A068443 -id:A068443 - cp's OEIS frontend

A156592 Product pq of two primes with q = 2p + 1.

10, 21, 55, 253, 1081, 1711, 3403, 5671, 13861, 15931, 25651, 34453, 60031, 64261, 73153, 108811, 114481, 126253, 158203, 171991, 258121, 351541, 371953, 392941, 482653, 518671, 703891, 822403, 853471, 869221, 933661, 1034641, 1104841

Offset: 1

Reinhard Zumkeller, Feb 10 2009

nonn

Subsequence of A068443.

Products of Sophie Germain primes p with their corresponding safe primes 2p+1. The smallest prime factor of a(n) is (a(n) - phi(a(n)))/3 and the largest prime factor of a(n) is 2(a(n) - phi(a(n)))/3 + 1. - Wesley Ivan Hurt, Oct 03 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..4000
Math Marathon (MM95) (in Russian) [from Vladimir Joseph Stephan Orlovsky, May 12 2010]

Cf. A005384, A005385. Subset of A001358.

lst={};Do[p=Prime[n];q=2*p+1;If[PrimeQ[q],AppendTo[lst,p*q]],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, May 12 2010 *)

a(n) = A005384(n)*A005385(n).

A164977 Numbers m such that the set {1..m} has only one nontrivial decomposition into subsets with equal element sum.

Original entry on oeis.org

3, 4, 5, 6, 10, 13, 22, 37, 46, 58, 61, 73, 82, 106, 157, 166, 178, 193, 226, 262, 277, 313, 346, 358, 382, 397, 421, 457, 466, 478, 502, 541, 562, 586, 613, 661, 673, 718, 733, 757, 838, 862, 877, 886, 982, 997, 1018, 1093, 1153, 1186, 1201, 1213, 1237, 1282

Offset: 1

Alois P. Heinz, Sep 03 2009

Numbers m such that the triangular number T(m) = m*(m+1)/2 has exactly two divisors >= m.
Also numbers m such that m*(m+1)/2 is the product of two primes.
Contains all numbers in A005383. - Harry Richman, Jan 09 2025
Contains all numbers in A077065. - Alois P. Heinz, Jan 19 2025

			10 is in the sequence, because there is only one nontrivial decomposition of {1..10} into subsets with equal element sum: {1,10}, {2,9}, {3,8}, {4,7}, {5,6}; 11|55.
13 is in the sequence with decomposition of {1..13}: {1,12}, {2,11}, {3,10}, {4,9}, {5,8}, {6,7}, {13}; 13|91.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A000217, A001222, A164978, A035470, A068443, A069904, A001358.

Cf. A005383, A077065 (distinct subsequences).

a:= proc(n) option remember; local k;
      for k from 1+ `if`(n=1, 2, a(n-1))
      while not (andmap(isprime, [k, (k+1)/2]) or
                 andmap(isprime, [k+1, k/2]))
      do od; k
    end:
seq(a(n), n=1..100);

Select[Range@1304, PrimeOmega[#] + PrimeOmega[# + 1] == 3 &] (* Robert G. Wilson v, Jun 28 2010 and updated Sep 21 2018 *)

is(n)=if(isprime(n),bigomega(n+1)==2, isprime(n+1) && bigomega(n)==2) \\ Charles R Greathouse IV, Sep 08 2015

is(n)=if(n%2, isprime((n+1)/2) && isprime(n), isprime(n/2) && isprime(n+1)) \\ Charles R Greathouse IV, Mar 16 2022

list(lim)=my(v=List()); forprime(p=3,lim, if(isprime((p+1)/2), listput(v,p))); forprime(p=5,lim+1, if(isprime(p\2), listput(v,p-1))); Set(v) \\ Charles R Greathouse IV, Mar 16 2022

{ m :  A035470(m) = 2 }.
{ m :  A164978(m) = 2 }.
{ m : |{d|m*(m+1)/2 : d>=m}| = 2 }.
{ m :  m*(m+1)/2 in {A068443} }.
{ m :  m*(m+1)/2 in {A001358} }.
{ m :  A069904(m) = 2 }.
{ m :  A001222(n) + A001222(n+1) = 3 }. - Alois P. Heinz, Jan 08 2022
{ A005383 } union { A077065 }. - Alois P. Heinz, Jan 19 2025

A128896 Triangular numbers that are products of three distinct primes.

