A284576 -id:A284576 - cp's OEIS frontend

A285106 a(n) = A001222(A284576(n)).

1, 2, 2, 3, 4, 5, 3, 4, 5, 7, 7, 6, 7, 8, 4, 5, 6, 8, 8, 9, 10, 10, 8, 8, 8, 10, 10, 9, 10, 11, 5, 6, 7, 9, 11, 10, 14, 16, 14, 13, 14, 13, 15, 17, 14, 16, 10, 10, 10, 11, 15, 14, 15, 17, 15, 13, 11, 13, 13, 12, 13, 14, 6, 7, 8, 10, 14, 13, 15, 19, 21, 17, 18, 20, 24, 25, 22, 20, 20, 16, 17, 19, 25, 23, 25, 30, 20, 22, 20, 20, 26, 26

Offset: 0

Antti Karttunen, Apr 11 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8192

Cf. A001222, A003986, A007306, A284576, A285107, A285108.

(define (A285106 n) (A001222 (A284576 n)))
;; A more practical version, needing only an implementation of A003986bi (bitwise-or, A003986) and memoization-macro definec:
(define (A285106 n) (apply + (bitwise_or_of_exp_lists (A260443as_coeff_list n) (A260443as_coeff_list (+ 1 n)))))
(define (bitwise_or_of_exp_lists nums1 nums2) (let ((len1 (length nums1)) (len2 (length nums2))) (cond ((< len1 len2) (bitwise_or_of_exp_lists nums2 nums1)) (else (map A003986bi nums1 (append nums2 (make-list (- len1 len2) 0)))))))
(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) = A001222(A284576(n)).
a(n) = A285107(n) + A285108(n).
Other identities. For all n >= 0:
A007306(1+n) = a(n) + A285108(n).

A059896 The set of Fermi-Dirac factors of A(n,k) is the union of the Fermi-Dirac factors of n and k. Symmetric square array read by antidiagonals.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 4, 6, 6, 4, 5, 8, 3, 8, 5, 6, 10, 12, 12, 10, 6, 7, 6, 15, 4, 15, 6, 7, 8, 14, 6, 20, 20, 6, 14, 8, 9, 8, 21, 24, 5, 24, 21, 8, 9, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 11, 10, 27, 8, 35, 6, 35, 8, 27, 10, 11, 12, 22, 30, 36, 40, 42, 42, 40, 36, 30, 22, 12, 13, 24

Offset: 1

Marc LeBrun, Feb 06 2001

Every positive integer, m, is the product of a unique subset, S(m), of the numbers listed in A050376 (primes, squares of primes etc.) The Fermi-Dirac factors of m are the members of S(m). So T(n,k) is the product of the members of (S(n) U S(k)).
Old name: Table a(i,j) = product prime(k)^(Ei(k) OR Ej(k)) where Ei and Ej are the vectors of exponents in the prime factorizations of i and j; OR is the bitwise operation on binary representation of the exponents.
Analogous to LCM, with OR replacing MAX.
A003418-analog seems to be A066616. - Antti Karttunen, Apr 12 2017
Considered as a binary operation, the result is the lowest common multiple of the squarefree parts of its operands multiplied by the square of the operation's result when applied to the square roots of the square parts of its operands. - Peter Munn, Mar 02 2022

			A(864,1944) = A(2^5*3^3,2^3*3^5) = 2^(5 OR 3) * 3^(3 OR 5) = 2^7*3^7 = 279936.
The top left 12 X 12 corner of the array:
   1,  2,  3,  4,  5,  6,  7,  8,   9,  10,  11,  12
   2,  2,  6,  8, 10,  6, 14,  8,  18,  10,  22,  24
   3,  6,  3, 12, 15,  6, 21, 24,  27,  30,  33,  12
   4,  8, 12,  4, 20, 24, 28,  8,  36,  40,  44,  12
   5, 10, 15, 20,  5, 30, 35, 40,  45,  10,  55,  60
   6,  6,  6, 24, 30,  6, 42, 24,  54,  30,  66,  24
   7, 14, 21, 28, 35, 42,  7, 56,  63,  70,  77,  84
   8,  8, 24,  8, 40, 24, 56,  8,  72,  40,  88,  24
   9, 18, 27, 36, 45, 54, 63, 72,   9,  90,  99, 108
  10, 10, 30, 40, 10, 30, 70, 40,  90,  10, 110, 120
  11, 22, 33, 44, 55, 66, 77, 88,  99, 110,  11, 132
  12, 24, 12, 12, 60, 24, 84, 24, 108, 120, 132,  12

Antti Karttunen, Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of the array
Eric Weisstein's World of Mathematics, Square Part.
Eric Weisstein's World of Mathematics, Squarefree Part.

