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 1-4 of 4 results.

A080364 Composite numbers whose least prime factor appears with multiplicity 1.

Original entry on oeis.org

6, 10, 14, 15, 18, 21, 22, 26, 30, 33, 34, 35, 38, 39, 42, 46, 50, 51, 54, 55, 57, 58, 62, 65, 66, 69, 70, 74, 75, 77, 78, 82, 85, 86, 87, 90, 91, 93, 94, 95, 98, 102, 105, 106, 110, 111, 114, 115, 118, 119, 122, 123, 126, 129, 130, 133, 134, 138, 141, 142, 143, 145, 146
Offset: 1

Views

Author

Labos Elemer, Feb 21 2003

Keywords

Comments

Density is Sum_{p >= 2} 1/p * Product_{q <= p} (1 - 1/q) which is around 0.65. (In the sum and product, p and q are restricted to primes.) - Charles R Greathouse IV, Jan 09 2022

Examples

			50 = 2^1 * 5^2; least prime factor is 2, whose exponent is 1, so 50 is a term.

Links

Crossrefs

Cf. A034444, A056169, A020639, A080363.

Programs

  • Mathematica
    mi[x_] := Part[Flatten[FactorInteger[x]], 1] k=0; Do[s=mi[n]; If[Equal[GCD[s, n/s], 1]&&!PrimeQ[n], Print[n]], {n, 2, 256}]
Select[Range[150],CompositeQ[#]&&FactorInteger[#][[1,2]]==1&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 23 2021 *)
  • PARI
    is(n,f=factor(n))=n>1 && f[1,2]==1 \\ Charles R Greathouse IV, Jan 09 2022
  • PARI
    is(n)=forprime(p=2,97, if(n%p==0, return(n%p^2>0))); !ispower(n) && factor(n)[1,2]==1 \\ Charles R Greathouse IV, Jan 09 2022

Extensions

Edited by Jon E. Schoenfield, Jul 10 2018

A080365 Composite numbers k whose smallest and largest prime factors are unitary prime factors.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 38, 39, 42, 46, 51, 55, 57, 58, 62, 65, 66, 69, 70, 74, 77, 78, 82, 85, 86, 87, 90, 91, 93, 94, 95, 102, 105, 106, 110, 111, 114, 115, 118, 119, 122, 123, 126, 129, 130, 133, 134, 138, 141, 142, 143, 145, 146, 154, 155, 158
Offset: 1

Views

Author

Labos Elemer, Feb 21 2003

Keywords

Examples

			k=90 is not a prime; 90 = 2*3*3*5; extremal prime factors are 2 and 5; gcd(2, 90/2) = gcd(5, 90/5) = 1, so 2 and 5 are unitary prime divisors of 90, thus 90 is in the sequence.

Links

Crossrefs

Cf. A034444, A056169, A020639, A006530, A080363, A080364.

Programs

  • GAP
    D:=List(Filtered([2..160],i->not IsPrime(i)),Factors);;
a:=[];;  for i in D do if Gcd(i[1],Product(i)/i[1])=1 and Gcd(i[Length(i)],Product(i)/i[Length(i)])=1 then Add(a,Product(i)); fi; od; a;  # Muniru A Asiru, Jul 10 2018~
  • Mathematica
    ma[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] mi[x_] := Part[Flatten[FactorInteger[x]], 1] k=0; Do[s=mi[n]; s1=ma[n]; If[Equal[GCD[s, n/s], 1]&&Equal[GCD[s1, n/s1], 1]&&!PrimeQ[n], Print[n]], {n, 2, 256}]
  • PARI
    lista(nn) = {forcomposite(n=1, nn, my(f=factor(n)[,1], p = vecmin(f), q = vecmax(f)); if ((gcd(p, n/p) == 1) && (gcd(q, n/q) == 1), print1(n, ", ")););} \\ Michel Marcus, Jul 09 2018

A080366 Numbers k whose least and greatest prime divisors are non-unitary.

Original entry on oeis.org

4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 216, 225, 243, 256, 288, 289, 300, 324, 343, 361, 392, 400, 432, 441, 484, 500, 512, 529, 576, 588, 600, 625, 648, 675, 676, 729, 784, 800, 841, 864, 900, 961, 968, 972
Offset: 1

Views

Author

Labos Elemer, Feb 21 2003

Keywords

Examples

			n=300: it is not a prime, 300 = 2*2*3*5*5; extremal prime factors are 2 and 5; gcd(2, 300/2) > 1 and gcd(5, 300/5) > 1, so neither 2 nor 5 is a unitary prime divisor of 300, thus 300 is in this sequence. - _Labos Elemer_, corrected by _Jeppe Stig Nielsen_, Jun 27 2017

Links

Crossrefs

Cf. A006530, A020639, A034444, A056169, A080363, A080364, A080365.

Programs

  • Mathematica
    ma[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] mi[x_] := Part[Flatten[FactorInteger[x]], 1] k=0; Do[s=mi[n]; s1=ma[n]; If[ !Equal[GCD[s, n/s], 1]&&!Equal[GCD[s1, n/s1], 1]&&!PrimeQ[n], Print[n]], {n, 2, 1000}]
  • PARI
    isA080366(n) = e=factor(n)[,2];e&&e[1]>1&&e[#e]>1 \\ Jeppe Stig Nielsen, Jun 27 2017

A101257 Remainder when the least divisor of n greater than or equal to the square root of n (A033677(n)) is divided by the greatest divisor of n less than or equal to the square root of n (A033676(n)).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 2, 0, 0, 0, 0, 1, 1, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 0, 2, 1, 2, 0, 0, 1, 1, 3, 0, 1, 0, 3, 4, 1, 0, 2, 0, 0, 2, 1, 0, 3, 1, 1, 1, 1, 0, 4, 0, 1, 2, 0, 3, 5, 0, 1, 2, 3, 0, 1, 0, 1, 0, 3, 4, 1, 0, 2, 0, 1, 0, 5, 2, 1, 2, 3, 0, 1, 6, 3, 1, 1, 4, 4, 0, 0, 2, 0, 0, 5, 0, 5, 1
Offset: 1

Views

Author

Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 17 2004

Keywords

Comments

Given n points, sort them into the most-square rectangular point lattice possible. Now sort the points into square point lattices of dimension equal to the lesser dimension of the earlier rectangle. a(n) is the number of points left over. a(n) is trivially 0 for prime numbers n (the most-square and only rectangular point lattice on a prime number of points is a linear point lattice). a(n) != 0 iff n is a member of A080363.

Examples

			a(6)=1 because the least divisor of 6 greater than sqrt(6) is 3, the greater divisor of 6 less than sqrt(6) is 2 and 3 mod 2 = 1

Links

Crossrefs

Cf. A033676, A033677, A080363.

Programs

  • Mathematica
    num[n_] := If[OddQ[DivisorSigma[0, n]], Sqrt[n], Divisors[n][[DivisorSigma[0, n]/2 + 1]]] den[n_] := If[OddQ[DivisorSigma[0, n]], Sqrt[n], Divisors[n][[DivisorSigma[0, n]/2]]] Table[Mod[num[n], den[n]], {n, 1, 128}]
  • PARI
    A033676(n) = if(n<2, 1, my(d=divisors(n)); d[(length(d)+1)\2]); \\ From A033676
A033677(n) = (n/A033676(n));
A101257(n) = (A033677(n)%A033676(n)); \\ Antti Karttunen, Sep 23 2018

Extensions

Definition corrected by Antti Karttunen, Sep 23 2018
Showing 1-4 of 4 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.