A273671 -id:A273671 - cp's OEIS frontend

A278261 a(n) = A046523(A273671(n)).

1, 8, 2, 8, 24, 24, 2, 64, 24, 24, 192, 72, 24, 24, 4, 8, 24, 96, 24, 72, 96, 24, 72, 192, 24, 3456, 192, 24, 3456, 24, 2, 216, 192, 24, 1080, 72, 24, 8640, 576, 192, 8640, 3456, 24, 1080, 3456, 192, 1080, 120, 72, 96, 120, 24, 96, 360, 72, 576, 360, 24, 192, 120, 24, 72, 6, 8, 24, 1080, 24, 5400, 8640, 24, 72, 1080, 24, 432000, 8640, 24, 3456, 12288, 24, 120

Offset: 1

Antti Karttunen, Nov 16 2016

nonn

This sequence works as a "sentinel" for A273671 by matching to any sequence that is obtained as f(A273671(n)), where f(n) is any function that depends only on the prime signature of n (see the index entry for "sequences computed from exponents in ..."). The only other sequence that as of Nov 11 2016 seems to match is A106347, although more terms of the latter would be needed to better ascertain whether the connection is spurious or genuine.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n

Cf. A046523, A273671, A278243.

Sequences that seem to partition N into same or coarser equivalence classes: A106347

(define (A278261 n) (A046523 (A273671 n)))

a(n) = A046523(A273671(n)).

A270419 Denominator of the rational number obtained when the exponents in prime factorization of n are reinterpreted as alternating binary sums (A065620).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 23 2016

Map n -> A270418(n)/A270419(n) is a bijection from N (1, 2, 3, ...) to the set of positive rationals.

Antti Karttunen, Table of n, a(n) for n = 1..12167

Cf. A270418 (gives the numerators).
Cf. A270428 (indices of ones).
Cf. A000069, A001969, A010059, A020639, A028234, A065620, A067029.
Cf. also A270420, A270421, A270436, A270437 and permutation pair A273671/A273672.
Differs from A055229 for the first time at n=32, where a(32)=8, while A055229(32)=2.

s[n_] := s[n] = If[OddQ[n], -2*s[(n - 1)/2] - 1, 2*s[n/2]]; s[0] = 0; f[p_, e_] := p^If[OddQ[DigitCount[e, 2, 1]], 0, s[e]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 01 2023 *)

A270419(n)={n=factor(n);n[,2]=apply(A065620,n[,2]);denominator(factorback(n))} \\ M. F. Hasler, Apr 16 2018

Multiplicative with a(p^e) = p^(-A065620(e)) for evil e, a(p^e)=1 for odious e, or equally, a(p^e) = p^(A010059(e) * -A065620(e)).
a(1) = 1, for n > 1, a(n) = a(A028234(n)) * A020639(n)^( A010059(A067029(n)) * -A065620(A067029(n)) ).
Other identities. For all n >= 1:
a(A270436(n)) = 1, a(A270437(n)) = n.

A270418 Numerator of the rational number obtained when exponents in prime factorization of n are reinterpreted as alternating binary sums (A065620).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 1, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 3, 25, 26, 1, 28, 29, 30, 31, 1, 33, 34, 35, 36, 37, 38, 39, 5, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 2, 55, 7, 57, 58, 59, 60, 61, 62, 63, 1, 65, 66, 67, 68, 69, 70, 71, 9, 73, 74, 75, 76, 77, 78, 79, 80, 81

Offset: 1

Antti Karttunen, May 23 2016

Map n -> A270418(n)/A270419(n) is a bijection from N (1, 2, 3, ...) to the set of positive rationals.

Antti Karttunen, Table of n, a(n) for n = 1..12167

Cf. A270419 (gives the denominators).
Cf. A262675 (indices of ones).
Cf. A000069, A001969, A010060, A020639, A028234, A065620, A067029.
Cf. also A270420, A270421, A270436, A270437 and permutation pair A273671/A273672.
Differs from A056192 for the first time at n=32, which here a(32)=1, while A056192(32)=4.

s[0] = 0; s[n_]:= s[n]= If[OddQ[n], 1 - 2*s[(n-1)/2], 2*s[n/2]]; f[p_, e_] := p^(ThueMorse[e] * s[e]); a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Sep 05 2023 *)

A270418(n)={n=factor(n);n[,2]=apply(A065620,n[,2]);numerator(factorback(n))} \\ M. F. Hasler, Apr 16 2018

Multiplicative with a(p^e) = p^A065620(e) for odious e, a(p^e)=1 for evil e, or equally, a(p^e) = p^(A010060(e)*A065620(e)).
a(1) = 1, for n > 1, a(n) = a(A028234(n)) * A020639(n)^( A010060(A067029(n)) * A065620(A067029(n)) ).
Other identities. For all n >= 1:
a(A270436(n)) = n, a(A270437(n)) = 1.