Original entry on oeis.org

66, 78, 105, 190, 231, 406, 435, 465, 561, 595, 741, 861, 903, 946, 1378, 1653, 2211, 2278, 2485, 3081, 3655, 3741, 4371, 4465, 5151, 5253, 5995, 6441, 7021, 7503, 8515, 8911, 9453, 9591, 10011, 10153, 10585, 11026, 12561, 13366, 14878, 15051, 15753

Offset: 1

Zak Seidov, Apr 20 2007

nonn

			a(1)=T(11)=66=2*3*11, a(2)=T(12)=78=2*3*13, a(3)=T(14)=105=3*5*7, a(4)=T(19)=190=2*5*19, a(5)=T(21)=231=3*7*11, a(6)=T(28)=406=2*7*29.
T(15) = 120 = 2^3*3*5. The triangular 120 has three prime factors but is not a product of these factors. Thus, 120 is not in this sequence.

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

Cf. A000217, A068443, A069903, A076551, A127637.

Select[Table[n(n+1)/2,{n,1,210}],Transpose[FactorInteger[ # ]][[2]]=={1,1,1}&]
Select[Accumulate[Range[200]],PrimeNu[#]==PrimeOmega[#]==3&] (* Harvey P. Dale, Apr 23 2017 *)

a(n) = T(k) = k*(k+1)/2 = p*q*r for some k,p,q,r, where T(k) is triangular number and p, q, r are distinct primes.

Equals A000217 INTERSECT A007304 and A075875 INTERSECT A121478. - R. J. Mathar, Apr 22 2007

Name clarified by Tanya Khovanova, Sep 06 2022

A075875 Triangular numbers that are 3-almost primes.

Original entry on oeis.org

28, 45, 66, 78, 105, 153, 171, 190, 231, 325, 406, 435, 465, 561, 595, 741, 861, 903, 946, 1378, 1653, 2211, 2278, 2485, 3081, 3655, 3741, 4371, 4465, 4753, 5151, 5253, 5995, 6441, 7021, 7381, 7503, 8515, 8911, 9453, 9591, 10011, 10153, 10585, 11026

Offset: 1

Shyam Sunder Gupta, Oct 19 2002

			a(1)=28, 28 is a triangular number and 28 = 2*2*7, i.e., is a product of 3 prime factors so is 3-almost prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A000217, A014612, A068443, A128896 (subsequence).

Select[Accumulate[Range[200]],PrimeOmega[#]==3&] (* Harvey P. Dale, Jul 24 2012 *)

issemi(n)=bigomega(n)==2
ok(m,n)=if(isprime(m), issemi(n), isprime(n) && issemi(m))
list(lim)=my(v=List()); lim\=1; for(n=7,(sqrt(8*lim+1)-1)\2, if(if(n%2, ok(n,(n+1)/2), ok(n/2,n+1)), listput(v, n*(n+1)/2))); Vec(v) \\ Charles R Greathouse IV, Jun 12 2017

q:= n-> is(numtheory[bigomega](n)=3):

select(q, [i*(i+1)/2$i=0..200])[]; # Alois P. Heinz, Mar 27 2024

A119663 Triangular numbers with at most two distinct prime factors.

Original entry on oeis.org

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 91, 136, 153, 171, 253, 325, 351, 496, 703, 1081, 1225, 1431, 1711, 1891, 2701, 3321, 3403, 4753, 5671, 7381, 8128, 12403, 13203, 13861, 15931, 18721, 25651, 29161, 29403, 31375, 32896, 34453, 38503, 49141

Offset: 1

Greg Huber, Jul 28 2006

1 and 3 are the only terms with less than two prime factors.

			a(6) = 3 * 7, a(7) = 2^2 * 7, a(8) = 2^2 * 3^2.

Klaus Brockhaus, Table of n, a(n) for n=1..1000

Cf. A000217, A068443, A005384.

Select[Accumulate[Range[0, 320]] ,PrimeNu[#]<3&] (* James C. McMahon, Sep 17 2024 *)

for(n=1, 320, k=binomial(n+1, 2); if(omega(k)<=2, print1(k, ", "))) \\ Klaus Brockhaus, Jul 30 2006

More terms from Klaus Brockhaus, Jul 30 2006 and May 21 2008

A076578 Triangular numbers which are 4-almost primes.

Original entry on oeis.org

36, 136, 210, 276, 351, 666, 820, 1035, 1225, 1275, 1326, 1431, 1770, 1830, 1953, 2145, 2346, 2415, 2775, 2926, 3003, 3486, 3916, 4005, 4186, 4278, 5050, 5356, 5565, 6105, 6555, 6670, 6903, 7626, 8001, 8385, 8646, 9316, 9730, 10731, 11175, 11325, 11476, 11935

Offset: 1

Shyam Sunder Gupta, Oct 19 2002

			36 is a term because it is a triangular number and 36 = 2*2*3*3, i.e., is a product of 4 prime factors so is a 4-almost prime.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Intersection of A000217 and A014613.

