A072411 -id:A072411 - cp's OEIS frontend

A290107 a(1) = 1; for n > 1, a(n) = product of distinct exponents in the prime factorization of n.

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

Offset: 1

Antti Karttunen, Aug 13 2017

nonn

			For n = 36 = 2^2 * 3^2, the only distinct exponent that occurs is 2, thus a(36) = 2.
For n = 144 = 2^4 * 3^2, the distinct exponents are 2 and 4, thus a(144) = 2*4 = 8.
For n = 4500 = 2^2 * 3^2 * 5^3, the distinct exponents are 2 and 3, thus a(4500) = 2*3 = 6.

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

Cf. A156061, A181819.
Differs from A005361 for the first time at n=36.
Differs from A072411 for the first time at n=144, and also from A157754 for the second time (after the initial term).

Table[If[n == 1, 1, Apply[Times, Union[FactorInteger[n][[All, -1]] ]]], {n, 120}] (* Michael De Vlieger, Aug 14 2017 *)

A290107(n) = factorback(vecsort((factor(n)[, 2]), ,8));

(define (A290107 n) (A156061 (A181819 n)))

a(n) = A156061(A181819(n)).

A277326 LCM of nonzero coefficients of the n-th Stern polynomial.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 3, 1, 2, 1, 1, 1, 2, 2, 3, 2, 12, 3, 6, 1, 6, 2, 3, 1, 2, 1, 1, 1, 2, 2, 6, 2, 20, 3, 12, 2, 4, 12, 30, 3, 20, 6, 6, 1, 6, 6, 15, 2, 60, 3, 12, 1, 6, 2, 3, 1, 2, 1, 1, 1, 2, 2, 6, 2, 60, 6, 12, 2, 12, 20, 40, 3, 140, 12, 12, 2, 12, 4, 40, 12, 60, 30, 140, 3, 60, 20, 40, 6, 20, 6, 6, 1

Offset: 0

Antti Karttunen, Oct 13 2016

nonn

a(n) = the least common multiple of nonzero terms on the n-th row of A125184.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for sequences related to lcm's

Cf. A072411, A125184, A260443.
Differs from A277325 for the first time at n=13, where a(13) = 2, while A277325(13) = 4.
After n=0, differs from A277315 for the first time at n=21, where a(21) = 12, while A277315(21) = 4.

(define (A277326 n) (A072411 (A260443 n)))
;; A standalone implementation:
(define (A277326 n) (reduce lcm 1 (filter positive? (A260443as_coeff_list n))))
(definec (A260443as_coeff_list n) (cond ((zero? n) (list)) ((= 1 n) (list 1)) ((even? n) (cons 0 (A260443as_coeff_list (/ n 2)))) (else (add_two_lists (A260443as_coeff_list (/ (- n 1) 2)) (A260443as_coeff_list (/ (+ n 1) 2))))))
(define (add_two_lists nums1 nums2) (let ((len1 (length nums1)) (len2 (length nums2))) (cond ((< len1 len2) (add_two_lists nums2 nums1)) (else (map + nums1 (append nums2 (make-list (- len1 len2) 0)))))))

a(n) = A072411(A260443(n)).
a(2n) = a(n).
a(n) <= A277325(n).

A284569 a(n) = LCM of the lengths of runs of 1-bits in binary representation of n.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Apr 14 2017

			For n = 27, in binary A007088(27) = "11011", the lengths of runs of 1-bits are [2,2], thus a(27) = lcm(2,2) = 2.
For n = 55, in binary A007088(55) = "110111", the lengths of runs of 1-bits are [2,3], thus a(55) = lcm(2,3) = 6.

Antti Karttunen, Table of n, a(n) for n = 0..10922
Index entries for sequences related to binary expansion of n

Cf. A005940, A072411, A227349, A284562, A284559.
Cf. A003714 (positions of ones).
Differs from A227349 for the first time at n=27, where a(27)=2, while A227349(27)= 4.
Differs from A038374 for the first time at n=55, where a(55) = 6, while A038374(55) = 3.

(define (A284569 n) (apply lcm (bisect (reverse (binexp->runcount1list n)) (- 1 (modulo n 2))))) ;; For bisect and binexp->runcount1list, see the Program section of A227349.
(define (A284569 n) (A072411 (A005940 (+ 1 n))))

a(n) = A072411(A005940(1+n)).

a(n) = A227349(n) / A284562(n).

A072414 Non-Achilles numbers for which LCM of the exponents in the prime factorization of n is not equal to the maximum of the same exponents.

