A289618 -id:A289618 - cp's OEIS frontend

A289619 Positions of ones in A289618.

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 36, 38, 39, 42, 46, 51, 55, 57, 58, 62, 65, 66, 69, 70, 72, 74, 77, 78, 82, 85, 86, 87, 91, 93, 94, 95, 96, 100, 102, 105, 106, 108, 110, 111, 114, 115, 118, 119, 122, 123, 129, 130, 133, 134, 138, 141, 142, 143, 145, 146, 154, 155, 158, 159, 160, 161, 165, 166, 170, 174

Offset: 1

Antti Karttunen, Jul 08 2017

nonn

Numbers n such that A289617(n) = A005187(A001222(n)) is equal to 1 + A046645(n). Whether a number is included depends only on its prime signature, thus whenever any n is present in the sequence, so is also A046523(n).

			6 = 2^1 * 3^1, thus A001222(6) = 1+1 = 2, and A005187(2) = 3. On the other hand, A005187(1) = 1, and 1+1 = 2, which is one less than 3, thus 6 is included like all nonsquare semiprimes.
30 = 2^1 * 3^1 * 5^1, thus A001222(30) = 3, while A005187(3) = 4, thus 30 is included like all products of three distinct primes.
72 = 2^3 * 3^2, thus A001222(72) = 3+2 = 5, and A005187(5) = 8. On the other hand, A005187(3)+A005187(2) = 4+3 = 7, and 8 = 7+1, thus 72 is included in the sequence.

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

Cf. A001222, A005187, A046523, A046645, A289617, A289618.

Differs from A182853 for the first time at n=26, where a(26) = 72, while A182853(26) = 74.

A046645 a(n) = log_2(A046644(n)); also the 2-adic valuation of A046644(n).

Original entry on oeis.org

0, 1, 1, 3, 1, 2, 1, 4, 3, 2, 1, 4, 1, 2, 2, 7, 1, 4, 1, 4, 2, 2, 1, 5, 3, 2, 4, 4, 1, 3, 1, 8, 2, 2, 2, 6, 1, 2, 2, 5, 1, 3, 1, 4, 4, 2, 1, 8, 3, 4, 2, 4, 1, 5, 2, 5, 2, 2, 1, 5, 1, 2, 4, 10, 2, 3, 1, 4, 2, 3, 1, 7, 1, 2, 4, 4, 2, 3, 1, 8, 7, 2, 1, 5, 2, 2, 2, 5, 1, 5, 2, 4, 2

Offset: 1

N. J. A. Sloane

A268375 gives numbers n for which a(n) = A289617(n) = A005187(A001222(n)). - Antti Karttunen, Jul 08 2017

Antti Karttunen, Table of n, a(n) for n = 1..65536
Index entries for sequences computed from exponents in factorization of n.

See A046643, A046644 for more details.

Cf. A001222, A005187, A007814, A077761, A268375, A289617, A289618, A293442, A293447.

f[p_, e_] := 2*e - DigitCount[2*e, 2, 1]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 28 2023 *)

A007814(n) = (valuation(n,2));
A046643perA046644(n) = { my(c=1); if(1==n,c,fordiv(n,d, if((d>1)&&(dA046643perA046644(d)*A046643perA046644(n/d)))); (c/2)); } \\ After the Maple-program given in A046643.
A046645(n) = A007814(denominator(A046643perA046644(n))); \\ Antti Karttunen, Jul 08 2017

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A046645(n) = vecsum(apply(e -> A005187(e), factorint(n)[, 2])); \\ A faster implementation. - Antti Karttunen, Jul 08 2017

a(n) = A007814(A046644(n)). - Michel Marcus, Apr 16 2015
Additive with a(p^n) = A005187(n). - Antti Karttunen, Jul 08 2017
a(n) = A293447(A293442(n)). - Antti Karttunen, Nov 10 2017
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{p prime} f(1/p) = 1.410258867603361890498..., where f(x) = -x + Sum_{k>=0} (2^(k+1)-1)*x^(2^k)/(1+x^(2^k)). - Amiram Eldar, Sep 29 2023

