subsequence of A316430.

Cf. A056239, A067538, A074761, A143773, A289508, A289509, A290103, A290104, A316431, A316432, A319328, A319330, A319333.

Select[Range[2,10000],With[{m=If[#==1,{},Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]},And[LCM@@m==Total[m],GCD@@m==Length[m]]]&]

More terms from Max Alekseyev, Jul 25 2024

cp's OEIS Frontend

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

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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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A290105 a(n) = A156061(n) / A290103(n).

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A316438 Heinz numbers of integer partitions whose product is strictly greater than the LCM of the parts.

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A330225 Position of first appearance of n in A290103 = LCM of prime indices.

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A316436 Sum divided by GCD of the integer partition with Heinz number n > 1.

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A319329 Heinz numbers of integer partitions, whose length is equal to the GCD of the parts and whose sum is equal to the LCM of the parts, in increasing order.

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