A130800 -id:A130800 - cp's OEIS frontend

A245365 Semiprimes of the form n(3n-1)/2.

22, 35, 51, 145, 247, 287, 1247, 1717, 2147, 2501, 3151, 4187, 5017, 7957, 11051, 13207, 15251, 16801, 17767, 20827, 26867, 33227, 49051, 63551, 68587, 71177, 76501, 81317, 96647, 112477, 118301, 128627, 147737, 159251, 182527, 232657, 237407, 241001, 250717

Offset: 1

K. D. Bajpai, Jul 19 2014

nonn

Semiprimes among pentagonal numbers A000326 = { (3*n^2-n)/2; n >= 0 }.

We can have an odd prime n = 2k + 1 and (3n - 1)/2 = 3k + 1 also prime, i.e., k in A130800, or n = 2p with p prime and 3n - 1 = 6p - 1 also prime, i.e., p in A158015. Considering the ratio of the two prime factors, the two possibilities are mutually exclusive, so this is the disjoint union of {A033570(n)=(2n+1)(3n+1); n in A130800} = A255584 and {p*(6p-1); p in A158015}. - M. F. Hasler, Dec 13 2019

			n=6: (3*n^2-n)/2 = 51 = 3 * 17 which is semiprime. Hence, 51 appears in the sequence.
n=10: (3*n^2-n)/2 = 145 = 5 * 29 which is semiprime. Hence, 145 appears in the sequence.

K. D. Bajpai, Table of n, a(n) for n = 1..10400

Cf. A001358, A000326.

Select[Table[(3*n^2 - n)/2, {n, 500}], PrimeOmega[#] == 2 &]

select(n->bigomega(n)==2, vector(1000, n, (3*n^2-n)/2)) \\ Colin Barker, Jul 20 2014

A255584 Semiprimes of the form (4n + 1)(6n + 1) = 24n^2 + 10*n + 1.

Original entry on oeis.org

35, 247, 1247, 2501, 4187, 7957, 15251, 17767, 33227, 49051, 81317, 118301, 128627, 182527, 241001, 250717, 265651, 302177, 318551, 438751, 485357, 563347, 655051, 679057, 736751, 753667, 886657, 981317, 1010651, 1090987, 1163801, 1361837, 1563151

Offset: 1

Vincenzo Librandi, Feb 27 2015

nonn

The first few values of n such that both n and n+1 give semiprimes in the sequence begin: 2607, 4017, 4062, 5967, 7107, 8472, 8892, ... In such cases, numbers of the form 10n+8 can always be expressed as the sum of the two primes 4n+1 and 6n+7. - Wesley Ivan Hurt, Feb 27 2015

			35 is in the sequence because 35 = 5*7 and 5, 7 are primes of the form 4*k+1 and 6*k+1 respectively.
247 is in the sequence because 247 = 13*19: both 13, 19 are primes of the form 6*k+1 and 13 also has the form 4*k+1.

Subsequence of A245365.
Cf. A001358, A002144, A002476, A113941, A255607 (associated n).
Equals A033570(A130800). - M. F. Hasler, Dec 13 2019

[(4*n+1)*(6*n+1): n in [1..300] | IsPrime(4*n+1) and IsPrime(6*n+1)];

IsSemiprime:=func; [s: n in [1..300] | IsSemiprime(s) where s is 24*n^2+10*n+1];

Select[Table[24 n^2 + 10 n + 1, {n, 300}], PrimeOmega[#] == 2 &] (* or *) f[n_] := Last /@ FactorInteger[n] == {1, 1}; Select[Array[24 #^2 + 10 # + 1 &, 300], f[#] &]

for(n=1,250,if(bigomega(s=24*n^2+10*n+1)==2,print1(s,", "))) \\ Derek Orr, Feb 28 2015

a(n) = A033570(A130800(n)) = A033570(2*A255607(n)). - M. F. Hasler, Dec 13 2019

A255607 Numbers n such that both 4n+1 and 6n+1 are primes.

Original entry on oeis.org

1, 3, 7, 10, 13, 18, 25, 27, 37, 45, 58, 70, 73, 87, 100, 102, 105, 112, 115, 135, 142, 153, 165, 168, 175, 177, 192, 202, 205, 213, 220, 238, 255, 258, 277, 282, 298, 300, 312, 322, 325, 352, 357, 363, 370, 373, 417, 423, 447, 465, 472, 475, 513, 520

Offset: 1

Vincenzo Librandi, Feb 28 2015

Numbers n such that A033570(2n) is semiprime.