Original entry on oeis.org

360, 504, 540, 600, 756, 792, 936, 1176, 1188, 1224, 1350, 1368, 1400, 1404, 1440, 1500, 1656, 1836, 1960, 2016, 2052, 2088, 2160, 2200, 2232, 2250, 2400, 2484, 2520, 2600, 2646, 2664, 2904, 2952, 3024, 3096, 3132, 3168, 3240, 3348, 3384, 3400, 3500

Offset: 1

Labos Elemer, Jun 17 2002

nonn

Most members of this sequence fail to be Achilles numbers because they have at least one prime factor with multiplicity 1. There are also numbers in the sequence that fail to be Achilles numbers because they are perfect powers: these are precisely the proper powers of members of A072412, so the smallest such is 5184 = 2^6*3^4 = 72^2. - Franklin T. Adams-Watters, Oct 09 2006

			m = 504 = 2*2*2*3*3*7: exponent-set = E = {3,2,1}, max(E) = 3 < lcm(E) = 6, gcd(E) = min(E) = 1.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A005361, A051903, A051904, A052409, A072411, A072412, A052486.

Select[Range@ 3500, And[LCM @@ # != Max@ #, GCD @@ # == Min@ # == 1] &[FactorInteger[#][[All, -1]] ] &] (* Michael De Vlieger, Jul 18 2017 *)

is(n)=my(f=factor(n)[,2]); n>9 && lcm(f)!=vecmax(f) && (#f==1 || vecmin(f)<2) \\ Charles R Greathouse IV, Oct 16 2015

A051903(a(n)) is not equal A072411(a(n)) but the numbers are not in A052486.

A328582 Least common multiple of nonzero digits in primorial base expansion of n.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Oct 21 2019

nonn

a(0) = 1 as an empty product.

Cf. A072411, A276086, A328581.

Cf. A276156 (positions of 1's).

A328582(n) = { my(m=1, p=2); while(n, if(n%p, m = lcm(m,n%p)); n = n\p; p = nextprime(1+p)); (m); };

a(n) = A072411(A276086(n)).

A329378 Least common multiple of exponents of prime factors of A108951(n), where A108951 is fully multiplicative with a(prime(i)) = prime(i)# = Product_{i=1..i} A000040(i).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 17 2019

nonn

Cf. A019565, A034386, A072411, A108951, A284002, A329382, A329600, A329605.

Differs from related A329617 for the first time at n=36.

A034386(n) = prod(i=1, primepi(n), prime(i));
A072411(n) = lcm(factor(n)[, 2]); \\ From A072411
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A329378(n) = A072411(A108951(n));

a(n) = A072411(A108951(n)) = A072411(A329600(n)).
a(n) <= A329617(n) <= A329382(n) <= A329605(n).
a(A019565(n)) = A284002(n).

A072413 Numbers k such that the LCM of exponents in the prime factorization of k does not equal the product of the exponents.

Original entry on oeis.org

36, 100, 144, 180, 196, 216, 225, 252, 300, 324, 396, 400, 441, 450, 468, 484, 576, 588, 612, 676, 684, 700, 720, 784, 828, 882, 900, 980, 1000, 1008, 1044, 1080, 1089, 1100, 1116, 1156, 1200, 1225, 1260, 1296, 1300, 1332, 1444, 1452, 1476, 1512, 1521

Offset: 1

Labos Elemer, Jun 17 2002

nonn

The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 2, 29, 348, 3548, 35761, 358258, 3583892, 35843109, 358440763, ... . Apparently, the asymptotic density of this sequence exists and equals 0.03584... . - Amiram Eldar, Sep 09 2022

			k = 36 = 2*2*3*3; exponent set = {2,2}; LCM = 2, product = 4.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A005361, A051903, A051904, A052409, A052486, A072411-A072414.

Select[Range@ 1600, LCM @@ # != Times @@ # &@ Map[Last, FactorInteger@ #] &] (* Michael De Vlieger, May 15 2016 *)

is(n)=my(f=factor(n)[,2]); n>9 && lcm(f)!=factorback(f) \\ Charles R Greathouse IV, Jan 14 2017

A005361(a(n)) != A072411(a(n)).

A157754 a(1) = 0, a(n) = lcm(A051904(n), A051903(n)) for n >= 2.

Original entry on oeis.org

0, 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, 2, 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, 6, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 5, 1, 2, 2, 2

Offset: 1

Jaroslav Krizek, Mar 05 2009

nonn

a(n) for n >= 2 equals LCM of minimum and maximum exponents in the prime factorization of n.

a(n) for n >= 2 deviates from A072411, first different term is a(360), a(360) = 3, A072411(360) = 6.

