A161510 -id:A161510 - cp's OEIS frontend

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

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

Amarnath Murthy, Apr 23 2004

nonn

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]

			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.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A093891, A093892.

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

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 *)

Corrected and extended by Ryan Propper, Jun 04 2006

A331665 Numbers k with a record number of divisors d < sqrt(k) such that d + k/d is prime.

Original entry on oeis.org

1, 2, 6, 30, 210, 2310, 3570, 4830, 11550, 30030, 43890, 111930, 131670, 510510, 690690, 870870, 1021020, 2459730, 9699690, 13123110, 17160990, 40750710, 146006070, 223092870, 340510170, 358888530, 688677990, 1462190730, 2445553110, 2911018110, 6469693230

Offset: 1

Amiram Eldar, Jan 23 2020

nonn

The corresponding record values are 0, 1, 2, 4, 8, 12, 13, 14, 15, 21, 24, 25, 29, 40, 41, 46, 49, 51, 70, 77, 88, 89, 90, 117, 120, 147, 153, 154, 155, 161, 263, ...

Apparently all the primorial numbers (A002110) are terms. The record values of terms that are primorial numbers are terms of A103787.

			2 has one divisor below sqrt(2), 1, such that 1 + 2/1 = 3 is prime.
6 has 2 divisors below sqrt(6), 1 and 2, such that 1 + 6/1 = 7 and 2 + 6/2 = 5 are primes.
30 has 4 divisors below sqrt(30), 1, 2, 3, and 5 such that 1 + 30/1 = 31, 2 + 30/2 = 17, 3 + 30/3 = 13 and 5 + 30/5 = 11 are primes.

Cf. A002110, A093890, A103787, A161510.

divCount[n_] := DivisorSum[n, Boole @ PrimeQ[# + n/#] &, #^2 < n &]; seq = {}; dm = -1; Do[d1 = divCount[n]; If[d1 > dm, dm = d1; AppendTo[seq, n]], {n, 1,10^6}]; seq

cp's OEIS Frontend

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

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