A268375 -id:A268375 - cp's OEIS frontend

A289618 a(n) = A289617(n) - A046645(n) = A005187(A001222(n)) - A046645(n).

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 1, 2, 1, 1, 0, 2, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 2, 1, 1, 1, 2, 0, 2, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 2

Offset: 1

Antti Karttunen, Jul 08 2017

nonn

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

Cf. A046645, A289617.

Cf. A268375 (positions of zeros), A289619 (of ones).

(define (A289618 n) (- (A289617 n) (A046645 n)))

a(n) = A289617(n) - A046645(n) = A005187(A001222(n)) - A046645(n).

A091862 a(n) = 1 if the sum of all exponents of the prime-factorization of n has no carries when summed in base 2, or a(n) = 0 if there are any carries.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0

Offset: 1

Leroy Quet, Mar 13 2004

Characteristic function for A268375. - Antti Karttunen, Nov 23 2017

			a(12) = 1 because 12 = 2^2 *3^1 and, in base 2, 2 = '10', 1 = '1' and '10' and '1' have their ones in different positions. But a(24) = 0 because 24 = 2^3 *3^1 and in base 2 3 = '11', 1 = '1', which both share a rightmost one.

Cf. A091863, A267116, A268374, A289618.

Cf. A268375 (positions of ones), A268376 (of zeros).

f[e_] := Position[Reverse[IntegerDigits[e, 2]], 1] // Flatten; a[n_] := Boole[UnsameQ @@ Flatten[f /@ FactorInteger[n][[;; , 2]]]]; Array[a, 100] (* Amiram Eldar, Dec 23 2023 *)

a(n) = {my(e = factor(n)[,2], b = 0); for(i=1, #e, b = bitor(b, e[i])); n == 1 || b == vecsum(e);} \\ Amiram Eldar, Dec 23 2023

If A268374(n) = 0, then a(n) = 1, 0 otherwise. - Antti Karttunen, Nov 23 2017

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 25 2004

A268374 a(n) = A001222(n) - A267116(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 1, 2, 0, 1, 1, 1, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 2, 0, 0, 1, 2, 0, 2, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 2, 0, 2, 0, 1, 2, 1, 0, 2, 0, 2, 1, 0, 0, 2, 1, 0, 0, 1, 1, 2

Offset: 1

Antti Karttunen, Feb 03 2016

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

Cf. A001222, A267116.

Cf. A268375 (indices of zeros), A268376 (of nonzeros).

Table[PrimeOmega@ n - BitOr @@ Map[Last, FactorInteger@ n], {n, 120}] /. -1 -> 0 (* Michael De Vlieger, Feb 04 2016 *)

(define (A268374 n) (- (A001222 n) (A267116 n)))

a(n) = A001222(n) - A267116(n).

A318363 Multiplicative with a(prime(i)^k) = prime(k)^2^(i-1).

Original entry on oeis.org

1, 2, 4, 3, 16, 8, 256, 5, 9, 32, 65536, 12, 4294967296, 512, 64, 7, 18446744073709551616, 18, 340282366920938463463374607431768211456, 48, 1024, 131072, 115792089237316195423570985008687907853269984665640564039457584007913129639936, 20, 81, 8589934592, 25

Offset: 1

Rémy Sigrist, Aug 24 2018

This sequence has similarities with A048767.
This sequence is injective (all terms are distinct).
This sequence is a permutation of A268375.

Cf. A005117, A007947, A048767, A048672, A268375.

Array[Apply[Times, FactorInteger[#] /. {p_, k_} /; p > 1 :> Prime[k]^2^(PrimePi[p] - 1)] &, 27] /. {1, 1} -> 1 (* Michael De Vlieger, Aug 25 2018 *)

a(n) = my (f=factor(n)); prod(i=1, #f~, prime(f[i,2])^2^(primepi(f[i,1])-1))

A007947(a(n)) = A007947(A048767(n)) for any n > 0.

a(A005117(n)) = 2^A048672(n) for any n > 0.

A355935 Dirichlet inverse of A091862, characteristic function of numbers for which A267116(n) = bigomega(n), where A267116 is the bitwise-OR of the exponents of primes in the prime factorization of n.

Original entry on oeis.org

1, -1, -1, 0, -1, 2, -1, 0, 0, 2, -1, -2, -1, 2, 2, 0, -1, -2, -1, -2, 2, 2, -1, 2, 0, 2, 0, -2, -1, -6, -1, 0, 2, 2, 2, 6, -1, 2, 2, 2, -1, -6, -1, -2, -2, 2, -1, -2, 0, -2, 2, -2, -1, 2, 2, 2, 2, 2, -1, 10, -1, 2, -2, 0, 2, -6, -1, -2, 2, -6, -1, -8, -1, 2, -2, -2, 2, -6, -1, -2, 0, 2, -1, 10, 2, 2, 2, 2, -1, 10, 2, -2, 2, 2, 2, 2, -1, -2, -2, 6, -1, -6, -1, 2, -6

Offset: 1

Antti Karttunen, Jul 21 2022

sign

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Cf. A091862, A267116, A268375.

s[n_] := If[n == 1 || PrimeOmega[n] == BitOr @@ FactorInteger[n][[;; , 2]], 1, 0]; a[1] = 1; a[n_] := a[n] = -DivisorSum[n, s[n/#]*a[#] &, # < n &]; Array[a, 100] (* Amiram Eldar, Jul 21 2022 *)

A267116(n) = if(1==n, 0, fold(bitor, factor(n)[, 2]));
A091862(n) = (bigomega(n)==A267116(n));
memoA355935 = Map();
A355935(n) = if(1==n,1,my(v); if(mapisdefined(memoA355935,n,&v), v, v = -sumdiv(n,d,if(dA091862(n/d)*A355935(d),0)); mapput(memoA355935,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA091862(n/d) * a(d).

cp's OEIS Frontend

A289618 a(n) = A289617(n) - A046645(n) = A005187(A001222(n)) - A046645(n).

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A091862 a(n) = 1 if the sum of all exponents of the prime-factorization of n has no carries when summed in base 2, or a(n) = 0 if there are any carries.

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A268374 a(n) = A001222(n) - A267116(n).

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A318363 Multiplicative with a(prime(i)^k) = prime(k)^2^(i-1).

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A355935 Dirichlet inverse of A091862, characteristic function of numbers for which A267116(n) = bigomega(n), where A267116 is the bitwise-OR of the exponents of primes in the prime factorization of n.

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