A129521
Numbers of the form p*q, p and q prime with q=2*p-1.
Original entry on oeis.org
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
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a129521 n = p * (2 * p - 1) where p = a005382 n
-- Reinhard Zumkeller, Nov 10 2013
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[2*n^2-n: n in [0..1000]|IsPrime(n) and IsPrime(2*n-1)]; // Vincenzo Librandi, Dec 27 2010
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p = Select[Prime[Range[155]], PrimeQ[2# - 1] &]; p (2p - 1) (* Robert G. Wilson v, Sep 11 2011 *)
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forprime(p=2,10000,q=2*p-1;if(isprime(q),print1(p*q,", ")))
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
The heptagonal number 34 is in the sequence because 34 = 2 * 17.
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IsSemiprime:=func; [s: n in [2..300] | IsSemiprime(s) where s is n*(5*n-3) div 2]; // Vincenzo Librandi, Jul 04 2015
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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 *)
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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)))
A382231
Octagonal numbers that are the product of three distinct primes.
Original entry on oeis.org
645, 1045, 1281, 2465, 2821, 3201, 3605, 7701, 8965, 12545, 15841, 17633, 18565, 20501, 23585, 24661, 25761, 26885, 30401, 34133, 36741, 45141, 51221, 52801, 57685, 59361, 62785, 66305, 68101, 71765, 73633, 89441, 95765, 100101, 116033, 120801, 123221, 125665, 138245
Offset: 1
645 is a term because 645=3*5*43 is a sphenic number and is the 15th octagonal number.
1045 is a term because 1045=5*11*19 is a sphenic number and is the 19th octagonal number.
1281 is a term because 1281=3*7*61 is a sphenic number and is the 21st octagonal number.
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N:= 10^6: # for terms <= N
isoct:= proc(n) issqr(1+3*n) and sqrt(1+3*n) mod 3 = 2 end proc:
P:= select(isprime,[seq(i,i=3..N/15,2)]): nP:= nops(P):
R:= NULL:
for i from 1 to nP while P[i]*P[i+1]*P[i+2] <= N do
for j from i+1 to nP while P[i]*P[j]*P[j+1] <= N do
for k from j+1 to nP do
v:= P[i]*P[j]*P[k];
if v > N then break fi;
if isoct(v) then R:= R,v fi;
od od od:
sort([R]); # Robert Israel, Mar 19 2025
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Select[Table[n*(3*n-2), {n, 1, 220}], FactorInteger[#][[;;, 2]] == {1, 1, 1} &] (* Amiram Eldar, Mar 19 2025 *)
Showing 1-3 of 3 results.
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