A270418 -id:A270418 - cp's OEIS frontend

A270420 Numbers n for which A270418(n) > A270419(n).

2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81

Offset: 1

Antti Karttunen, May 23 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Complement: A270421 (apart from 1 which is in neither sequence).
Cf. A270428 (a subsequence after initial 1).
Cf. A270418, A270419.

A270421 Numbers n for which A270418(n) < A270419(n).

Original entry on oeis.org

8, 27, 32, 54, 64, 96, 125, 160, 192, 216, 224, 243, 250, 343, 375, 486, 500, 512, 686, 729, 864, 972, 1000, 1024, 1029, 1080, 1215, 1331, 1372, 1458, 1536, 1701, 1715, 1728, 1944, 2058, 2197, 2430, 2560, 2662, 2673, 2744, 2916, 3000, 3072, 3125, 3159, 3375, 3402, 3584, 3645, 3888, 3993, 4000, 4096, 4131, 4320, 4394

Offset: 1

Antti Karttunen, May 23 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..720

Complement: A270420 (apart from 1 which is in neither sequence).
Cf. A262675 (a subsequence after initial 1).
Cf. A270418, A270419.

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))))))

A262675 Exponentially evil numbers.

Original entry on oeis.org

1, 8, 27, 32, 64, 125, 216, 243, 343, 512, 729, 864, 1000, 1024, 1331, 1728, 1944, 2197, 2744, 3125, 3375, 4000, 4096, 4913, 5832, 6859, 7776, 8000, 9261, 10648, 10976, 12167, 13824, 15552, 15625, 16807, 17576, 19683, 21952, 23328, 24389, 25000, 27000, 27648, 29791

Offset: 1

Vladimir Shevelev, Sep 27 2015

Or the numbers whose prime power factorization contains primes only in evil exponents (A001969): 0, 3, 5, 6, 9, 10, 12, ...
If n is in the sequence, then n^2 is also in the sequence.
A268385 maps each term of this sequence to a unique nonzero square (A000290), and vice versa. - Antti Karttunen, May 26 2016

			864 = 2^5*3^3; since 5 and 3 are evil numbers, 864 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Reinhard Zumkeller)

Subsequence of A036966.
Apart from 1, a subsequence of A270421.
Indices of ones in A270418.
Sequence A270437 sorted into ascending order.
Cf. A001969, A209061, A138302, A197680, A000578, A000584, A001014, A001017, A008456, A010803, A010805, A010806, A010808, A010811, A010812, A001221, A010059, A124010, A268385, A270428.

a262675 n = a262675_list !! (n-1)
a262675_list = filter
   (all (== 1) . map (a010059 . fromIntegral) . a124010_row) [1..]
-- Reinhard Zumkeller, Oct 25 2015

{1}~Join~Select[Range@ 30000, AllTrue[Last /@ FactorInteger[#], EvenQ@ First@ DigitCount[#, 2] &] &] (* Michael De Vlieger, Sep 27 2015, Version 10 *)
expEvilQ[n_] := n == 1 || AllTrue[FactorInteger[n][[;; , 2]], EvenQ[DigitCount[#, 2, 1]] &]; With[{max = 30000}, Select[Union[Flatten[Table[i^2*j^3, {j, Surd[max, 3]}, {i, Sqrt[max/j^3]}]]], expEvilQ]] (* Amiram Eldar, Dec 01 2023 *)

isok(n) = {my(f = factor(n)); for (i=1, #f~, if (hammingweight(f[i,2]) % 2, return (0));); return (1);} \\ Michel Marcus, Sep 27 2015

use ntheory ":all"; sub isok { my @f = factor_exp($[0]); return scalar(grep { !(hammingweight($->[1]) % 2) } @f) == @f; } # Dana Jacobsen, Oct 26 2015

Product_{k=1..A001221(n)} A010059(A124010(n,k)) = 1. - Reinhard Zumkeller, Oct 25 2015

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + Sum_{k>=2} 1/p^A001969(k)) = Product_{p prime} f(1/p) = 1.2413599378..., where f(x) = (1/(1-x) + Product_{k>=0} (1 - x^(2^k)))/2. - Amiram Eldar, May 18 2023, Dec 01 2023

More terms from Michel Marcus, Sep 27 2015

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.

