cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

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

Views

Author

Shyam Sunder Gupta, Oct 19 2002

Keywords

Examples

			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.

Links

Crossrefs

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

Programs

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

Formula

q:= n-> is(numtheory[bigomega](n)=3):
select(q, [i*(i+1)/2$i=0..200])[]; # Alois P. Heinz, Mar 27 2024

A128905 Numbers k such that the k-th triangular number has exactly four distinct prime factors.

Original entry on oeis.org

20, 51, 59, 60, 65, 68, 69, 76, 77, 83, 91, 92, 105, 110, 114, 115, 123, 129, 131, 139, 154, 156, 165, 182, 185, 186, 187, 194, 210, 212, 221, 227, 228, 235, 236, 237, 246, 254, 258, 265, 266, 267, 273, 276, 286, 290, 291, 307, 309, 318, 321, 322, 330, 345
Offset: 1

Views

Author

Zak Seidov, Apr 22 2007

Keywords

Comments

Or, indices of triangular numbers with exactly four distinct prime factors.

Examples

			In order of increasing p (the least prime factor of T(k)):
  a(1)  =  20 because T(20)  =    210 =  2* 3* 5* 7,
  a(5)  =  65 because T(65)  =   2145 =  3* 5*11*13,
  a(21) = 154 because T(154) =  11935 =  5* 7*11*31,
  a(45) = 286 because T(286) =  41041 =  7*11*13*41,
  a(143)= 781 because T(781) = 305371 = 11*17*23*71,
  a(91) = 493 because T(493) = 121771 = 13*17*19*29, etc.

Crossrefs

Cf. A000217, A068443, A069903, A076551, A127637, A128896.

Programs

  • Mathematica
    lim=346;tn=Rest[Array[ #*(# - 1)/2 &,lim]];Select[Range[lim-1],PrimeNu[tn[[#]]]==PrimeOmega[tn[[#]]]==4&] (* James C. McMahon, Jan 12 2025 *)

Formula

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

A333771 Triangular numbers that are the product of four distinct primes.

Original entry on oeis.org

210, 1326, 1770, 1830, 2145, 2346, 2415, 2926, 3003, 3486, 4186, 4278, 5565, 6105, 6555, 6670, 7626, 8385, 8646, 9730, 11935, 12246, 13695, 16653, 17205, 17391, 17578, 18915, 22155, 22578, 24531, 25878, 26106, 27730, 27966, 28203, 30381, 32385, 33411, 35245
Offset: 1

Views

Author

Jon E. Schoenfield, Apr 04 2020

Keywords

Comments

The maximum exponent for each prime in the factorization of each term is one. - Harvey P. Dale, Jul 21 2021

Examples

			The 20th triangular number, T(20) = 20*21/2 = 210 = 2 * 3 * 5 * 7, so 210 is a term.
T(1333) = 889111 = 23 * 29 * 31 * 43, so 889111 is a term.

Links

Crossrefs

Cf. A000217 (triangular numbers), A068443 (triangular numbers that are the product of 2 distinct primes), A128896 (triangular numbers that are the product of 3 distinct primes).
Cf. A076578, A121479.

Programs

  • Maple
    q:= n-> map(i-> i[2], ifactors(n)[2])=[1$4]:
select(q, [seq(n*(n+1)/2, n=0..300)])[];  # Alois P. Heinz, Apr 04 2020
  • Mathematica
    Select[Accumulate[Range[300]],PrimeNu[#]==PrimeOmega[#]==4&] (* Harvey P. Dale, Jul 21 2021 *)

A348185 Smallest number k in a set of three consecutive triangular numbers that are sphenic.

Original entry on oeis.org

406, 861, 39621, 2166321, 3924201, 11146281, 14804961, 19198306, 73951041, 83417986, 97951006, 209643526, 310415986, 522339681, 526225461, 583333246, 611153241, 801460666, 1601581906, 2520251506, 2690954841, 4455349606, 6681853401, 9895642221, 13878029901
Offset: 1

Views

Author

G. L. Honaker, Jr., Oct 05 2021

Keywords

Comments

a(2)-a(9) from Chuck Gaydos.

Examples

			a(1)=406 because 406 is the smallest number in the first set of three consecutive triangular numbers that are sphenic, i.e., {406=2*7*29, 435=3*5*29, 465=3*5*31}.

Links

Crossrefs

Cf. A000217 (triangular numbers), A007304 (sphenic numbers), A128896 (sphenic triangular numbers). Subsequence of A349696.

Programs

  • Mathematica
    t[n_] := n*(n+1)/2; spQ[n_] := FactorInteger[n][[;;,2]] == {1,1,1}; Select[Partition[t /@ Range[170000], 3, 1], AllTrue[#, spQ] &][[;; , 1]] (* Amiram Eldar, Oct 06 2021 *)

Extensions

a(10)-a(25) from Alois P. Heinz, Oct 05 2021

A349696 Smallest number in a set of three consecutive triangular numbers each with three prime factors (counted with multiplicity).