Cf. A068443, A075875.

q:= n-> is(numtheory[bigomega](n)=4):
select(q, [i*(i+1)/2$i=0..200])[];  # Alois P. Heinz, Mar 27 2024

q[n_] := PrimeOmega[n] == 4;
Select[Table[i*(i+1)/2, {i, 0, 200}], q] (* Jean-François Alcover, Jan 13 2025, after Alois P. Heinz *)

A109927 First primes p connected to two primes either by 2p+1 or 2p-1 upward [downward (p-1)/2 or (p+1)/2].

Original entry on oeis.org

3, 5, 11, 23, 37, 83, 157, 179, 359, 661, 719, 877, 997, 1019, 1237, 1439, 1657, 2039, 2063, 2137, 2459, 2557, 2819, 2903, 2963, 3023, 3061, 3623, 3779, 3803, 3863, 4177, 4261, 4357, 4621, 4919, 5399, 5581, 5639, 6037, 6121, 6217, 6361, 6899, 6983, 7079

Offset: 1

Alexandre Wajnberg, Aug 31 2005

These primes may be part of Cunningham chains longer than three terms. It seems the two operators are never mixed, except for 3, 5 and 7: -for 3, we have: 2 through <2p-1> -> 3 through <2p+1> -> 7 -for 5: 3 <2p-1> -> 5 <2p+1> -> 11 -for 7: 3 <2p+1> -> 7 <2p-1> -> 13
For p > 7, such a mixed chain with p in the middle is impossible because the number 3 would be a nontrivial factor of either the smallest or the largest term. - Jeppe Stig Nielsen, May 05 2019
Primes (excluding 2 and 7) that divide more than one semiprime triangular number A068443. - Jeppe Stig Nielsen, May 05 2019
The disjoint union of A059455 and A109835. - Jeppe Stig Nielsen, May 05 2019

			a(3)=11 is here because 5->11->23 through <2p+1>;
a(4)=23 because 11->23->47 through <2p+1>;
a(5)=37 because 19->37->73 through <2p-1>.

Chris Caldwell's Prime Glossary, Cunningham chains.

Cf. A005382, A005383, A005384, A005385, A059455, A068497.

forprime(p=3,10^6,if(p%3==2,isprime((p-1)/2)&&isprime(2*p+1),isprime((p+1)/2)&&isprime(2*p-1))&&print1(p,", ")) \\ Jeppe Stig Nielsen, May 05 2019

A113940 Triangular numbers that are also brilliant (A078972).

Original entry on oeis.org

6, 10, 15, 21, 253, 703, 1081, 1711, 1891, 2701, 3403, 25651, 34453, 38503, 49141, 60031, 64261, 73153, 79003, 88831, 104653, 108811, 114481, 126253, 146611, 158203, 171991, 188191, 218791, 226801, 258121, 269011, 286903, 351541, 371953, 385003, 392941

Offset: 1

Giovanni Resta, Jan 31 2006

Smallest value where each factor has n digits for n = 1, 2, 3, 4, 5, are: 6 = 2 * 3; 253 = 11 * 23; 25651 = 113 * 227; 2035153 = 1009 * 2017; 202457503 = 10061 * 20123. [From Jonathan Vos Post, Apr 04 2009]

			253 = T(22) and 253 = 11*23 is brilliant.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A068443, A078972, A000217.

brilQ[n_]:=Module[{fin=FactorInteger[n]},Total[Transpose[fin][[2]]]==2&& Length[Union[IntegerLength[Transpose[fin][[1]]]]]==1]
Intersection[Accumulate[Range[850]],Select[Range[362000],brilQ]]  (* Harvey P. Dale, Feb 06 2011 *)

A000217 INTERSECTION A078972. Subset of A068443. [From Jonathan Vos Post, Apr 04 2009]

A121479 Triangular numbers with more than three distinct prime factors.

Original entry on oeis.org

210, 630, 780, 990, 1326, 1540, 1596, 1770, 1830, 2145, 2346, 2415, 2850, 2926, 3003, 3486, 3570, 3828, 4095, 4186, 4278, 4560, 4950, 5460, 5565, 6105, 6216, 6555, 6670, 6786, 7140, 7260, 7626, 8385, 8646, 8778, 9180, 9730, 9870, 10296, 10440, 10878

Offset: 1

Klaus Brockhaus, Aug 01 2006

			20*21/2 = 2*3*5*7 = 210 is the smalles triangular number with more than three distinct prime factors, hence a(1) = 210.