A367168 The largest unitary divisor of n that is a term of A138302.

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

Offset: 1

Amiram Eldar, Nov 07 2023

First differs from A056192 at n = 32 and from A270418 at n = 128.

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

Cf. A048298, A065465, A138302, A367169, A367170, A367171.
Cf. A056192, A270418.
Similar sequences: A350388, A350389, A360539.

f[p_, e_] := If[e == 2^IntegerExponent[e, 2], p^e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1 << valuation(f[i, 2], 2), f[i, 1]^f[i, 2], 1));}

from math import prod
from sympy import factorint
def A367168(n): return prod(p**e for p,e in factorint(n).items() if not(e&-e)^e) # Chai Wah Wu, Nov 10 2023

Multiplicative with a(p^e) = p^A048298(e).
a(n) <= n, with equality if and only if n is in A138302.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 0.881513... (A065465).

A273671 Permutation of natural numbers: a(n) = A270436(A007305(n+1)) * A270437(A047679(n-1)).

Original entry on oeis.org

1, 8, 2, 27, 54, 24, 3, 64, 250, 375, 192, 108, 135, 40, 4, 125, 686, 96, 1029, 1372, 160, 1715, 500, 320, 875, 16000, 448, 189, 3456, 56, 5, 216, 1458, 3993, 3000, 5324, 10985, 8640, 2916, 3645, 12096, 281216, 9317, 7000, 170368, 5103, 1080, 750, 3087, 352, 3430, 3773, 416, 4116, 1125, 576, 1500, 1625, 704, 270, 297, 72, 6, 343

Offset: 1

Antti Karttunen, May 27 2016

Permutation of natural numbers induced by looking up the position of A007305(n+1)/A047679(n-1) [each fraction in the full Stern-Brocot tree] in the set of positive rationals as ordered by A270418(n)/A270419(n).

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences that are permutations of the natural numbers

Inverse: A273672.

Cf. A007305, A047679, A270418, A270419, A270436, A270437.

(define (A273671 n) (* (A270436 (A007305 (+ 1 n))) (A270437 (A047679 (- n 1)))))

a(n) = A270436(A007305(n+1)) * A270437(A047679(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

A367514 The exponentially odious part of n: the largest unitary divisor of n that is an exponentially odious number (A270428).

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

Offset: 1

Amiram Eldar, Nov 21 2023

First differs from A056192 at n = 32, and from A270418 and A367168 at n = 128.

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

Cf. A000069, A001221, A010060, A034444, A102392, A262675, A270428, A293439, A366901, A366903, A367515.
Cf. A056192, A270418.
Similar sequences: A350388, A350389, A366905, A367168, A367513.

f[p_, e_] := p^(e*ThueMorse[e]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(hammingweight(f[i, 2])%2, f[i, 1]^f[i, 2], 1));}

from math import prod
from sympy import factorint
def A367514(n): return prod(p**e for p, e in factorint(n).items() if e.bit_count()&1) # Chai Wah Wu, Nov 23 2023

Multiplicative with a(p^e) = p^(e*A010060(e)) = p^A102392(e).
a(n) = n/A367513(n).
A001221(a(n)) = A293439(n).
A034444(a(n)) = A367515(n).
a(n) >= 1, with equality if and only if n is an exponentially evil number (A262675).
a(n) <= n, with equality if and only if n is an exponentially odious number (A270428).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} f(1/p) = 0.88585652437242918295..., and f(x) = (x+2)/(2*(x+1)) + (x/2) * Product_{k>=0} (1 - x^(2^k)).

cp's OEIS Frontend

A270420 Numbers n for which A270418(n) > A270419(n).

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A270421 Numbers n for which A270418(n) < A270419(n).

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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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A262675 Exponentially evil numbers.

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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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A367168 The largest unitary divisor of n that is a term of A138302.

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A273671 Permutation of natural numbers: a(n) = A270436(A007305(n+1)) * A270437(A047679(n-1)).

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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.