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.

A048969 Composite numbers k such that sigma(k) / d(k) is prime.

Original entry on oeis.org

6, 20, 45, 49, 150, 169, 361, 832, 961, 1445, 1734, 1849, 5329, 8405, 9409, 9477, 10086, 10609, 14580, 14641, 16129, 17405, 20886, 24649, 25205, 30246, 39601, 39605, 47526, 49729, 51005, 58081, 61206, 73441, 85805, 102966, 139445, 149645, 167281, 167334
Offset: 1

Views

Author

G. L. Honaker, Jr.

Keywords

Examples

			For k=6 (composite), sigma(6)=12, d(6)=4 and 12/4 = 3 is prime.

Links

Crossrefs

Cf. A000203, A000005, A000043, A048968.

Programs

  • Mathematica
    nn=62000;With[{comps=Complement[Range[nn],Prime[Range[PrimePi[nn]]]]}, Select[ comps, PrimeQ[DivisorSigma[1,#]/DivisorSigma[0,#]]&]] (* Harvey P. Dale, Feb 19 2012 *)

Extensions

More terms from Michel ten Voorde

A089765 Composite n whose sum of distinct divisors, s(d), ignoring divisors n and 1, divided by the count of divisors (not counting n and 1), c(d), are primes. Duplicate divisors, as in 2*2=4 are counted just once.

Original entry on oeis.org

4, 8, 9, 18, 21, 25, 33, 49, 57, 69, 81, 85, 93, 121, 129, 133, 145, 169, 177, 205, 213, 217, 237, 249, 253, 265, 273, 289, 309, 361, 393, 417, 445, 469, 489, 493, 505, 517, 529, 553, 565, 573, 597, 633, 669, 685, 697, 753, 777, 781, 793, 813, 817, 841, 865
Offset: 1

Views

Author

Enoch Haga, Jan 09 2004

Keywords

Examples

			a(1)= 8 because its factors are 8, 1, 2, 4. Ignoring 8 and 1, the sum of 2+4=6. The count of factors is 2 and 6/2=3, a prime.

References

  • Glenn James and Robert C. James, Mathematics Dictionary, Princeton, N.J.: D. Van Nostrand Co., Inc., 1959; page 154 (factor of an integer).

Links

Crossrefs

Cf. A002808, A048968, A048969.

Programs

  • Mathematica
    aQ[n_] := CompositeQ[n] && PrimeQ[(DivisorSigma[1, n] - n - 1)/(DivisorSigma[0, n] - 2)]; Select[Range[865], aQ] (* Amiram Eldar, Sep 07 2019 *)

Formula

Factor n into its distinct divisors, ignore n and 1, add the divisors and divide by the number of divisors. If s(d) / c(d) [sum divided by count] is prime, add to sequence.

A348659 Numbers whose numerator and denominator of the harmonic mean of their divisors are both prime numbers.

Original entry on oeis.org

3, 5, 13, 14, 15, 37, 42, 61, 66, 73, 92, 114, 157, 182, 193, 258, 277, 308, 313, 397, 402, 421, 457, 476, 477, 541, 546, 570, 613, 661, 673, 733, 744, 757, 812, 877, 978, 997, 1093, 1148, 1153, 1201, 1213, 1237, 1266, 1278, 1321, 1381, 1428, 1453, 1621, 1657
Offset: 1

Views

Author

Amiram Eldar, Oct 28 2021

Keywords

Comments

The prime terms of this sequence are the primes p such that (p+1)/2 is also a prime (A005383).
If p is in A109835, then p*(2*p-1) is a semiprime term.

Examples

			3 is a term since the harmonic mean of its divisors is 3/2 and both 2 and 3 are primes.

Links

Crossrefs

Cf. A005383, A099377, A099378, A109835.
Similar sequences: A023194, A048968, A074266, A348659.

Programs

  • Mathematica
    q[n_] := Module[{h = DivisorSigma[0, n]/DivisorSigma[-1, n]}, And @@ PrimeQ[{Numerator[h], Denominator[h]}]]; Select[Range[2000], q]

A048977 Primes arising in A048969.

Original entry on oeis.org

3, 7, 13, 19, 31, 61, 127, 127, 331, 307, 307, 631, 1801, 1723, 3169, 1093, 1723, 3571, 1093, 3221, 5419, 3541, 3541, 8269, 5113, 5113, 13267, 8011, 8011, 16651, 10303, 19441, 10303, 24571, 17293, 17293, 28057, 30103, 55897, 28057, 59221, 30103, 8191, 19531
Offset: 1

Views

Author

G. L. Honaker, Jr.

Keywords

Crossrefs

Cf. A048968, A048969.

Programs

  • Mathematica
    p[n_]:=Module[{c=DivisorSigma[1,n]/DivisorSigma[0,n]},If[PrimeQ[c],c,0]]; nn=62000;With[{comps=Complement[Range[nn],Prime[Range[PrimePi[nn]]]]}, Select[ p/@comps,#!=0&]] (* Harvey P. Dale, Feb 19 2012 *)

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 17 2001
More terms from Sean A. Irvine, Jul 16 2021
Showing 1-4 of 4 results.

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