The On-Line Encyclopedia of Integer Sequences (OEIS)

More Web Proxy on the site http://driver.im/

The OEIS is supported by the many generous donors to the OEIS Foundation.

Revision History for A000028 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
Showing entries 1-10 | older changes

Let k = p_1^e_1 p_2^e_2 p_3^e_3 ... be the prime factorization of n. Sequence gives k such that the sum of the numbers of 1's in the binary expansions of e_1, e_2, e_3, ... is odd.
(history; published version)

#80 by Alois P. Heinz at Fri Feb 14 19:26:18 EST 2025

COMMENTS

Presumably, that a(n) = A037992(n + 1)/ A037992(n). - Juri-Stepan Gerasimov, Feb 14 2025

CROSSREFS

Cf. A037992, A380836.

KEYWORD

nonn,nice,easy,changed

STATUS

editing

approved

#79 by Alois P. Heinz at Fri Feb 14 16:31:26 EST 2025

STATUS

proposed

editing

Discussion

Fri Feb 14

16:32

Alois P. Heinz: not the same:
2, 3, 4, 5, 7, 9, 11, 13, 16, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49 
2, 3, 4, 5, 7, 9, 11, 13, 16, 17, 19, 23, 24, 25, 29, 30, 31, 37, 40, 41

16:33

Alois P. Heinz: if there was a connection ... it should have been in A037992, not here ...

16:34

Alois P. Heinz: and definitely: crossref to A380836 should not be here ...

16:43

Alois P. Heinz: please check your conjectures before you enter them ... especially when there is enough data available ...

19:26

Alois P. Heinz: reverting now ...

#78 by Juri-Stepan Gerasimov at Fri Feb 14 16:27:13 EST 2025

STATUS

editing

proposed

Discussion

Fri Feb 14

16:29

Stefano Spezia: To change into “It appears that a(n) = …” !?

16:31

Alois P. Heinz: but it is easy to see that this is not correct ...

#77 by Juri-Stepan Gerasimov at Fri Feb 14 16:26:40 EST 2025

COMMENTS

Presumably, that a(n) = A037992(n + 1)/ A037992(n). - Juri-Stepan Gerasimov, Feb 14 2025

CROSSREFS

Cf. A037992, A380836.

STATUS

approved

editing

#76 by Hugo Pfoertner at Wed Oct 02 04:47:05 EDT 2024

STATUS

reviewed

approved

#75 by Joerg Arndt at Wed Oct 02 04:19:05 EDT 2024

STATUS

proposed

reviewed

#74 by Amiram Eldar at Wed Oct 02 03:11:02 EDT 2024

STATUS

editing

proposed

#73 by Amiram Eldar at Wed Oct 02 03:03:16 EDT 2024

COMMENTS

For example, if nm = 96, then the maximal exponent of 2 that divides 96 is 5, for 3 it is 1, for 4 it is 2, for 16 it is 1. Thus k(96) = 3 and 96 is a term.

#72 by Amiram Eldar at Wed Oct 02 02:52:53 EDT 2024

COMMENTS

Numbers whose exponentially odious part (A367514) has an odd number of distinct prime factors, i.e., numbers k such that A092248(A001221(A367514(k))) = 1. (End)

CROSSREFS

Cf. A001221, A030230, A037445, A092248, A367514.

#71 by Amiram Eldar at Wed Oct 02 02:50:50 EDT 2024

NAME

Let n k = p_1^e_1 p_2^e_2 p_3^e_3 ... be the prime factorization of n. Sequence gives n k such that the sum of the numbers of 1's in the binary expansions of e_1, e_2, e_3, ... is odd.

COMMENTS

Numbers n m such that infinitary Moebius function of n m (A064179) equals -1. This follows from the definition of A064179.

Number n m is in the sequence if and only if the number k = k(nm) of terms of A050376 which divide n m with odd maximal exponent is odd.

For example, if n=96, then the maximal exponent of 2 that divides 96 is 5, for 3 it is 1, for 4 it is 2, for 16 it is 1. Thus k(96) = 3 and 96 is a term.

REFERENCES

J. Joe Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 22.

LINKS

<a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.

EXAMPLE

If nk = 96 then the maximal exponent of 2 that divides 96 is 5, for 3 it is 1. 5 in binary is 101_2 and has so has a sum of binary digits of 1 + 0 + 1 = 2. 1 in binary is 1_2 and so has a sum of binary digits of 1. Thus the sum of digits of binary exponents is 2 + 1 = 3 which is odd and so 96 is a term. - Vladimir Shevelev, Oct 28 2013, edited by David A. Corneth, Mar 20 2019