			For n = 12 = 2^2 * 3^1 we have a(12) = lcm(2,1) = 2.
For n = 144 = 2^4 * 3^2 we have a(144) = lcm(4,2) = 4.

Cf. A000040, A003990, A006881, A033150, A120944, A000961, A000027, A072411, A051904, A051903, A158378.

Table[LCM @@ {Min@ #, Max@ #} - Boole[n == 1] &@ FactorInteger[n][[All, -1]], {n, 100}] (* Michael De Vlieger, Jul 12 2017 *)

a(n) = if(n == 1, 0, my(e = factor(n)[,2]); lcm(vecmin(e), vecmax(e))); \\ Amiram Eldar, Sep 11 2024

a(1) = 0, a(p) = 1, a(pq) = 1, a(pq...z) = 1, a(p^k) = k, for p = primes (A000040), pq = product of two distinct primes (A006881), pq...z = product of k (k > 2) distinct primes p, q, ..., z (A120944), p^k = prime powers (A000961(n) for n > 1) k = natural numbers (A000027).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A033150. - Amiram Eldar, Sep 11 2024

A072412 Numbers k such that the LCM of exponents in the prime factorization of k does not equal the largest exponent.

Original entry on oeis.org

72, 108, 200, 288, 360, 392, 432, 500, 504, 540, 600, 648, 675, 756, 792, 800, 864, 936, 968, 972, 1125, 1152, 1176, 1188, 1224, 1323, 1350, 1352, 1368, 1372, 1400, 1404, 1440, 1500, 1568, 1656, 1800, 1836, 1944, 1960, 2000, 2016, 2052, 2088, 2160

Offset: 1

Labos Elemer, Jun 17 2002

nonn

This sequence differs from the Achilles numbers (A052486).

			k = 360 = 2*2*2*3*3*5, exponent set = {3,2,1}; LCM=6, max=3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005361, A051903, A051904, A052409, A052486, A072411, A072413, A072414.

Select[Range[2000], LCM @@ (e = FactorInteger[#][[;; , 2]]) != Max[e] &] (* Amiram Eldar, Jul 30 2022 *)

is(n)=my(f=factor(n)[,2]); n>9 && vecmax(f)!=lcm(f) \\ Charles R Greathouse IV, Oct 16 2015

A051903(a(n)) != A072411(a(n)).

A273058 Numbers having pairwise coprime exponents in their canonical prime factorization.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70

Offset: 1

Giuseppe Coppoletta, May 14 2016

nonn

The complement of A072413.

			36 is not a term because 36 = 2^2 * 3^2 and gcd(2,2) = 2 > 1.
360 is a term because 360 = 2^3 * 3^2 * 5 and gcd(3,2) = gcd(2,1) = 1.
10800 is not a term because 10800 = 2^4 * 3^3 * 5^2 and gcd(4,2) > 1

Giuseppe Coppoletta, Table of n, a(n) for n = 1..10000

Cf. A005361, A072411, A130091, A072413.

Select[Range@ 120, LCM @@ # == Times @@ # &@ Map[Last, FactorInteger@ #] &] (* Michael De Vlieger, May 15 2016 *)

is(n)=my(f=factor(n)[,2]); factorback(f)==lcm(f) \\ Charles R Greathouse IV, Jan 14 2017

def d(n):
    v=factor(n)[:]; L=len(v); diff=prod(v[j][1] for j in range(L)) - lcm([v[j][1] for j in range(L)])
    return diff
[k for k in (1..100) if d(k)==0]

A005361(a(n)) = A072411(a(n)).

cp's OEIS Frontend

A290107 a(1) = 1; for n > 1, a(n) = product of distinct exponents in the prime factorization of n.

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A277326 LCM of nonzero coefficients of the n-th Stern polynomial.

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A284569 a(n) = LCM of the lengths of runs of 1-bits in binary representation of n.

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A072414 Non-Achilles numbers for which LCM of the exponents in the prime factorization of n is not equal to the maximum of the same exponents.

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A328582 Least common multiple of nonzero digits in primorial base expansion of n.

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A329378 Least common multiple of exponents of prime factors of A108951(n), where A108951 is fully multiplicative with a(prime(i)) = prime(i)# = Product_{i=1..i} A000040(i).

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A072413 Numbers k such that the LCM of exponents in the prime factorization of k does not equal the product of the exponents.

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A157754 a(1) = 0, a(n) = lcm(A051904(n), A051903(n)) for n >= 2.

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