A157973 Numbers n > 1 with spf(n) < n such that there is a number k, n < k <= 2*n, with k - spf(k) = n - spf(n).

9, 12, 20, 21, 24, 25, 28, 35, 40, 44, 49, 52, 54, 55, 56, 57, 63, 65, 72, 77, 81, 88, 90, 91, 92, 93, 95, 96, 98, 105, 108, 111, 115, 116, 119, 121, 125, 126, 128, 130, 133, 143, 144, 145, 147, 148, 150, 155, 161, 164, 165, 169, 170, 172, 180, 184, 185, 187, 188, 189

Offset

1,1

Comments

spf(k) is the sum of the prime factors of k with multiplicity, cf. A001414.

Condition spf(n) < n excludes 4 and the primes.

It appears that if a number k > n with k - spf(k) = n - spf(n) exists at all, then k <= 2*n. If this can be proved, the corresponding condition in the definition could be omitted.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Example

For n = 77, n-spf(n) = 77-18 = 59; for k = 93, k-spf(k) = 93-34 =59. Hence 77 is in the sequence.

Mathematica

f[1] = 1; f[n_] := n - Plus @@ Times @@@ FactorInteger@n; aQ[n_] := (fn = f[n]) > 0 && Module[{ans = False}, Do[If[f[k] == fn, ans = True; Break[]], {k, n + 1, 2 n}]; ans]; Select[Range[200], aQ] (* Amiram Eldar, Sep 08 2019 *)

Prog

(Magma) spf:=func< n | &+[a[1]*a[2]: a in Factorization(n)] >; [ n: n in [2..190] | d gt 0 and exists(t){ k: k in [n+1..2*n] | d eq k-spf(k) } where d is n-spf(n) ];

Crossrefs

Cf. A001414 (sum of primes dividing n).

Sequence in context: A366865 A366861 A364343 * A057577 A328069 A014766

Adjacent sequences: A157970 A157971 A157972 * A157974 A157975 A157976

Keyword

nonn

Author

J. M. Bergot, Mar 10 2009

Extensions

Edited, missing terms inserted and extended by Klaus Brockhaus, Mar 30 2009

Status

approved