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.

A046684 Numbers k such that k and sum of squares of divisors of k are relatively prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 18, 19, 21, 23, 25, 27, 29, 31, 32, 33, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 64, 67, 69, 71, 72, 73, 77, 79, 81, 83, 87, 89, 91, 93, 95, 97, 98, 99, 100, 101, 103, 107, 109, 111, 113, 119, 121, 123, 125, 127, 128, 129
Offset: 1

Views

Author

Labos Elemer

Keywords

Comments

All even terms are either squares or doubled squares. - Ivan Neretin, Dec 30 2015
The asymptotic density of this sequence is 0 (Dressler, 1974). - Amiram Eldar, Jul 23 2020

Links

Crossrefs

Cf. A001157, A046686, A046687.

Programs

  • Mathematica
    Select[Range[130], GCD[#, DivisorSigma[2, #]] == 1 &] (* Ivan Neretin, Dec 30 2015 *)
  • PARI
    isok(n) = gcd(n, sigma(n, 2)) == 1; \\ Michel Marcus, Jan 10 2017

A046686 Numbers k such that k and sum of cubes of divisors of k are relatively prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 36, 37, 39, 41, 43, 47, 49, 50, 53, 55, 57, 59, 61, 63, 64, 65, 67, 71, 73, 75, 77, 79, 81, 83, 85, 89, 93, 97, 98, 100, 101, 103, 107, 109, 111, 113, 115, 117, 121, 125, 127, 128, 129, 131, 137, 139, 143
Offset: 1

Views

Author

Labos Elemer

Keywords

Comments

All even terms are either squares or doubled squares. - Ivan Neretin, Dec 30 2015
The asymptotic density of this sequence is 0 (Dressler, 1974). - Amiram Eldar, Jul 23 2020

Links

Crossrefs

Cf. A001158, A046684, A046687.

Programs

  • Mathematica
    Select[Range[143], GCD[#, DivisorSigma[3, #]] == 1 &] (* Ivan Neretin, Dec 30 2015 *)

A088948 Numbers k such that (A006530(k) + A020639(k))/2 is an integer; that is, arithmetic mean of least and largest prime factor is an integer.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 64, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125
Offset: 1

Views

Author

Labos Elemer, Nov 20 2003

Keywords

Comments

Union of odd numbers and powers of 2 minus {1}. - Ivan Neretin, Dec 30 2015
In other words, the symmetric difference of sets A005408 (all prime factors are odd) and A000079 (all prime factors are even). If we had allowed 1 as a member, it would have been the union of A005408 and A000079, as stated. - Jeppe Stig Nielsen, Dec 27 2019

Examples

			Primes and prime powers are here.
Also other composites: n=105, (3+7)/2 = 5 is an integer (and, moreover, divides n).

Links

Crossrefs

Different from A046687.
Cf. A006530, A020639, A088949, A088949, A046687.

Programs

  • Mathematica
    Rest@ Select[Range@ 125, IntegerQ[(FactorInteger[#][[1, 1]] + FactorInteger[#][[-1, 1]])/2] &] (* Michael De Vlieger, Mar 28 2015 *)
amintQ[n_]:=Module[{fi=FactorInteger[n][[;;,1]]},IntegerQ[Mean[{fi[[1]],fi[[-1]]}]]]; Select[Range[150],amintQ] (* Harvey P. Dale, Mar 02 2023 *)
  • PARI
    is_a088948(n) = {local (f);f=factor(n);if(Mod(vecmin(f[,1])+vecmax(f[,1]),2)==0,1,0)} \\ Michael B. Porter, Mar 28 2015

A088949 Composite numbers k such that (A006530(k) + A020639(k))/2 is an integer that divides k; special terms of A088948.

Original entry on oeis.org

4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 105, 121, 125, 128, 169, 231, 243, 256, 289, 315, 343, 361, 512, 525, 529, 625, 627, 693, 729, 735, 841, 897, 935, 945, 961, 1024, 1155, 1331, 1369, 1575, 1581, 1617, 1681, 1729, 1849, 1881, 2048, 2079, 2187, 2197, 2205, 2209
Offset: 1

Views

Author

Labos Elemer, Nov 20 2003

Keywords

Examples

			k = 315 = 3*3*5*7 (composite); (3 + 7)/2 = 5, which divides k.

Links

Crossrefs

Cf. A006530, A020639, A088948, A088595, A046687.

Programs

  • Mathematica
    q[k_] := CompositeQ[k] && Module[{p = FactorInteger[k][[;;, 1]], m}, m = (p[[1]] + p[[-1]]); EvenQ[m] && Divisible[k, m/2]]; Select[Range[2500], q] (* Amiram Eldar, Mar 01 2025 *)
  • PARI
    lista(nn) = {forcomposite(n=1, nn, my(f = factor(n)[,1], x = (vecmin(f) + vecmax(f))/2); if ((denominator(x)==1) && !(n % x), print1(n, ", ")););} \\ Michel Marcus, Jul 09 2018

Extensions

Edited by Jon E. Schoenfield, Jul 08 2018
More terms from Michel Marcus, Jul 09 2018
Showing 1-4 of 4 results.

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

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