			10 is in this sequence because 4*10+1=41 and 6*10+1=61 are primes.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001358, A033570, A130800, A186721.

Cf. A255584: semiprimes of the form (4*n+1)*(6*n+1).

[n: n in [1..600] | IsPrime(6*n+1) and IsPrime(4*n+1)];

A255607:=n->`if`(isprime(4*n+1) and isprime(6*n+1), n, NULL): seq(A255607(n), n=1..600); # Wesley Ivan Hurt, Feb 28 2015

Select[Range[600], PrimeQ[4 # + 1] && PrimeQ[6 # + 1] &]
Select[Range[600],AllTrue[{4#,6#}+1,PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 22 2020 *)

for(n=1,10^3,if(isprime(4*n+1)&&isprime(6*n+1),print1(n,", "))) \\ Derek Orr, Mar 01 2015

select( is_A255607(n)=isprime(4*n+1)&&isprime(6*n+1), [1..555]) \\ M. F. Hasler, Dec 13 2019

a(n) = A130800(n)/2.

A330409 Semiprimes of the form p(6p - 1).

Original entry on oeis.org

22, 51, 145, 287, 1717, 2147, 3151, 5017, 11051, 13207, 16801, 20827, 26867, 63551, 68587, 71177, 76501, 96647, 112477, 147737, 159251, 232657, 237407, 308947, 314417, 342487, 433897, 480251, 587501, 602617, 722107, 772927, 834401, 861467, 879751, 907537, 945257, 1155887, 1177051

Offset: 1

M. F. Hasler, Dec 13 2019

nonn

			A158015(1) = 2 is the smallest prime p such that 6p - 1 = 12 - 1 = 11 is also prime, whence a(1) = A049452(2) = 2*(6*2 - 1) = 22.
prime(5) = 11 is the smallest prime not in A024898 or A158015, because 6p - 1 is not a prime, therefore A049452(11) = 11*(6*11 - 1) is not in the sequence, and idem for A049452(13).
But prime(7) = 17 is in A024898 and A158015, so a(5) = A024898(A158015(5)) = A024898(17) = 17*(6*17 - 1).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A024898 (6n-1 is prime), A158015 (primes), A049452 = {n(6n-1)}.

Complement of A255584 = A033570(A130800) (semiprimes (2n+1)(3n+1)) in A245365 (primes of the form n(3n-1)/2).

Select[Table[p(6p-1),{p,500}],PrimeOmega[#]==2&] (* Harvey P. Dale, Apr 27 2022 *)

[p*(6*p-1) | p<-primes(99), isprime(6*p-1)]

a(n) = A049452(A158015(n)) = p(6p - 1) with p = A158015(n).

A330410 a(n) = 6*prime(n) - 1.

Original entry on oeis.org

11, 17, 29, 41, 65, 77, 101, 113, 137, 173, 185, 221, 245, 257, 281, 317, 353, 365, 401, 425, 437, 473, 497, 533, 581, 605, 617, 641, 653, 677, 761, 785, 821, 833, 893, 905, 941, 977, 1001, 1037, 1073, 1085, 1145, 1157, 1181, 1193, 1265, 1337, 1361, 1373, 1397, 1433, 1445

Offset: 1

M. F. Hasler, Dec 13 2019

nonn

Composite terms are a(k) with k in {5, 6, 11, 12, 13, 18, 20, 21, ...} = indices of primes missing in A158015. Primes are A016969(A158015 - 1).

Cf. A000040 (primes), A016969 (6n+5), A024898 (6n-1 is prime), A158015 (primes in A024898), A049452 = {n(6n-1)}, A255584 = A033570(A130800) (semiprimes (2n+1)(3n+1)), A245365 (primes of the form n(3n-1)/2).

PARI
```
apply( a(n)=6*prime(n)-1, [1..99])
```
PARI
```
apply( n->6*n-1, primes(99))
```

a(n) = A016969(A000040(n)-1) = 6p - 1 with p = A000040(n) = prime(n).

cp's OEIS Frontend

A245365 Semiprimes of the form n*(3*n-1)/2.

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A255584 Semiprimes of the form (4*n + 1)*(6*n + 1) = 24*n^2 + 10*n + 1.

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A255607 Numbers n such that both 4*n+1 and 6*n+1 are primes.

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A330409 Semiprimes of the form p(6p - 1).

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A330410 a(n) = 6*prime(n) - 1.

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A245365 Semiprimes of the form n(3n-1)/2.

A255584 Semiprimes of the form (4n + 1)(6n + 1) = 24n^2 + 10*n + 1.

A255607 Numbers n such that both 4n+1 and 6n+1 are primes.