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 1-5 of 5 results.

A115957 Numbers k having exactly 3 distinct prime factors, the largest of which is greater than or equal to sqrt(k) (i.e., sqrt(k)-rough numbers with exactly 3 distinct prime factors).

Original entry on oeis.org

42, 66, 78, 102, 110, 114, 130, 138, 156, 170, 174, 186, 190, 204, 222, 228, 230, 238, 246, 255, 258, 266, 276, 282, 285, 290, 310, 318, 322, 342, 345, 348, 354, 366, 370, 372, 402, 406, 410, 414, 426, 430, 434, 435, 438, 444, 460, 465, 470, 474, 483, 492
Offset: 1

Views

Author

Emeric Deutsch, Feb 02 2006

Keywords

Examples

			156 is in the sequence because it has 3 distinct prime factors (2, 3 and 13) and 13 > sqrt(156).

Links

Crossrefs

Cf. A115956, A115958, A115959, A115960, A115961.

Programs

  • Maple
    with(numtheory): a:=proc(n) if nops(factorset(n))=3 and factorset(n)[3]^2>=n then n else fi end: seq(a(n),n=1..530);
  • Mathematica
    Select[Range[500],PrimeNu[#]==3&&FactorInteger[#][[-1,1]]>=Sqrt[#]&] (* Harvey P. Dale, Apr 09 2019 *)

A115959 Numbers k having exactly 5 distinct prime factors, the largest of which is greater than or equal to sqrt(k) (i.e., sqrt(k)-rough numbers with exactly 5 distinct prime factors).

Original entry on oeis.org

44310, 46830, 47670, 48090, 48930, 50190, 50610, 52710, 53970, 55230, 56490, 56910, 58170, 59010, 59430, 61530, 64470, 65310, 65730, 66570, 69510, 70770, 72870, 73290, 74130, 75390, 77070, 78330, 79590, 80430, 81690, 83370, 84210
Offset: 1

Views

Author

Emeric Deutsch, Feb 02 2006

Keywords

Examples

			46830 is in the sequence because it has 5 distinct prime factors (2, 3, 5, 7 and 223) and 223 > sqrt(46830).

Links

Crossrefs

Cf. A115956, A115957, A115958, A115960, A115961.

Programs

  • Maple
    with(numtheory): a:=proc(n) if nops(factorset(n))=5 and factorset(n)[5]^2>=n then n else fi end: seq(a(n),n=1..93000);

A115956 Numbers k having exactly 2 distinct prime factors, the largest of which is greater than or equal to sqrt(k) (i.e., sqrt(k)-rough numbers with exactly 2 distinct prime factors).

Original entry on oeis.org

6, 10, 14, 15, 20, 21, 22, 26, 28, 33, 34, 35, 38, 39, 44, 46, 51, 52, 55, 57, 58, 62, 65, 68, 69, 74, 76, 77, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 99, 104, 106, 111, 115, 116, 117, 118, 119, 122, 123, 124, 129, 133, 134, 136, 141, 142, 143, 145, 146, 148, 152, 153
Offset: 1

Views

Author

Emeric Deutsch, Feb 02 2006

Keywords

Examples

			20 is in the sequence because it has 2 distinct prime factors (2 and 5) and 5 > sqrt(20).

Crossrefs

Cf. A115957, A115958, A115959, A115960, A115961.

Programs

  • Maple
    with(numtheory): a:=proc(n) if nops(factorset(n))=2 and factorset(n)[2]^2>=n then n else fi end: seq(a(n),n=1..170);
  • Mathematica
    tdpfQ[n_]:=Module[{fi=FactorInteger[n]},Length[fi]==2&&fi[[2,1]]>Sqrt[n]]; Select[Range[ 200],tdpfQ] (* Harvey P. Dale, Aug 07 2023 *)

A115960 Numbers k having exactly 6 distinct prime factors, the largest of which is greater than or equal to sqrt(k) (i.e., sqrt(k)-rough numbers with exactly 6 distinct prime factors).

Original entry on oeis.org

5338410, 5389230, 5403090, 5407710, 5421570, 5430810, 5444670, 5477010, 5490870, 5500110, 5504730, 5518590, 5527830, 5541690, 5569410, 5583270, 5597130, 5629470, 5638710, 5652570, 5680290, 5698770, 5712630, 5721870
Offset: 1

Views

Author

Emeric Deutsch, Feb 02 2006

Keywords

Examples

			5389230 is in the sequence because it has 6 distinct prime factors (2, 3, 5, 7, 11 and 2333) and 2333 > sqrt(5389230).

Crossrefs

Cf. A115956, A115957, A115958, A115959, A115961.

Programs

  • Maple
    with(numtheory): a:=proc(n) if nops(factorset(n))=6 and factorset(n)[6]^2>=n then n else fi end: seq(a(n),n=(2*3*5*7*11)^2..5850000);
  • Mathematica
    dpf6Q[n_]:=Module[{pf=FactorInteger[n][[All,1]]},Length[pf]==6 && pf[[6]]>=Sqrt[n]]; Select[Range[6*10^6],dpf6Q] (* Harvey P. Dale, Mar 24 2017 *)

A115961 a(n)=least number having exactly n distinct prime factors, the largest of which is greater than or equal to sqrt(a(n)).

Original entry on oeis.org

2, 6, 42, 930, 44310, 5338410, 902311410, 260630159790, 94084209188970, 49770436899273090, 41856930884959119930, 40224510201386387907030, 55067354465876062759959510, 92568222856398333359120816010
Offset: 1

Views

Author

Emeric Deutsch, Feb 02 2006

Keywords

Examples

			a(3)=42; indeed, 42=2*3*7, 7>sqrt(42) and 2*3*5 does not qualify because
5<sqrt(30).

Crossrefs

Cf. A115956, A115957, A115958, A115959, A115960.
Cf. A002110.

Programs

  • Maple
    a:=n->product(ithprime(j),j=1..n-1)*nextprime(product(ithprime(j),j=1..n-1)): seq(a(n),n=1..16);

Formula

a(n)=y*(smallest prime that is larger than y), where y is the product of first n-1 consecutive primes.
a(n) = (n-1)# * NextPrime((n-1)#). a(n) = A002110(n-1) * NextPrime(A002110(n-1)). E.g. a(15) = 14# * 13082761331670077 = (2 * 3 * 5 * 7 * 11 * 13 * 17 * 19 * 23 * 29 * 31 * 37 * 41 * 43) + 13082761331670077, since 13082761331670077 = 14# + 47 is the least prime > 14#. - Jonathan Vos Post, Feb 13 2006
Showing 1-5 of 5 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.