A245365 -id:A245365 - cp's OEIS frontend

A129521 Numbers of the form pq, p and q prime with q=2p-1.

6, 15, 91, 703, 1891, 2701, 12403, 18721, 38503, 49141, 79003, 88831, 104653, 146611, 188191, 218791, 226801, 269011, 286903, 385003, 497503, 597871, 665281, 721801, 736291, 765703, 873181, 954271, 1056331, 1314631, 1373653, 1537381

Offset: 1

Reinhard Zumkeller, Apr 19 2007

nonn

All terms are Fermat 4-pseudoprimes, i.e., satisfy 4^n == 4 (mod n). See A020136 and A122781.

David W. Wilson, Table of n, a(n) for n = 1..1000
D. E. Iannucci, When the small divisors of a natural number are in arithmetic progression, INTEGERS, Electronic Journal of Combinatorial Number Theory, #77, 2018. See p. 9.

Cf. A005382, A005383.
Subsequence of A006881, A129510, and A122781.
Intersection of A000384 and A001358, "hexagonal semiprimes". - Wesley Ivan Hurt, Jul 04 2013
Cf. A141768, A191311, A191592, A245365, A259676, A259677.

a129521 n = p * (2 * p - 1) where p = a005382 n
-- Reinhard Zumkeller, Nov 10 2013

[2*n^2-n: n in [0..1000]|IsPrime(n) and IsPrime(2*n-1)]; // Vincenzo Librandi, Dec 27 2010

p = Select[Prime[Range[155]], PrimeQ[2# - 1] &]; p (2p - 1) (* Robert G. Wilson v, Sep 11 2011 *)

forprime(p=2,10000,q=2*p-1;if(isprime(q),print1(p*q,", ")))

a(n) = A005382(n)*A005383(n).

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

A259677 Octagonal numbers (A000567) that are semiprimes (A001358).

Original entry on oeis.org

21, 65, 133, 341, 481, 1541, 4033, 5461, 6533, 8321, 11041, 13333, 14981, 31621, 38081, 48133, 56033, 79381, 83333, 97921, 109061, 111361, 133141, 188501, 197633, 206981, 219781, 229633, 256961, 282133, 293281, 328021, 340033, 360533, 416641, 481601, 556421

Offset: 1

Colin Barker, Jul 03 2015

nonn

			The octagonal number 21 is in the sequence because 21 = 3 * 7.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A129521, A245365, A259676.

IsSemiprime:=func; [s: n in [2..500] | IsSemiprime(s) where s is n*(3*n-2) ]; // Vincenzo Librandi, Jul 04 2015

a={}; Do[If[PrimeOmega[n (3 n - 2)]==2, AppendTo[a, n(3 n - 2)]], {n, 1, 200}]; a (* Vincenzo Librandi, Jul 04 2015 *)
Select[PolygonalNumber[8,Range[500]],PrimeOmega[#]==2&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 15 2019 *)

pg(m, n) = (n^2*(m-2)-n*(m-4))/2 \\ n-th m-gonal number
select(n->bigomega(n)==2, vector(2000, n, pg(8, n)))

Equals A000567 intersect A001358.

A259676 Heptagonal numbers (A000566) that are semiprimes (A001358).

Original entry on oeis.org

34, 55, 235, 403, 469, 697, 1177, 1651, 2059, 2839, 4141, 5221, 6943, 9211, 9517, 13213, 13579, 21949, 23377, 25351, 29539, 31753, 34633, 37027, 53071, 62173, 68641, 74563, 78943, 93799, 96727, 118483, 130759, 144841, 164737, 171217, 187279, 191407, 196981

Offset: 1

Colin Barker, Jul 03 2015

nonn

For these semiprimes k*(5*k-3)/2, the corresponding k are listed in A114517.

			The heptagonal number 34 is in the sequence because 34 = 2 * 17.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A129521, A245365, A259677.

IsSemiprime:=func; [s: n in [2..300] | IsSemiprime(s) where s is n*(5*n-3) div 2]; // Vincenzo Librandi, Jul 04 2015

a={}; Do[If[PrimeOmega[n (5 n - 3) / 2]==2, AppendTo[a, n(5 n - 3) / 2]], {n, 1, 200}]; a (* Vincenzo Librandi, Jul 04 2015 *)
Select[PolygonalNumber[7,Range[300]],PrimeOmega[#]==2&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 07 2021 *)

pg(m, n) = (n^2*(m-2)-n*(m-4))/2 \\ n-th m-gonal number
select(n->bigomega(n)==2, vector(2000, n, pg(7, n)))

Equals A000566 intersect A001358.