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

Cf. A000217, A068443, A119663, A121478.

Select[Accumulate[Range[200]],PrimeNu[#]>3&] (* Harvey P. Dale, Jun 06 2013 *)

for(n=1,100,k=binomial(n+1,2);if(omega(k)>3,print1(k,",")))

A298855 Squarefree semiprimes pq for which the symmetric representation of sigma(pq) has four parts, in increasing order.

Original entry on oeis.org

21, 33, 39, 51, 55, 57, 65, 69, 85, 87, 93, 95, 111, 115, 119, 123, 129, 133, 141, 145, 155, 159, 161, 177, 183, 185, 201, 203, 205, 213, 215, 217, 219, 235, 237, 249, 253, 259, 265, 267, 287, 291, 295, 301, 303, 305, 309, 319, 321, 327, 329, 335, 339, 341, 355, 365, 371, 377, 381, 393, 395

Offset: 1

Hartmut F. W. Hoft, Jan 27 2018

All numbers in this sequence are odd since the symmetric representation of 2*p, p prime > 3, has two parts each of size 3*(p+1)/2, and that for 6 has one part of size 12.
A number in this sequence has the form p*q, p and q prime, 3 <= p and 2*p < q, since in this case 2*p <= floor((sqrt(8*p*q + 1) - 1)/2) < q so that 1's in row p*q of A237048 occur only in positions 1, 2, p and 2*p.
This sequence is a subsequence of A046388, hence of A006881, as well as of A174905, A241008 and A280107.
The two central parts of the symmetric representation of sigma(p*q), each of size (p+q)/2, meet on the diagonal when q = 2*p + 1 since in this case 2*p = floor((sqrt(8*p*q + 1) - 1)/2). These triangular numbers p*(2p+1) form sequence A156592, except for its first element 10, and form a subsequence of the diagonal in the associated irregular triangle of this sequence given in the Example section. They also are a subsequence of A264104. A function to compute the coordinates on the diagonal where the two central parts meet is defined in sequence A240542.
Except for missing 10 the intersection of this sequence and A298856 equals A156592.

			21=3*7 is the smallest number in the sequence since 2*3<7.
1081=23*(2*23+1) is in the sequence; its central parts meet at 751 on the diagonal.
The semiprimes p*q can be arranged as an irregular triangle with rows and columns labeled by the respective odd primes:
  q\p|   3    5    7   11   13   17   19   23
  ---+---------------------------------------
   7 |  21
  11 |  33   55
  13 |  39   65
  17 |  51   85  119
  19 |  57   95  133
  23 |  69  115  161  253
  29 |  87  145  203  319  377
  31 |  93  155  217  341  403
  37 | 111  185  259  407  481  629
  41 | 123  205  287  451  533  697  779
  43 | 129  215  301  473  559  731  817
  47 | 141  235  329  517  611  799  893 1081

Cf. A001358, A005384, A005385, A006881, A046388, A068443, A156592, A174905, A237048, A237270, A237593, A240542, A241008, A264104, A280107, A298856.

(* Function a237270[] is defined in A237270 *)
a006881Q[n_] := Module[{f=FactorInteger[n]}, Length[f]==2 && AllTrue[Last[Transpose[f]], #==1&]]
a298855[m_, n_] := Select[Range[m, n], a006881Q[#] && Length[a237270[#]]==4 &]
a298855[1, 400] (* data *)
(* column for prime p through number n *)
stalk[n_, p_] := Select[a298855[1, n], First[First[FactorInteger[#]]]==p&]

cp's OEIS Frontend

A156592 Product p*q of two primes with q = 2*p + 1.

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A164977 Numbers m such that the set {1..m} has only one nontrivial decomposition into subsets with equal element sum.

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A128896 Triangular numbers that are products of three distinct primes.

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A075875 Triangular numbers that are 3-almost primes.

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A119663 Triangular numbers with at most two distinct prime factors.

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A076578 Triangular numbers which are 4-almost primes.

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Maple

Mathematica

A109927 First primes p connected to two primes either by 2p+1 or 2p-1 upward [downward (p-1)/2 or (p+1)/2].

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A156592 Product pq of two primes with q = 2p + 1.

A298855 Squarefree semiprimes pq for which the symmetric representation of sigma(pq) has four parts, in increasing order.