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 1-10 of 12 results. Next

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

Views

Author

Zak Seidov, Apr 20 2007

Keywords

Examples

			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.

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

Formula

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

Extensions

Name clarified by Tanya Khovanova, Sep 06 2022

A358862 a(n) is the smallest n-gonal number with exactly n distinct prime factors.

Original entry on oeis.org

66, 44100, 11310, 103740, 3333330, 185040240, 15529888374, 626141842326, 21647593547580, 351877410344460, 82634328555218440, 2383985537862979050, 239213805711830629680
Offset: 3

Views

Author

Ilya Gutkovskiy, Dec 03 2022

Keywords

Comments

The corresponding indices of n-gonal numbers are 11, 210, 87, 228, 1155, 7854, 66612, 395646, 2193303, ...

Examples

			a(3) = 66, because 66 is a triangular number with 3 distinct prime factors {2, 3, 11} and this is the smallest such number.

Links

Crossrefs

Cf. A001221, A076551, A156236, A156237, A156238, A156239, A358863, A358864, A358865.

Programs

  • Mathematica
    Table[SelectFirst[PolygonalNumber[n,Range[400000]],PrimeNu[#]==n&],{n,3,10}] (* The program generates the first 8 terms of the sequence. *) (* Harvey P. Dale, Sep 09 2023 *)
  • PARI
    a(n) = if(n<3, return()); for(k=1, oo, my(t=(k*(n*k - n - 2*k + 4))\2); if(omega(t) == n, return(t))); \\ Daniel Suteu, Dec 04 2022
  • PARI
    omega_polygonals(A, B, n, k) = A=max(A, vecprod(primes(n))); (f(m, p, j) = my(list=List()); forprime(q=p, sqrtnint(B\m, j), my(v=m*q, r=nextprime(q+1)); while(v <= B, if(j==1, if(v>=A && ispolygonal(v,k), listput(list, v)), if(v*r <= B, list=concat(list, f(v, r, j-1)))); v *= q)); list); vecsort(Vec(f(1, 2, n)));
a(n, k=n) = if(n < 3, return()); my(x=vecprod(primes(n)), y=2*x); while(1, my(v=omega_polygonals(x, y, n, k)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Dec 04 2022

Extensions

a(12)-a(15) from Daniel Suteu, Dec 04 2022

A156238 Smallest heptagonal number with n distinct prime factors.

Original entry on oeis.org

7, 18, 286, 3010, 32890, 769230, 3333330, 159189030, 16015883940, 477463360374, 21643407275490, 1148540321999070, 18489352726664820, 4561561662153109614, 71000485538666794110, 14440652550858108745170, 927869754030522488795610
Offset: 1

Views

Author

Donovan Johnson, Feb 07 2009

Keywords

Comments

a(18) <= 150849873309136386205130310. - Donovan Johnson, Feb 15 2012

Examples

			a(9) = 16015883940 = 2^2*3^2*5*7*17*19*23*29*59. 16015883940 is the smallest heptagonal number with 9 distinct prime factors.

Links

Crossrefs

Cf. A000566, A076551, A156236, A156237, A156239.

Programs

  • Python
    from sympy import primefactors
def A000566(n): return n*(5*n-3)//2
def a(n):
    k = 1
    while len(primefactors(A000566(k))) != n: k += 1
    return A000566(k)
print([a(n) for n in range(1, 9)]) # Michael S. Branicky, Jul 18 2021
  • Python
    # faster version using heptagonal structure
from sympy import primefactors
def A000566(n): return n*(5*n-3)//2
def A000566_distinct_factors(n):
    pf1 = primefactors(n)
    pf2 = primefactors(5*n-3)
    combined = set(pf1) | set(pf2)
    return len(combined) if n%4 == 0 or (5*n-3)%4 == 0 else len(combined)-1
def a(n):
    k = 1
    while A000566_distinct_factors(k) != n: k += 1
    return A000566(k)
print([a(n) for n in range(1, 10)]) # Michael S. Branicky, Jul 18 2021

Extensions

a(17) from Donovan Johnson, Jul 02 2011

A127637 Smallest squarefree triangular number with exactly n prime factors.