A381650 Pentagonal numbers which are products of three distinct primes.

Original entry on oeis.org

70, 590, 651, 715, 782, 1001, 1162, 1335, 1426, 2035, 2882, 5551, 5735, 6305, 6501, 7107, 7526, 8177, 8626, 9087, 9322, 10795, 11837, 12927, 14065, 20126, 22265, 24897, 25285, 26467, 28085, 29751, 31901, 32782, 34126, 35497, 36895, 37367, 38801, 40262, 41251, 43265, 44807, 45327

Offset: 1

Massimo Kofler, Mar 03 2025

nonn

			A000326(7) = 70 = 7*(3*7-1)/2 = 2*5*7.
A000326(20) = 590 = 20*(3*20-1)/2 = 2*5*59.
A000326(21) = 651 = 21*(3*21-1)/2 = 3*7*31.

Robert Israel, Table of n, a(n) for n = 1..703

Intersection of A000326 and A007304.

Cf. A245365.

N:= 10^5: # for terms <= N
ispent:= proc(n) issqr(1+24*n) and sqrt(1+24*n) mod 6 = 5 end proc:
P:= select(isprime,[2,seq(i,i=3..N/6,2)]): R:= {}:
nP:= nops(P):
for i1 from 3 to nP do
   p1:= P[i1];
   for i2 from 1 to i1-1 while p1 * P[i2] <= N/2 do
     p1p2:= p1*P[i2];
     m:= ListTools:-BinaryPlace(P[1..i2-1],N/p1p2);
     V:=select(ispent, P[1..m] *~ p1p2);
     if V <> [] then
        R:= R union convert(V,set);
     fi
od od:
sort(convert(R,list));# Robert Israel, Mar 10 2025

Select[Table[n*(3*n-1)/2, {n, 1, 200}], FactorInteger[#][[;; , 2]] == {1, 1, 1} &] (* Amiram Eldar, Mar 03 2025 *)

lista(n)= my(i=0); vector(n, t, while(factor(t=i++*(3*i-1)/2)[, 2]~ != [1, 1, 1], ); t); \\ Ruud H.G. van Tol, Mar 10 2025

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).

A381919 Pentagonal numbers which are products of four distinct primes.

Original entry on oeis.org

210, 330, 2262, 3290, 4030, 4510, 4845, 5370, 6902, 7315, 8855, 10542, 13490, 15555, 15862, 16485, 18095, 18426, 19437, 21182, 23002, 24130, 28497, 29330, 30602, 31465, 36426, 44290, 46905, 49595, 50142, 54626, 60501, 67310, 67947, 72490, 77862, 79235, 83426, 84135

Offset: 1

Massimo Kofler, Mar 10 2025

nonn

			A000326(12) = 210 = 12*(3*12-1)/2 = 2*3*5*7.
A000326(15) = 330 = 15*(3*15-1)/2 = 2*3*5*11.
A000326(57) = 4845 = 57*(3*57-1)/2 = 3*5*17*19.

Intersection of A000326 and A046386.

Cf. A245365, A381650.

N:= 10^5: # for terms <= N
P:= select(isprime,[2,seq(i,i=3..N/30,2)]): R:= {}:
nP:= nops(P):
for i1 from 3 to nP do
   p1:= P[i1];
   for i2 from 1 to i1-1 while p1 * P[i2] <= N/6 do
     p1p2:= p1*P[i2];
   for i3 from 1 to i2-1 while p1p2 * P[i3] <= N/2 do
     p1p2p3:= p1p2 * P[i3];
     m:= ListTools:-BinaryPlace(P[1..i3-1],N/p1p2p3);
     V:=select(ispent, P[1..m] *~ p1p2p3);
     if V <> [] then
        R:= R union convert(V,set);
     fi
od od od:
sort(convert(R,list)); # Robert Israel, Mar 10 2025

Select[Table[n*(3*n-1)/2, {n, 1, 250}], FactorInteger[#][[;; , 2]] == {1, 1, 1, 1} &] (* Amiram Eldar, Mar 10 2025 *)

cp's OEIS Frontend

A129521 Numbers of the form p*q, p and q prime with q=2*p-1.

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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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Magma

Magma

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A259677 Octagonal numbers (A000567) that are semiprimes (A001358).

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Magma

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A259676 Heptagonal numbers (A000566) that are semiprimes (A001358).

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Magma

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A381650 Pentagonal numbers which are products of three distinct primes.

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Maple

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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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A129521 Numbers of the form pq, p and q prime with q=2p-1.

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