A270436 a(1) = 1, for n > 1, a(n) = A020639(n)^A065621(A067029(n)) * a(A028234(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 128, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 384, 25, 26, 2187, 28, 29, 30, 31, 8192, 33, 34, 35, 36, 37, 38, 39, 640, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 4374, 55, 896, 57, 58, 59, 60, 61, 62, 63, 16384, 65, 66, 67, 68, 69, 70, 71, 1152, 73, 74, 75

Offset: 1

Antti Karttunen, May 27 2016

Antti Karttunen, Table of n, a(n) for n = 1..12167

Cf. A020639, A028234, A065621, A067029.
Cf. A270428 (same sequence sorted into ascending order).
Cf. also A270418, A270419, A270437 and permutation A273671.

f[p_, e_] := p^BitXor[e - 1, 2*e - 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 13 2023 *)

Multiplicative with a(p^e) = p^A065621(e).
a(1) = 1, for n > 1, a(n) = A020639(n)^A065621(A067029(n)) * a(A028234(n)).
Other identities. For all n >= 1:
A270418(a(n)) = n, A270419(a(n)) = 1.

A270437 Multiplicative with a(p^e) = p^(e XOR 2e), where XOR is bitwise-xor.

Original entry on oeis.org

1, 8, 27, 64, 125, 216, 343, 32, 729, 1000, 1331, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859, 8000, 9261, 10648, 12167, 864, 15625, 17576, 243, 21952, 24389, 27000, 29791, 32768, 35937, 39304, 42875, 46656, 50653, 54872, 59319, 4000, 68921, 74088, 79507, 85184, 91125, 97336, 103823, 110592, 117649, 125000

Offset: 1

Antti Karttunen, May 27 2016

Multiplicative with a(p^e) = p^A048724(e), where A048724(e) = (e XOR 2e).

Multiples of 8 in the ring defined in A329329. - Peter Munn, Jan 17 2020

Antti Karttunen, Table of n, a(n) for n = 1..12167

Cf. A020639, A028234, A048724, A067029.
Cf. A262675 (same sequence sorted into ascending order).
Cf. also A270418, A270419, A270436 and permutation A273671.
Row 8 and column 8 of A329329.

f[p_, e_] := p^BitXor[2*e, e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Sep 07 2023 *)

a(1) = 1, for n > 1, a(n) = A020639(n)^A048724(A067029(n)) * a(A028234(n)).
Other identities. For all n >= 1:
A270418(a(n)) = 1, A270419(a(n)) = n.
a(n) = A329329(n,8) = A329329(8,n). - Peter Munn, Jan 17 2020

Name changed by Antti Karttunen, Sep 07 2023

A273672 Permutation of natural numbers induced by looking up the position of fraction A270418(n)/A270419(n) from the full Stern-Brocot tree A007305(n+1)/A047679(n-1).

Original entry on oeis.org

1, 3, 7, 15, 31, 63, 127, 2, 511, 1023, 2047, 4095, 8191, 16383, 32767, 65535, 131071, 262143, 524287, 1048575, 2097151, 4194303, 8388607, 6, 33554431, 67108863, 4, 268435455, 536870911, 1073741823, 2147483647, 128, 8589934591, 17179869183, 34359738367, 68719476735, 137438953471, 274877906943, 549755813887, 14

Offset: 1

Antti Karttunen, May 27 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences that are permutations of the natural numbers

Inverse: A273671.
Cf. A270418, A270419.
Cf. also A007305, A047679.

A065620(n, c=1) = sum(i=0, logint(n+!n, 2), if(bittest(n, i), (-1)^c++<A065620
SBtree_index(r) = { my(m=numerator(r),n=denominator(r),z=1); while(m!=n, if(mA273672(n) = { n=factor(n); n[, 2] = apply(A065620, n[, 2]); SBtree_index(factorback(n)); }; \\ Antti Karttunen, Mar 07 2020, based also on M. F. Hasler's code in A270418 and A270419

(define (A273672 n) (SBtree_index (A270418 n) (A270419 n)))
(define (SBtree_index m n) (let loop ((m m) (n n) (z 1)) (cond ((= m n) z) ((< m n) (loop m (- n m) (+ z z))) (else (loop (- m n) n (+ z z 1))))))

A120249 Numerator of cfenc[n] (see definition in comments).