Original entry on oeis.org

1, 3, 6, 66, 210, 3570, 207690, 930930, 56812470, 1803571770, 32395433070, 265257422430, 91348974206490, 24630635909489610, 438603767516904990, 14193386885746698630, 2378522762792139793830, 351206814022419685159830, 28791787439593010836313310
Offset: 0

Views

Author

Rick L. Shepherd, Jan 28 2007, Feb 03 2007

Keywords

Comments

The sequence of smallest squarefree triangular numbers with at least n prime factors has identical terms through 91348974206490 at least.
a(19) <= 8285055066500101241048306610. a(20) <= 120052594044654305809137933570. - Donovan Johnson, Feb 28 2012

Examples

			a(12) = 91348974206490 = 2*3*5*7*11*13*17*19*29*37*67*131 = A000217(13516580).

Links

Crossrefs

Cf. A075088, A076551, A005117, A000217.

Programs

  • PARI
    squarefree_omega_polygonals(A, B, n, k) = A=max(A, vecprod(primes(n))); (f(m, p, j) = my(list=List()); my(s=sqrtnint(B\m, j)); if(j==1, forprime(q=max(p, ceil(A/m)), s, if(ispolygonal(m*q, k), listput(list, m*q))), forprime(q=p, s, my(t=m*q); list=concat(list, f(t, q+1, j-1)))); list); vecsort(Vec(f(1, 2, n)));
a(n, k=3) = if(n==0, return(1)); my(x=vecprod(primes(n)), y=2*x); while(1, my(v=squarefree_omega_polygonals(x, y, n, k)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Jan 18 2023

Extensions

a(13)-a(16) from Donovan Johnson, Jan 28 2009
a(17) from Donovan Johnson, Feb 07 2009
a(18) from Donovan Johnson, Feb 28 2012

A156236 Smallest pentagonal number with n distinct prime factors.

Original entry on oeis.org

5, 12, 70, 210, 11310, 145860, 1560090, 23130030, 2491434660, 200736225690, 1906748513670, 281993526895080, 9427390580377770, 904832960818356570, 117584828859645537390, 2928413909631253063020, 400579656276180828206760
Offset: 1

Views

Author

Donovan Johnson, Feb 07 2009

Keywords

Comments

a(18) <= 44573764536301609937057730. - Donovan Johnson, Feb 15 2012

Examples

			a(9) = 2491434660 = 2^2*3*5*11*13*17*19*29*31. 2491434660 is the smallest pentagonal number with 9 distinct prime factors.

Links

Crossrefs

Cf. A000326, A076551, A156237, A156238, A156239.

Extensions

a(17) from Donovan Johnson, Jun 26 2011

A156237 Smallest hexagonal number with n distinct prime factors.

Original entry on oeis.org

6, 66, 630, 7140, 103740, 1272810, 56812470, 1722580860, 48098217090, 1850186768430, 139261952960130, 17743036637876550, 741902728913225880, 21549201398378163510, 2378522762792139793830, 351206814022419685159830
Offset: 2

Views

Author

Donovan Johnson, Feb 07 2009

Keywords

Comments

a(18) <= 45781615623002935783197090. - Donovan Johnson, Feb 15 2012

Examples

			a(9) = 1722580860 = 2^2*3*5*7*11*13*23*29*43. 1722580860 is the smallest hexagonal number with 9 distinct prime factors.

Links

Crossrefs

Cf. A000384, A076551, A156236, A156238, A156239.

Programs

  • Mathematica
    Module[{nn=167*10^5,c},c={#,PrimeNu[#]}&/@PolygonalNumber[6,Range[nn]];Table[ SelectFirst[ c,#[[2]]==n&],{n,2,12}]][[;;,1]] (* The program generates the first 11 terms of the sequence. *) (* Harvey P. Dale, Jan 19 2024 *)

A156239 Smallest octagonal number with n distinct prime factors.

Original entry on oeis.org

8, 21, 280, 1680, 38760, 326040, 10986360, 185040240, 4897368840, 383246454360, 13143876816840, 376306806515640, 27961718389364760, 3250163645572822440, 152582219844376633080, 6202664616058189439160, 1454199694916714984358120
Offset: 1

Views

Author

Donovan Johnson, Feb 07 2009

Keywords

Comments

a(18) <= 68286531655807008335271480. - Donovan Johnson, Feb 15 2012

Examples

			a(9) = 4897368840 = 2^3*3*5*7*13*17*23*31*37. 4897368840 is the smallest octagonal number with 9 distinct prime factors.

