A242332 -id:A242332

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A242331

Numbers k such that k^2 + 3 is a semiprime.

+10
10

1, 6, 16, 18, 20, 24, 26, 32, 34, 36, 40, 44, 46, 48, 56, 60, 66, 68, 78, 80, 88, 98, 100, 102, 104, 108, 116, 118, 120, 128, 136, 148, 152, 164, 170, 174, 176, 182, 188, 190, 192, 196, 200, 204, 212, 220, 226, 232, 234, 238, 246, 250, 252, 258, 260, 262, 266

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

OFFSET

1,2

COMMENTS

The semiprimes of this form are: 4, 39, 259, 327, 403, 579, 679, 1027, 1159, 1299, 1603, 1939, 2119, 2307, 3139, 3603, 4359, 4627, ...

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

MATHEMATICA

Select[Range[300], PrimeOmega[#^2 + 3] == 2 &]

PROG

(Magma) IsSemiprime:=func<i | &+[d[2]: d in Factorization(i)] eq 2>; [n: n in [0..300] | IsSemiprime(s) where s is n^2+3];

CROSSREFS

Cf. A049422, A049423, A085722, A242330, A242332, A242333.

KEYWORD

nonn,easy

AUTHOR

Vincenzo Librandi, May 14 2014

STATUS

approved

A242330

Numbers k such that k^2 + 2 is a semiprime.

+10
7

2, 6, 7, 11, 12, 17, 18, 27, 29, 35, 37, 42, 43, 48, 51, 53, 54, 55, 60, 65, 66, 69, 72, 73, 75, 79, 83, 84, 87, 90, 93, 97, 115, 119, 125, 132, 133, 135, 137, 141, 144, 150, 153, 155, 159, 161, 165, 169, 174, 183, 186, 187, 189, 191, 192, 195, 198

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

OFFSET

1,1

COMMENTS

The semiprimes of this form are: 6, 38, 51, 123, 146, 291, 326, 731, 843, 1227, 1371, 1766, 1851, 2306, 2603, 2811, 2918, 3027, 3602, ....

There are no four consecutive terms in this sequence, that is, a(n) > a(n-3) + 3 (check mod 6). Probably sieve theory can show that this sequence has density 0. - Charles R Greathouse IV, Feb 24 2023

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

FORMULA

a(n) > 2n for n > 1. - Charles R Greathouse IV, Feb 24 2023

MATHEMATICA

Select[Range[300], PrimeOmega[#^2 + 2] == 2 &]

PROG

(Magma) IsSemiprime:=func<i | &+[d[2]: d in Factorization(i)] eq 2>; [n: n in [2..200] | IsSemiprime(s) where s is n^2+2];

(PARI) issemi(n)=forprime(p=2, 997, if(n%p==0, return(isprime(n/p)))); bigomega(n)==2

is(n)=issemi(n^2+2) \\ Charles R Greathouse IV, Feb 24 2023

CROSSREFS

Cf. A056899, A067201, A085722, A242331, A242332, A242333.

KEYWORD

nonn,easy

AUTHOR

Vincenzo Librandi, May 14 2014

STATUS

approved

A242333

Numbers k such that k^2 + 5 is a semiprime.

+10
7

1, 2, 3, 4, 8, 9, 14, 18, 21, 22, 24, 26, 27, 28, 30, 33, 42, 44, 51, 54, 57, 58, 62, 63, 64, 68, 69, 82, 84, 86, 90, 93, 98, 99, 102, 104, 108, 111, 118, 132, 134, 138, 144, 152, 154, 156, 166, 174, 177, 180, 183, 184, 186, 188, 189, 194, 208, 210, 212, 216

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

OFFSET

1,2

COMMENTS

The semiprimes of this form are: 6, 9, 14, 21, 69, 86, 201, 329, 446, 489, 581, 681, 734, 789, 905, 1094, 1769, 1941, 2606, 2921, 3254, ...

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

MATHEMATICA

Select[Range[0, 300], PrimeOmega[#^2 + 5] == 2 &]

PROG

(Magma) IsSemiprime:=func<i | &+[d[2]: d in Factorization(i)] eq 2>; [n: n in [0..300] | IsSemiprime(s) where s is n^2+5];

CROSSREFS

Cf. A056905, A078402, A085722, A242330, A242331, A242332.

KEYWORD

nonn,easy

AUTHOR

Vincenzo Librandi, May 14 2014

STATUS

approved

A360741

Semiprimes of the form k^2 + 4.

+10
1

4, 85, 365, 445, 533, 629, 965, 1685, 1853, 2605, 2813, 3029, 3973, 4765, 5045, 5629, 5933, 6245, 6893, 8285, 8653, 11029, 11453, 11885, 12773, 14165, 15133, 16645, 17165, 17693, 20453, 21029, 22205, 22805, 23413, 24653, 27229, 29245, 29933, 30629, 32765, 34229

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

OFFSET

1,1

COMMENTS

A242332 gives the corresponding values of k.

Except for 4, all terms == 5 (mod 8). - Robert Israel, Feb 18 2023

LINKS

Table of n, a(n) for n=1..42.

FORMULA

a(n) = A242332(n)^2 + 4.

EXAMPLE

85 is a term because 9^2 + 4 = 85 = 5*17.

MAPLE

select(t -> numtheory:-bigomega(t)=2, [seq(i^2+4, i=0..1000)]); # Robert Israel, Feb 18 2023

MATHEMATICA

Select[Range[0, 200]^2 + 4, PrimeOmega[#] == 2 &] (* Amiram Eldar, Feb 18 2023 *)

CROSSREFS

Cf. A001358, A085722, A087475, A144255, A242330, A242331, A242332.

KEYWORD

nonn,easy

AUTHOR

Elmo R. Oliveira, Feb 18 2023

STATUS

approved

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