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.

A093890 Number of primes arising as the sum of one or more divisors of n.

Original entry on oeis.org

0, 2, 1, 4, 1, 5, 1, 6, 2, 7, 1, 9, 1, 5, 4, 11, 1, 12, 1, 13, 5, 5, 1, 17, 2, 5, 4, 16, 1, 20, 1, 18, 4, 6, 6, 24, 1, 5, 5, 24, 1, 24, 1, 18, 11, 5, 1, 30, 1, 15, 3, 18, 1, 30, 6, 30, 5, 7, 1, 39, 1, 3, 18, 31, 6, 34, 1, 16, 3, 34, 1, 44, 1, 4, 13, 16, 4, 39, 1, 42, 5, 5, 1, 48, 5, 5, 2, 41, 1, 51, 2
Offset: 1

Views

Author

Amarnath Murthy, Apr 23 2004

Keywords

Comments

a(2^n) = pi(2^(n+1)-1).
Except for n=3 and n=42, it appears that the records occur at the highly abundant numbers A002093. The record values appear to be pi(sigma(n)) for n in A002093, which means that these n are members of A093891. [T. D. Noe, Mar 19 2010]

Examples

			a(4) = 4, the divisors of 4 are 1, 2 and 4.
Primes arising are 2, 3 = 1 + 2, 5 = 1 + 4 and 7 = 1 + 2 + 4.

Links

Crossrefs

Cf. A093891, A093892.
Cf. A161510 (primes counted with repetition). [T. D. Noe, Mar 19 2010]

Programs

  • Mathematica
    Do[l = Subsets[Divisors[n]]; l = Union[Map[Plus @@ #&, l]]; Print[Length[Select[l, PrimeQ]]], {n, 100}] (* Ryan Propper, Jun 04 2006 *)
CountPrimes[n_] := Module[{d=Divisors[n],t,lim,x}, t=CoefficientList[Product[1+x^i, {i,d}], x]; lim=PrimePi[Length[t]-1]; Count[t[[1+Prime[Range[lim]]]], ?(#>0 &)]]; Table[CountPrimes[n], {n,100}] (* _T. D. Noe, Mar 19 2010 *)

Extensions

Corrected and extended by Ryan Propper, Jun 04 2006

A093891 Numbers k such that every prime up to sigma(k) is a sum of divisors of k.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 104, 108, 112, 120, 126, 128, 132, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192, 196, 198, 200, 204, 208, 210, 216, 220, 224, 228, 234, 240
Offset: 1

Views

Author

Amarnath Murthy, Apr 23 2004

Keywords

Comments

Sequence is infinite as sigma (2^n) = 2^(n+1)-1 and a(2^n) = pi(2^(n+1)-1).
Does this sequence include any non-members of A005153 other than 10, 70 and 836? - Franklin T. Adams-Watters, Apr 28 2006
The answer to the previous comment is yes, this sequence has many terms that are not in A005153. See A174434. - T. D. Noe, Mar 19 2010

Examples

			4 is a member as sigma(4) = 7 and all the primes up to 7 are a partial sum of divisors of 4, since divisors of 4 are 1, 2 and 4 and because primes arising are 2, 3 = 1+2, 5 = 1+4 and 7 = 1+2+4.

Links

Crossrefs

Cf. A093890, A093892.
Cf. A005153.

Programs

  • Mathematica
    Select[Range[240], SubsetQ[Total /@ Rest@ Subsets@ Divisors[#], Prime@ Range@ PrimePi@ DivisorSigma[1, #]] &] (* Michael De Vlieger, Mar 19 2021 *)
  • PARI
    isok(m) = {my(d=divisors(m), vp = primes(primepi(sigma(m)))); for (i=1, 2^#d - 1, my(b = Vecrev(binary(i)), x = sum(k=1, #b, b[k]*d[k])); if (vecsearch(vp, x), vp = setminus(vp, Set(x))); if (#vp == 0, return (1)););} \\ Michel Marcus, Mar 19 2021

Extensions

More terms from Franklin T. Adams-Watters, Apr 28 2006

A093893 Numbers n such that every sum of two or more divisors is composite.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 87, 89, 91, 97, 101, 103, 107, 109, 113, 121, 127, 131, 133, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 183, 191, 193, 197, 199, 211, 213, 217, 223, 227, 229, 233
Offset: 1

Views

Author

Amarnath Murthy, Apr 23 2004

Keywords

Comments

All terms are odd and very few are composite. Every odd prime is a trivial member.
Very few terms have more than four divisors. The smallest such term is 4753, which has six divisors: 1,7,49,97,679,4753. - Adam M. Kalman (mocha(AT)clarityconnect.com), Nov 11 2004

Crossrefs

Cf. A093890, A093891, A093892, A093894.

Programs

  • Mathematica
    For[a:=3, a<=500, s =Divisors[a];n := 1;d := False; While[(n<=2^Length[s])\[And]( ["not" character]d), If[Length[NthSubset[n, s]]>=2, If[ !PrimeQ[Plus@@NthSubset[n, s]], n++, d:= True], n++ ]]; If[ ["not" character]d, Print[a]];a+=2]; (Kalman)
fQ[n_] := Union@ PrimeQ[Plus @@@ Subsets[ Divisors@n, {2, Infinity}]] == {False}; Select[ Range[3, 235, 2], fQ@# &] (* Robert G. Wilson v, May 25 2009 *)

Extensions

More terms from Adam M. Kalman (mocha(AT)clarityconnect.com), Nov 11 2004
a(1)=1 prepended by Max Alekseyev, Mar 31 2015

A093894 Composite members of A093893.

Original entry on oeis.org

49, 87, 91, 121, 133, 169, 183, 213, 217, 247, 249, 259, 287, 301, 339, 343, 361, 403, 411, 427, 445, 469, 473, 481, 501, 511, 527, 529, 553, 559, 581, 589, 591, 633, 679, 699, 703, 713, 717, 721, 763, 789, 793, 817, 841, 843, 871, 889, 895, 949, 951, 961
Offset: 1

Views

Author

Amarnath Murthy, Apr 23 2004

Keywords

Comments

Comment: Most terms of this sequence have four divisors. Some terms (the squares of primes) have three divisors; very few terms have more than four divisors (the first such term is 4753, with six). Conjecture: This sequence is infinite. - Adam M. Kalman (mocha(AT)clarityconnect.com), Nov 11 2004

Examples

			133 is a term, the divisors are 1,7,19,133 and no sum of two or more gives a prime.

Links

Crossrefs

Cf. A093890, A093891, A093892, A093893.

Programs

  • Mathematica
    For[a:=4, a<=2000, s =Divisors[a];n := 1;d := False; While[(n<=2^Length[s])\[And]( ["not" character]d), If[Length[NthSubset[n, s]]>=2, If[ !PrimeQ[Plus@@NthSubset[n, s]], n++, d:= True], n++ ]]; If[ ["not" character]d, Print[a]];a++;While[PrimeQ[a], a+=2]]; (* Adam M. Kalman, Nov 11 2004 *)

Extensions

Corrected and extended by Adam M. Kalman (mocha(AT)clarityconnect.com), Nov 11 2004
Showing 1-4 of 4 results.

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