Cf. A003418, A003990, A007947, A028233, A028234, A066616, A284576.
Sequences used in a definition of this sequence: A003986, A000188/A007913/A008833, A052330/A052331.
Has simple/very significant relationships with A003961, A059895/A059897, A225546, A267116.

a[i_, i_] := i;
a[i_, j_] := Module[{f1 = FactorInteger[i], f2 = FactorInteger[j], e1, e2}, e1[] = 0; Scan[(e1[#[[1]]] = #[[2]])&, f1]; e2[] = 0; Scan[(e2[#[[1]]] = #[[2]])&, f2]; Times @@ (#^BitOr[e1[#], e2[#]]& /@ Union[f1[[All, 1]], f2[[All, 1]]])];
Table[a[i - j + 1, j], {i, 1, 15}, {j, 1, i}] // Flatten (* Jean-François Alcover, Jun 19 2018 *)

A059896(n,k) = if(n==k, n, lcm(core(n),core(k)) * A059896(core(n,1)[2],core(k,1)[2])^2) \\ Peter Munn, Mar 07 2022

(define (A059896 n) (A059896bi (A002260 n) (A004736 n)))
(define (A059896bi a b) (let loop ((a a) (b b) (m 1)) (cond ((= 1 a) (* m b)) ((= 1 b) (* m a)) ((equal? (A020639 a) (A020639 b)) (loop (A028234 a) (A028234 b) (* m (expt (A020639 a) (A003986bi (A067029 a) (A067029 b)))))) ((< (A020639 a) (A020639 b)) (loop (/ a (A028233 a)) b (* m (A028233 a)))) (else (loop a (/ b (A028233 b)) (* m (A028233 b)))))))
;; Antti Karttunen, Apr 11 2017

From Antti Karttunen, Apr 11 2017: (Start)
A(x,y) = A059895(x,y) * A059897(x,y).
A(x,y) * A059895(x,y) = x*y.
(End).
From Peter Munn, Mar 02 2022: (Start)
OR denotes the bitwise operation (A003986).
Limited multiplicative property: if gcd(n_1*k_1, n_2*k_2) = 1 then A(n_1*n_2, k_1*k_2) = A(n_1, k_1) * A(n_2, k_2).
For prime p, A(p^e_1, p^e_2) = p^(e_1 OR e_2).
A(n, A(m, k)) = A(A(n, m), k).
A(n, k) = A(k, n).
A(n, 1) = A(n, n) = n.
A(n^2, k^2) = A(n, k)^2.
A(n, k) = A(A007913(n), A007913(k)) * A(A008833(n), A008833(k)) = lcm(A007913(n), A007913(k)) * A(A000188(n), A000188(k))^2.
A007947(A(n, k)) = A007947(n*k).
Isomorphism: A(A052330(n), A052330(k)) = A052330(n OR k).
Equivalently, A(n, k) = A052330(A052331(n) OR A052331(k)).
A(A003961(n), A003961(k)) = A003961(A(n, k)).
A(A225546(n), A225546(k)) = A225546(A(n, k)).
(End)

New name from Peter Munn, Mar 02 2022

A277324 Odd bisection of A260443 (the even terms): a(n) = A260443((2*n)+1).

Original entry on oeis.org

2, 6, 18, 30, 90, 270, 450, 210, 630, 6750, 20250, 9450, 15750, 47250, 22050, 2310, 6930, 330750, 3543750, 1653750, 4961250, 53156250, 24806250, 727650, 1212750, 57881250, 173643750, 18191250, 8489250, 25467750, 2668050, 30030, 90090, 40020750, 1910081250, 891371250, 9550406250, 455814843750, 212713593750

Offset: 0

Antti Karttunen, Oct 10 2016

nonn

From David A. Corneth, Oct 22 2016: (Start)
The exponents of the prime factorization of a(n) are first nondecreasing, then nonincreasing.
The exponent of 2 in the prime factorization of a(n) is 1. (End)

			A method to find terms of this sequence, explained by an example to find a(7). To find k = a(7), we find k such that A048675(k) = 2*7+1 = 15. 7 has the binary partitions: {[7, 0, 0], [5, 1, 0], [3, 2, 0], [1, 3, 0], [3, 0, 1], [1, 1, 1]}. To each of those, we prepend a 1. This gives the binary partitions of 15 starting with a 1. For example, for the first we get [1, 7, 0, 0]. We see that only [1, 5, 1, 0], [1, 3, 2, 0] and [1, 1, 1, 1] start nondecreasing, then nonincreasing, so we only check those. These numbers will be the exponents in a prime factorization. [1, 5, 1, 0] corresponds to prime(1)^1 * prime(2)^5 * prime(3)^1 * prime(4)^0 = 2430. We find that [1, 1, 1, 1] gives k = 210 for which A048675(k) = 15 so a(7) = 210. - _David A. Corneth_, Oct 22 2016

Antti Karttunen, Table of n, a(n) for n = 0..1024

Cf. A005811, A005940, A007306, A007949, A156552, A260443, A277189, A277323, A283484, A283975, A284267, A284268, A284563, A284564, A284573.