Original entry on oeis.org

153, 406, 861, 39621, 2166321, 3924201, 11146281, 14804961, 19198306, 73951041, 83417986, 97951006, 209643526, 310415986, 522339681, 526225461, 583333246, 611153241, 801460666, 1601581906, 2520251506, 2690954841, 4455349606, 6681853401, 9895642221, 13878029901
Offset: 1

Views

Author

Shyam Sunder Gupta, Nov 25 2021

Keywords

Comments

153 is the only known number in the sequence which is not squarefree.
From Robert Israel, Mar 11 2025: (Start)
Terms are of the form A000217(k) with either
k prime == 5 (mod 12), k + 1 = 6 * prime, k + 2 prime, k + 3 = 4 * prime
or
k = 4 * prime == 4 (mod 12), k + 1 prime, k + 2 = 6 * prime, k + 3 prime.
In particular, for k > 17 (where k = 17 corresponds to a(1) = 153), the primes mentioned above are > 3, and so the terms except for 153 are squarefree. (End)

Examples

			a(1) = 153 because 153 is the smallest number in the first set of three consecutive triangular numbers with three prime factors (counted with multiplicity), i.e., (153 = 3*3*17, 171 = 3*3*19, 190 = 2*5*19).

Links

Crossrefs

Cf. A000217, A128896, A255200, A348185.

Programs

  • Maple
    R:= NULL: count:= 0:
for i from 1 while count < 100 do
  k:= 12*i+4;
  if isprime(k+1) and isprime((k+2)/6) and isprime(k+3) then
     if isprime(k/4) then R:= R, k*(k+1)/2; count:= count+1; fi;
     if isprime((k+4)/4) then R:= R, (k+1)*(k+2)/2; count:= count+1; fi;
  fi;
od:
R; # Robert Israel, Mar 11 2025
  • Mathematica
    t[n_] := n*(n + 1)/2; q[n_] := PrimeOmega[n] == 3; Select[Partition[t /@ Range[10^5], 3, 1], AllTrue[#, q] &][[;; , 1]] (* Amiram Eldar, Nov 25 2021 *)
(#(#+1))/2&/@SequencePosition[PrimeOmega[Accumulate[Range[170000]]],{3,3,3}][[;;,1]] (* Harvey P. Dale, Oct 20 2023 *)

Formula

a(n) = A000217(A255200(n)). - Michel Marcus, Dec 25 2021

Extensions

Definition clarified by Harvey P. Dale, Oct 20 2023

A349539 Smallest number m in a set of at least three consecutive triangular numbers with three distinct prime factors.

Original entry on oeis.org

378, 406, 528, 820, 861, 1953, 2485, 3081, 5050, 5151, 5778, 7750, 9316, 11026, 11175, 18145, 19306, 19503, 36046, 36315, 39621, 92665, 93096, 130816, 131328, 135981, 205120, 326836, 337431, 661825, 816003, 1439056, 1993006, 1995003, 2166321, 2835771
Offset: 1

Views

Author

Shyam Sunder Gupta, Nov 25 2021

Keywords

Examples

			a(1) = 378 because 378 is the smallest number in the first set of three consecutive triangular numbers with three distinct prime factors, i.e., (378 = 2*3^3*7, 406 = 2*7*29, 435 = 3*5*29).

Crossrefs

Cf. A000217, A128896, A348185.

Programs

  • Mathematica
    t[n_] := n*(n + 1)/2; q[n_] := PrimeNu[n] == 3; Select[Partition[t /@ Range[3*10^3], 3, 1], AllTrue[#, q] &][[;; , 1]] (* Amiram Eldar, Nov 26 2021 *)

Extensions

Name clarified by Michel Marcus, Dec 02 2021

A357590 Triangular numbers which are products of five distinct primes.

Original entry on oeis.org

3570, 8778, 9870, 12090, 13530, 20706, 20910, 21945, 24090, 24310, 26565, 33670, 40470, 40755, 47586, 54285, 57630, 57970, 63546, 66430, 69006, 72390, 76245, 87990, 88410, 91806, 92235, 94395, 94830, 98790, 121278, 130305, 132870, 133386, 141778, 148785, 154290, 159330, 163878, 167910
Offset: 1

Views

Author

Massimo Kofler, Oct 05 2022

Keywords

Comments

A squarefree subsequence of triangular numbers (T(n) = n*(n+1)/2).

Examples

			3570 = 2*3*5*7*17 = 84*85/2.
21945 = 3*5*7*11*19 = 209*210/2.
121278 = 2*3*17*29*41 = 492*493/2.
154290 = 2*3*5*37*139 = 555*556/2.

Crossrefs

Intersection of A000217 and A046387.
Cf. A068443, A128896, A333771.

Programs

  • Maple
    q:= n-> map(i-> i[2], ifactors(n)[2])=[1$5]:
select(q, [seq(n*(n+1)/2, n=0..1000)])[];  # Alois P. Heinz, Oct 05 2022
  • Mathematica
    Select[Accumulate @ Range[600], FactorInteger[#][[;; , 2]] == {1, 1, 1, 1, 1} &] (* Amiram Eldar, Oct 05 2022 *)
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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