A268375 Numbers k for which A001222(k) = A267116(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 25, 27, 28, 29, 31, 32, 37, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 59, 61, 63, 64, 67, 68, 71, 73, 75, 76, 79, 80, 81, 83, 89, 92, 97, 98, 99, 101, 103, 107, 109, 112, 113, 116, 117, 121, 124, 125, 127, 128, 131, 137, 139, 144, 147, 148, 149, 151, 153

Offset: 1

Antti Karttunen, Feb 03 2016

Numbers k whose prime factorization k = p_1^e_1 * ... * p_m^e_m contains no pair of exponents e_i and e_j (i and j distinct) whose base-2 representations have at least one shared digit-position in which both exponents have a 1-bit.
Equivalently, numbers k such that the factors in the (unique) factorization of k into powers of squarefree numbers with distinct exponents that are powers of two, are prime powers. For example, this factorization of 90 is 10^1 * 3^2, so 90 is not included, as 10 is not prime; whereas this factorization of 320 is 5^1 * 2^2 * 2^4, so 320 is included as 5 and 2 are both prime. - Peter Munn, Jan 16 2020
A225546 maps the set of terms 1:1 onto A138302. - Peter Munn, Jan 26 2020
Equivalently, numbers k for which A064547(k) = A331591(k). - Amiram Eldar, Dec 23 2023

			12 = 2^2 * 3^1 is included in the sequence as the exponents 2 ("10" in binary) and 1 ("01" in binary) have no 1-bits in the same position, and 18 = 2^1 * 3^2 is included for the same reason.
On the other hand, 24 = 2^3 * 3^1 is NOT included in the sequence as the exponents 3 ("11" in binary) and 1 ("01" in binary) have 1-bit in the same position 0.
720 = 2^4 * 3^2 * 5^1 is included as the exponents 1, 2 and 4 ("001", "010" and "100" in binary) have no 1-bits in shared positions.
Likewise, 10! = 3628800 = 2^8 * 3^4 * 5^2 * 7^1 is included as the exponents 1, 2, 4 and 8 ("0001", "0010", "0100" and "1000" in binary) have no 1-bits in shared positions. And similarly for any term of A191555.

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

Indices of zeros in A268374, also in A289618.
Cf. A001222, A064547, A267116, A046645, A289617, A331591.
Cf. A091862 (characteristic function), A268376 (complement).
Cf. A000961, A054753, A191555 (subsequences).
Related to A138302 via A225546.
Cf. also A318363 (a permutation).

{1}~Join~Select[Range@ 160, PrimeOmega@ # == BitOr @@ Map[Last, FactorInteger@ #] &] (* Michael De Vlieger, Feb 04 2016 *)

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

Original entry on oeis.org

0, 1, 1, 3, 1, 3, 1, 4, 3, 3, 1, 4, 1, 3, 3, 7, 1, 4, 1, 4, 3, 3, 1, 7, 3, 3, 4, 4, 1, 4, 1, 8, 3, 3, 3, 7, 1, 3, 3, 7, 1, 4, 1, 4, 4, 3, 1, 8, 3, 4, 3, 4, 1, 7, 3, 7, 3, 3, 1, 7, 1, 3, 4, 10, 3, 4, 1, 4, 3, 4, 1, 8, 1, 3, 4, 4, 3, 4, 1, 8, 7, 3, 1, 7, 3, 3, 3, 7, 1, 7, 3, 4, 3, 3, 3, 10, 1, 4, 4, 7, 1, 4, 1, 7, 4, 3, 1, 8, 1, 4, 3, 8, 1, 4, 3, 4, 4, 3, 3, 8

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. A001222, A005187, A289618.

Cf. A268375 (positions where coincides with A046645).

a(n) = my(bn=bigomega(n)); 2*bn-hammingweight(bn); \\ Michel Marcus, Jul 10 2017

(define (A289617 n) (A005187 (A001222 n)))

a(n) = A005187(A001222(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

cp's OEIS Frontend

A289619 Positions of ones in A289618.

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A046645 a(n) = log_2(A046644(n)); also the 2-adic valuation of A046644(n).

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A268375 Numbers k for which A001222(k) = A267116(k).

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A289617 a(n) = A005187(A001222(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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