Cf. A277200 (same sequence sorted into ascending order).

a[n_] := a[n] = Which[n < 2, n + 1, EvenQ@ n, Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1] &@ a[n/2], True, a[#] a[# + 1] &[(n - 1)/2]]; Table[a[2 n + 1], {n, 0, 38}] (* Michael De Vlieger, Apr 05 2017 *)

from sympy import factorint, prime, primepi
from operator import mul
def a003961(n):
    F=factorint(n)
    return 1 if n==1 else reduce(mul, [prime(primepi(i) + 1)**F[i] for i in F])
def a260443(n): return n + 1 if n<2 else a003961(a260443(n//2)) if n%2==0 else a260443((n - 1)//2)*a260443((n + 1)//2)
def a(n): return a260443(2*n + 1)
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 21 2017

a(n) = A260443((2*n)+1).
a(0) = 2; for n >= 1, a(n) = A260443(n) * A260443(n+1).
Other identities. For all n >= 0:
A007949(a(n)) = A005811(n). [See comments in A125184.]
A156552(a(n)) = A277189(n), a(n) = A005940(1+A277189(n)).
A048675(a(n)) = 2n + 1. - David A. Corneth, Oct 22 2016
A001222(a(n)) = A007306(1+n).
A056169(a(n)) = A284267(n).
A275812(a(n)) = A284268(n).
A248663(a(n)) = A283975(n).
A000188(a(n)) = A283484(n).
A247503(a(n)) = A284563(n).
A248101(a(n)) = A284564(n).
A046523(a(n)) = A284573(n).
a(n) = A277198(n) * A284008(n).
a(n) = A284576(n) * A284578(n) = A284577(n) * A000290(A284578(n)).

More linking formulas added by Antti Karttunen, Apr 16 2017

A284577 a(n) = A059897(A260443(n), A260443(1+n)).

Original entry on oeis.org

2, 6, 2, 30, 90, 270, 2, 210, 630, 6750, 2250, 378, 15750, 47250, 2, 2310, 6930, 6750, 630, 66150, 198450, 3402, 90, 14850, 24750, 92610, 30870, 14850, 8489250, 25467750, 2, 30030, 90090, 6750, 339570, 14850, 382016250, 372093750, 9843750, 1400726250, 4202178750, 3402, 198450, 20465156250, 7796250, 83531250, 90, 859950, 1433250, 1890

Offset: 0

Antti Karttunen, Apr 11 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..1024

Cf. A059897, A260443, A277324, A284576, A284578, A285107.

(define (A284577 n) (A059897bi (A260443 n) (A260443 (+ 1 n)))) ;; For A059897bi, see A059897.

a(n) = A059897(A260443(n), A260443(1+n)).
a(n) = A277324(n) / A000290(A284578(n)).
A001222(a(n)) = A285107(n).

A284578 a(n) = A059895(A260443(n), A260443(1+n)).

Original entry on oeis.org

1, 1, 3, 1, 1, 1, 15, 1, 1, 1, 3, 5, 1, 1, 105, 1, 1, 7, 75, 5, 5, 125, 525, 7, 7, 25, 75, 35, 1, 1, 1155, 1, 1, 77, 75, 245, 5, 35, 147, 1, 1, 30625, 13125, 7, 245, 245, 40425, 11, 11, 13475, 1029, 245, 245, 1715, 1617, 11, 77, 1225, 3675, 385, 1, 1, 15015, 1, 1, 1001, 75, 245, 2695, 1715, 3, 1, 7, 48125, 7203, 1, 35, 300125, 363, 1, 1, 75625

Offset: 0

Antti Karttunen, Apr 11 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..1024

Cf. A000196, A059895, A260443, A277324, A284576, A284577, A285108.

(define (A284578 n) (A059895bi (A260443 n) (A260443 (+ 1 n)))) ;; For A059895bi, see A059895.

a(n) = A059895(A260443(n), A260443(1+n)).
a(n) = A277324(n) / A284576(n) = A000196(A277324(n)/A284577(n)).
a(n) = A284576(n) / A284577(n).
A001222(a(n)) = A285108(n).

cp's OEIS Frontend

A285106 a(n) = A001222(A284576(n)).

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A059896 The set of Fermi-Dirac factors of A(n,k) is the union of the Fermi-Dirac factors of n and k. Symmetric square array read by antidiagonals.

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A277324 Odd bisection of A260443 (the even terms): a(n) = A260443((2*n)+1).

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A284577 a(n) = A059897(A260443(n), A260443(1+n)).

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A284578 a(n) = A059895(A260443(n), A260443(1+n)).

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