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 1-6 of 6 results.

A075088 Smallest triangular number with n prime factors (counted with multiplicity).

Original entry on oeis.org

1, 3, 6, 28, 36, 120, 528, 2080, 2016, 25200, 32640, 819840, 131328, 7874496, 20476800, 52433920, 8386560, 102767616, 536887296, 2147450880, 26306560000, 16240435200, 116802736128, 34359607296, 2199022206976, 549755289600
Offset: 0

Views

Author

Amarnath Murthy, Sep 13 2002

Keywords

Examples

			a(3)=28 because 28 is a triangular number with 3 prime factors: 28 = 2*2*7.

Links

Crossrefs

Cf. A000217, A076550.

Programs

  • PARI
    a(n)=if(n<0,0,s=1; while(abs(bigomega(s*(s+1)/2)-n)>0,s++); s*(s+1)/2)

Extensions

More terms from Benoit Cloitre, Sep 17 2002
a(21)-a(24) from Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 15 2004
Edited by Ray Chandler, Dec 17 2004
a(25) from Ray Chandler, Dec 22 2004

A101745 Indices of triangular numbers which are 10-almost primes. Indices of A101744.

Original entry on oeis.org

255, 384, 511, 575, 639, 728, 767, 896, 1088, 1295, 1376, 1407, 1599, 1700, 1727, 1792, 1919, 1920, 2015, 2024, 2375, 2431, 2672, 2815, 2880, 2915, 2944, 2975, 3104, 3159, 3199, 3327, 3375, 3392, 3456, 3583, 3744, 3999, 4031, 4032, 4160, 4223, 4256
Offset: 1

Views

Author

Jonathan Vos Post, Dec 14 2004

Keywords

Examples

			a(1) = 255 because that is the smallest index of a triangular number which is also a 10-almost prime; specifically T(255) = 255*(255+1)/2 = 32640 = 2^7 * 3 * 5 * 17.

Links

Crossrefs

Cf. A101744, A000217, A046314, A076550, A069904.

Programs

  • GAP
    F:=List([1..4300],n->Length(Factors(n*(n+1)/2)));; a:=Filtered([1..Length(F)],i->F[i]=10); # Muniru A Asiru, Dec 22 2018
  • Magma
    [n: n in [2..4500] | &+[d[2]: d in Factorization((n*(n+1)))] eq 11]; // Vincenzo Librandi, Dec 22 2018
  • Mathematica
    BigOmega[n_Integer]:=Plus@@Last[Transpose[FactorInteger[n]]]; Do[ t=n*(n+1)/2; If[BigOmega[t]==10, Print[n, " ", t];];, {n, 2, 5000}]; (* Ray Chandler, Dec 14 2004 *)
Flatten[Position[Accumulate[Range[5000]],?(PrimeOmega[#]==10&)]] (* _Harvey P. Dale, May 12 2011 *)

Formula

a(n)*(a(n)+1)/2 has exactly 10 prime factors.
{ m : A069904(m) = 10 }. - Alois P. Heinz, Aug 05 2019

Extensions

More terms from Ray Chandler, Dec 14 2004

A359090 a(n) is the index of the smallest tetrahedral number with exactly n prime factors (counted with multiplicity), or -1 if no such number exists.

Original entry on oeis.org

1, -1, 2, 4, 6, 8, 14, 30, 48, 62, 126, 160, 350, 510, 1022, 2046, 1024, 4095, 4094, 13310, 28672, 32768, 65534, 180224, 262142, 360448, 262143, 2097151, 3276800, 4194302, 2097150, 33554432, 16777214, 66715648, 33554430, 184549374, 134217728, 536870910, 1073741824
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 16 2022

Keywords

Links

Crossrefs

Cf. A000292, A001222, A076550, A358927, A359089.

Programs

  • PARI
    a(n) = if(n==1, return(-1)); for(k=1, oo, my(t=(k*(k+1)*(k+2))\6); if(bigomega(t) == n, return(k))); \\ Daniel Suteu, Dec 30 2022

Extensions

a(27)-a(34) from Daniel Suteu, Dec 30 2022
a(35)-a(38) from Lucas A. Brown, Sep 11 2024

A209048 Indices of the pentagonal numbers listed in A209049.

Original entry on oeis.org

1, 2, 4, 3, 12, 11, 32, 27, 75, 192, 427, 171, 1323, 2187, 6912, 8192, 4779, 10923, 49152, 109227, 60075, 170667, 786432, 699051, 1092267, 4893355, 5810859, 11184811, 25864875, 95070891, 101362347, 44739243
Offset: 0

Views

Author

Robert G. Wilson v, Mar 04 2012

Keywords

Comments

Pentagonal analog of A076550.

Crossrefs

A209049, A076550.

Programs

  • Mathematica
    k = 1; t = Table[0, {50}]; While[k < 500000001, a = PrimeOmega[k] + PrimeOmega[3 k - 1] - 1; If[ t[[a + 1]] == 0, t[[a + 1]] = k; Print[{k, a}]]; k++]; t

A359017 a(n) is the index of the smallest triangular number with exactly n distinct prime factors.

Original entry on oeis.org

1, 2, 3, 11, 20, 84, 455, 1364, 10659, 58695, 254540, 728364, 13516580, 133595384, 812646120, 5327923964, 68971338435, 838101203939, 7588384207404, 69322940121435, 490005293940084
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 12 2022

Keywords

Links

Crossrefs

Cf. A000217, A001221, A076550, A076551.

Programs

  • PARI
    a(n) = my(k=1); while(omega(k*(k+1)/2) != n, k++); k; \\ Michel Marcus, Feb 26 2023

Extensions

a(19)-a(20) from Michael S. Branicky, Feb 25 2023

A363847 Numbers k such that Omega(m*(m+1)) < Omega(k*(k+1)) for all m < k, where Omega(k) is the number of prime divisors of k counted with multiplicity (A001222).

Original entry on oeis.org

1, 2, 3, 7, 8, 15, 32, 63, 224, 255, 512, 3968, 4095, 14336, 32768, 65535, 180224, 262143, 1048575, 14680064, 16777215, 134217728, 268435455, 1073741823, 8589934592, 12884901887, 34359738368, 68719476735, 1099511627775, 4398046511103, 17592186044415, 35184372088832
Offset: 1

Views

Author

Amiram Eldar, Jun 24 2023

Keywords

Comments

Terms a(2)-a(18) were found by Erdős and Nicolas (1978-1979).
Equivalently, numbers k such that Omega(m) + Omega(m+1) < Omega(k) + Omega(k+1), for all m < k.
The corresponding record values are 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 17, 18, 19, 20, 22, 24, 26, 27, 31, 33, 34, 37, 38, 39, 40, 46, 48, 50, 51, 52, ... .

Links

Crossrefs

Cf. A002378, A001222, A059958, A075088, A076550.

Programs

  • Mathematica
    seq[kmax_] := Module[{o1 = 0, o2, om = 0, s = {}}, Do[o2 = PrimeOmega[k]; o = o1 + o2; If[o > om, om = o; AppendTo[s, k - 1]]; o1 = o2, {k, 2, kmax}]; s]; seq[10^5]
  • PARI
    lista(kmax) = {my(o1 = 0, o2, om = 0); for(k = 2, kmax, o2 = bigomega(k); o = o1 + o2; if(o > om, om = o; print1(k-1, ", ")); o1 = o2); }

Extensions

a(29)-a(32) from Martin Ehrenstein, Jul 08 2023
Showing 1-6 of 6 results.

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

Content is available under The OEIS End-User License Agreement.