A270420 -id:A270420 - cp's OEIS frontend

A270428 Exponentially odious numbers: 1 together with positive integers n such that all exponents in prime factorization of n are odious numbers (A000069).

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84

Offset: 1

Antti Karttunen, May 26 2016

A268385 maps each term of this sequence to a unique term of A268335, and vice versa.

The asymptotic density of this sequence is Product_{p prime} f(1/p) = 0.87686263163054480657..., where f(x) = 1 - x + (1 - (1-x) * Product_{k>=0} (1-x^(2^k)))/2. - Amiram Eldar, Oct 27 2023

Antti Karttunen, Table of n, a(n) for n = 1..10000
Vladimir Shevelev, S-exponential numbers, Acta Arithmetica, Vol. 175 (2016), 385-395.

Apart from 1, a subsequence of A270420.
Indices of ones in A270419.
Sequence A270436 sorted into ascending order.
Cf. A010060, A028234, A067029, A355825 (characteristic function).
Cf. also A262675, A268335, A268385.
Differs from its subsequence A138302 for the first time at n=113, where a(113) = 128 = 2^7, a value which does not occur in A138302.

odiousQ[n_] := OddQ[DigitCount[n, 2, 1]]; Select[Range[100], AllTrue[FactorInteger[#][[;;, 2]], odiousQ] &] (* Amiram Eldar, May 18 2023 *)

A355825(n) = factorback(apply(e->(hammingweight(e)%2), factor(n)[, 2]));
isA270428(n) = A355825(n); \\ Antti Karttunen, Jul 21 2022

;; Requires Antti Karttunen's IntSeq-library.
(define A270428 (ZERO-POS 1 1 (COMPOSE sub1 A270419)))

;; Requires Antti Karttunen's IntSeq-library.
(definec (chA270428 n) (cond ((= 1 n) 1) (else (* (A010060 (A067029 n)) (chA270428 (A028234 n))))))
(define A270428 (NONZERO-POS 1 1 chA270428))

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.

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.

cp's OEIS Frontend

A270428 Exponentially odious numbers: 1 together with positive integers n such that all exponents in prime factorization of n are odious numbers (A000069).

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

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