A356452 -id:A356452

Search: a356452 -id:a356452

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A356448

Even numbers k such that k^2 is in A014567.

+10
7

2, 4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 72, 74, 76, 80, 82, 86, 88, 90, 92, 94, 96, 100, 102, 104, 106, 108, 110, 116, 118, 120, 122, 128, 130, 132, 134, 136, 138, 140, 142, 144, 146, 148

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

OFFSET

1,1

COMMENTS

Even numbers k such that k and sigma(k^2) are coprime, sigma = A000203.

LINKS

Jianing Song, Table of n, a(n) for n = 1..7127 (all terms <= 20000)

FORMULA

a(n) = 2*A356451(n).

EXAMPLE

12 is a term since 12 and sigma(12^2) = 403 are coprime.

PROG

(PARI) isA356448(n) = !(n%2) && gcd(n, sigma(n^2))==1

CROSSREFS

Cf. A014567, A000203, A356382, A356449, A356451, A356452 (complement in the even numbers).

KEYWORD

nonn,easy

AUTHOR

Jianing Song, Aug 07 2022

STATUS

approved

A356453

Numbers k such that 2*k^2 is not in A014567; complement of A356449.

+10
5

3, 6, 9, 10, 12, 15, 18, 21, 24, 27, 28, 30, 33, 36, 39, 40, 42, 45, 48, 50, 51, 54, 57, 60, 63, 66, 69, 70, 72, 75, 77, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 110, 111, 114, 117, 120, 123, 126, 129, 130, 132, 133, 135, 136, 138, 140, 141, 144, 147, 150, 153, 154, 155

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

OFFSET

1,1

COMMENTS

Numbers k such that k and sigma(2*k^2) are not coprime, sigma = A000203.

Includes all multiples of 3 since 3 divides sigma(m) if m is twice a square (cf. A065766).

LINKS

Jianing Song, Table of n, a(n) for n = 1..8914 (all terms <= 20000)

EXAMPLE

10 is a term since 10 and sigma(2*10^2) = 465 have a common factor 5.

MATHEMATICA

Select[Range[155], GCD[#, DivisorSigma[1, 2#^2]]>1 &] (* Stefano Spezia, Aug 07 2024 *)

PROG

(PARI) isA356453(n) = gcd(n, sigma(2*n^2))>1

CROSSREFS

Cf. A014567, A000203, A356448, A356449, A356452, A356454, A356456 (all multiples of 3 removed), A065766.

KEYWORD

nonn,easy

AUTHOR

Jianing Song, Aug 07 2022

STATUS

approved

A356454

Numbers k such that 4*k^2 is not in A014567; complement of A356451.

+10
4

7, 21, 35, 39, 42, 49, 56, 57, 62, 63, 77, 78, 84, 91, 93, 99, 105, 111, 114, 117, 119, 126, 129, 133, 147, 154, 155, 156, 161, 168, 171, 175, 183, 186, 189, 195, 198, 201, 203, 209, 210, 217, 219, 222, 228, 231, 234, 237, 245, 252, 258, 259, 266, 273, 275, 279, 280, 285

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

OFFSET

1,1

COMMENTS

Numbers k such that k and sigma(4*k^2) are not coprime, sigma = A000203.

LINKS

Jianing Song, Table of n, a(n) for n = 1..12241 (all terms <= 40000)

FORMULA

a(n) = A356452(n)/2.

EXAMPLE

7 is a term since 7 and sigma(4*7^2) = 399 have a common factor 7.

PROG

(PARI) isA356454(n) = gcd(n, sigma(4*n^2))>1

CROSSREFS

Cf. A014567, A000203, A356448, A356452, A356453.

KEYWORD

nonn,easy

AUTHOR

Jianing Song, Aug 07 2022

STATUS

approved

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