Links

Crossrefs

Cf. A000567, A076551, A156236, A156237, A156238.

Programs

  • Mathematica
    f[n_] := PrimeNu@ n; nn = 10; k = 1; t = Table[0, {nn}]; While[Times@@t == 0, oct = k(3k-2); a = f@ oct; If[ a <= nn && t[[a]] == 0, t[[a]] = k; Print[{a, oct}]]; k++]; t (* Robert G. Wilson v, Aug 23 2012 *)
  • Python
    from sympy import primefactors
def octagonal(n): return n*(3*n - 2)
def a(n):
    k = 1
    while len(primefactors(octagonal(k))) != n: k += 1
    return octagonal(k)
print([a(n) for n in range(1, 10)]) # Michael S. Branicky, Aug 21 2021
  • Python
    # faster version using octagonal structure
from sympy import primefactors, primorial
def A000567(n): return n*(3*n-2)
def A000567_distinct_factors(n):
    return len(set(primefactors(n)) | set(primefactors(3*n-2)))
def a(n):
    k, lb = 1, primorial(n)
    while A000567(k) < lb: k += 1
    while A000567_distinct_factors(k) != n: k += 1
    return A000567(k)
print([a(n) for n in range(1, 10)]) # Michael S. Branicky, Aug 21 2021

Extensions

a(17) from Donovan Johnson, Jul 03 2011

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.

A358928 a(n) is the smallest centered triangular number with exactly n distinct prime factors.

Original entry on oeis.org

1, 4, 10, 460, 9010, 772210, 20120860, 1553569960, 85507715710, 14932196985010, 1033664429333260, 197628216951078460, 21266854897681220860, 7423007155473283614010, 3108276166302017120182510, 851452464506763307285599610, 32749388246772812069108696710
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 06 2022

Keywords

Comments

a(n) cannot be divisible by a bunch of primes like 3, 7, 11, 13, ... as (3*k^2 + 3*k + 2)/2 is never a multiple of any of them. - David A. Corneth, Dec 12 2022
a(16) <= 1421044357661885128003268103460. - David A. Corneth, Dec 14 2022

Examples

			a(4) = 9010, because 9010 is a centered triangular number with 4 distinct prime factors {2, 5, 17, 53} and this is the smallest such number.

Links

Crossrefs

Cf. A001221, A005448, A076551, A156329, A358894, A358929.

Programs

  • Mathematica
    c[k_] := (3*k^2 + 3*k + 2)/2; a[n_] := Module[{k = 0, ck}, While[PrimeNu[ck = c[k]] != n, k++]; ck]; Array[a, 9, 0] (* Amiram Eldar, Dec 09 2022 *)
  • PARI
    a(n) = for(k=0, oo, my(t=3*k*(k+1)/2 + 1); if(omega(t) == n, return(t))); \\ Daniel Suteu, Dec 10 2022

Extensions

a(9)-a(11) from Daniel Suteu, Dec 10 2022
a(12)-a(13) from David A. Corneth, Dec 12 2022
a(13) corrected by Daniel Suteu, Dec 13 2022
a(14)-a(15) from David A. Corneth, Dec 14 2022
a(16) from Daniel Suteu, Dec 14 2022
a(15) corrected by Daniel Suteu, Dec 15 2022

A321852 a(n) is the smallest m for which binomial(m, 6) has exactly n distinct prime factors.

Original entry on oeis.org

6, 7, 8, 9, 10, 18, 26, 40, 77, 120, 210, 477, 715, 2227, 3290, 9065, 17020, 49915, 139195, 240465, 721929, 1124840, 4455445, 16319578, 26683220, 105655905, 134879176, 677868170, 3290262264
Offset: 0

Views

Author

Zachary M Franco, Nov 19 2018

Keywords

Crossrefs

Cf. A076551, A156329.

Programs

  • Mathematica
    a[n_] := Module[{m=6, t=1}, While[PrimeNu[t] != n, m++; t*=m/(m-6)]; m]; Array[a, 20] (* Amiram Eldar, Nov 27 2018 *)
  • PARI
    a(n)={my(m=6, t=1); while(omega(t)<>n, m++; t*=m/(m-6)); m} \\ Andrew Howroyd, Nov 26 2018

Extensions

a(22)-a(28) from Giovanni Resta, Nov 27 2018
a(0) prepended by Jianing Song, Dec 31 2018
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