A369965 -id:A369965

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A369966

Numbers of the form 4m+2 that have an even number of prime factors with multiplicity.

+10
4

6, 10, 14, 22, 26, 34, 38, 46, 54, 58, 62, 74, 82, 86, 90, 94, 106, 118, 122, 126, 134, 142, 146, 150, 158, 166, 178, 194, 198, 202, 206, 210, 214, 218, 226, 234, 250, 254, 262, 274, 278, 294, 298, 302, 306, 314, 326, 330, 334, 342, 346, 350, 358, 362, 382, 386, 390, 394, 398, 414, 422, 446, 454, 458, 462, 466

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Numbers that have one even prime factor (2) and an odd number of odd prime factors with multiplicity.

Numbers that have an even number of prime factors with multiplicity and whose arithmetic derivative (A003415) is odd.

LINKS

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

FORMULA

a(n) = 2*A067019(n).

PROG

(PARI) \\ See A369965

CROSSREFS

Intersection of A016825 and A028260.

Intersection of A028260 and A235991.

Cf. A003415, A067019, A369668 (subsequence), A369965 (characteristic function).

Cf. A369661 (numbers whose arithmetic derivative is in this sequence).

KEYWORD

nonn

AUTHOR

Antti Karttunen, Feb 08 2024

STATUS

approved

A369660

a(n) = 1 if n' is of the form 4k+2, and n' has an even number of prime factors, otherwise a(n) = 0, where n' stands for the arithmetic derivative of n, A003415.

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..100000

Index entries for characteristic functions

FORMULA

a(0) = a(1) = 0, and for n > 1, a(n) = A353495(n) * A065043(A003415(n)).

a(n) = 1 if n' is congruent to 2 modulo 4 and A276085(n') is congruent to 3 modulo 4, otherwise a(n) = 0.

a(n) = A369965(A003415(n)).

PROG

(PARI)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));