Original entry on oeis.org

1, 2, 3, 3, 5, 5, 8, 4, 4, 8, 13, 7, 21, 13, 7, 5, 34, 7, 55, 11, 11, 21, 89, 9, 7, 34, 5, 18, 144, 12, 233, 6, 18, 55, 12, 10, 377, 89, 29, 14, 610, 19, 987, 29, 9, 144, 1597, 11, 11, 11, 47, 47, 2584, 9, 19, 23, 76, 233, 4181, 17, 6765, 377, 14, 7, 31, 31, 10946, 76, 123, 19

Offset: 1

Joseph Biberstine (jrbibers(AT)indiana.edu), Jun 12 2006, Jun 25 2006

a[n] := numerator of cfenc[n]. cfenc[n] := number given by interpreting as a continued fraction expansion (indexed from 1) the sequence whose i-th entry is one plus the exponent on the i-th prime factor of n (fix cfenc[1]=1). a[2^k] = cfenc[2^k] = k+1.

			a(2646) = numerator[cfenc[2646]]= numerator[cfenc[2^1 * 3^3 * 7^2]] = numerator[FromContinuedFraction[{2; 4, 1, 3}]] = numerator[2 + 1/(4 + 1/(1 + 1/3))] = numerator[42/19] = 42.
From _Antti Karttunen_, Oct 29 2019: (Start)
a(6) = 3 because 6 = 2^1 * 3^1, and the numerator of the continued fraction 1+1 + 1/(1+1) = 5/2 is 5.
a(12) = 7 because 12 = 2^2 * 3^1, and the numerator of the continued fraction 2+1 + 1/(1+1) = 7/2 is 7.
a(15) = 7 because 15 = 2^0 * 3^1 * 5^1, and the numerator of the continued fraction 0+1 + 1/(1+1 + 1/(1+1)) = 1 + 1/(2 + 1/2) = 1 + 2/5 = 7/5 is 7.
(End)

Hans Havermann, Table of n, a(n) for n = 1..10000
Wikipedia, Continued fraction

Corresponding denominators in A120250. Numerators modulo respective denominators in A120251.
Cf. A000040, A000045, A001511, A003961, A061395, A286561, A323080.
Cf. also A007305, A075158, A273671, A273672.

Table[If[n == 1, 1, (fl = FactorInteger[n]; pq = Table[1, {i, 1, PrimePi[Last[fl][[1]]]}]; While[Length[fl] > 0, pp = First[fl]; fl = Drop[fl, 1]; pq[[PrimePi[pp[[1]]]]] = pp[[2]] + 1;]; Numerator[FromContinuedFraction[pq]])],{n,1,80}]

A120249(n) = if(1==n,n, my(pi=primepi(vecmax(factor(n)[, 1])), cf=1+valuation(n,prime(pi))); pi--; while(pi, cf = (1+valuation(n,prime(pi)))+(1/cf); pi--); numerator(cf)); \\ Antti Karttunen, Oct 26 2019

a(2^k) = k+1.
a(A000040(n)) = A000045(n+2).
From Antti Karttunen, Oct 29 2019: (Start)
The following formula employs Gauss's notation for continued fractions (see the section "Notations" in the Wikipedia-article), for example, K_{i=1..3} u(i) stands for 1/(u(1) + 1/(u(2) + 1/u(3))):
Let c(n) = A001511(n) + K_{i=2..A061395(n)} 1/(1+A286561(n,A000040(i))). Then a(n) is the numerator of c(n), and A120250(n) is the denominator of c(n).
For all n >= 2, a(2n) = a(A003961(n)), thus a(n) = f(A323080(n)) for some function f.
(End)

cp's OEIS Frontend

A278261 a(n) = A046523(A273671(n)).

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A270419 Denominator of the rational number obtained when the exponents in prime factorization of n are reinterpreted as alternating binary sums (A065620).

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A270418 Numerator of the rational number obtained when exponents in prime factorization of n are reinterpreted as alternating binary sums (A065620).

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A270436 a(1) = 1, for n > 1, a(n) = A020639(n)^A065621(A067029(n)) * a(A028234(n)).

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A270437 Multiplicative with a(p^e) = p^(e XOR 2e), where XOR is bitwise-xor.

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A273672 Permutation of natural numbers induced by looking up the position of fraction A270418(n)/A270419(n) from the full Stern-Brocot tree A007305(n+1)/A047679(n-1).

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A120249 Numerator of cfenc[n] (see definition in comments).

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