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A056240 Smallest number whose prime divisors (taken with multiplicity) add to n. 30
2, 3, 4, 5, 8, 7, 15, 14, 21, 11, 35, 13, 33, 26, 39, 17, 65, 19, 51, 38, 57, 23, 95, 46, 69, 92, 115, 29, 161, 31, 87, 62, 93, 124, 155, 37, 217, 74, 111, 41, 185, 43, 123, 86, 129, 47, 215, 94, 141, 188, 235, 53, 329, 106, 159, 212, 265, 59, 371, 61, 177, 122 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,1

COMMENTS

a(n) = index of first occurrence of n in A001414.

From David James Sycamore and Michel Marcus, Jun 16 2017, Jun 28 2017: (Start)

Recursive calculation of a(n):

For prime p, a(p) = p.

For even composite n, let P_n denote the largest prime < n-1 such that n-P_n is prime (except if n=6).

For odd composite n, let P_n denote the largest prime < n-1 such that n-3-P_n is prime.

Conjecture: a(n) = Min{q*a(n-q); q prime, P_n <= q < n-1}.

Examples:

For n=9998, P_9998=9967, and a(9998) = Min{9973*a(25),9967*a(31)}= 9967*31=308977.

For n=875, P_875=859, and a(875) = Min{863*a(12),859*a(16)}=863*35=30205.

Note: A000040 and A288313 are both subsequences of this sequence. (End)

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 2..10000

H. Havermann, Tables of sum-of-prime-factors sequences (overview with links to the first 50000 sums).

FORMULA

Trivial but essential; a(n) >= n. - David A. Corneth, Mar 23 2018

EXAMPLE

a(8) = 15 = 3*5 because 15 is the smallest number whose prime divisors sum to 8.

a(10000) = 586519. Let gpf(n) be the largest prime < n and the candidate being the current value that might be a(10000). Then we see that gpf(10000 - 1) = 9967, giving a candidate 9967 * a(10000 - 9967) = 9967 * 62. gpf(9967) = 9949, giving the candidate 9949 * a(10000 - 9949) = 9962 * 188. This is larger than our candidate so we keep 9967 * 62 as our candidate. gpf(9949) = 9941, giving a candidate 9941 * gpf(10000 - 9941) = 9941 * 59. We see that (n - p) * a(p) >= (n - p) * p > candidate = 9941 * 59 for p > 59 so we stop iterating to conclude a(10000) = 9941 * 59 = 586519. - David A. Corneth, Mar 23 2018

MATHEMATICA

a = Table[0, {75}]; Do[b = Plus @@ Flatten[ Table[ #1, {#2}] & @@@ FactorInteger[n]]; If[b < 76 && a[[b]] == 0, a[[b]] = n], {n, 2, 1000}]; a (* Robert G. Wilson v, May 04 2002 *)

b[n_] := b[n] = Total[Times @@@ FactorInteger[n]];

a[n_] := For[k = 2, True, k++, If[b[k] == n, Return[k]]];

Table[a[n], {n, 2, 63}] (* Jean-Fran├žois Alcover, Jul 03 2017 *)

PROG

(Haskell)

a056240 = (+ 1) . fromJust . (`elemIndex` a001414_list)

-- Reinhard Zumkeller, Jun 14 2012

(PARI) isok(k, n) = my(f=factor(k)); sum(j=1, #f~, f[j, 1]*f[j, 2]) == n;

a(n) = my(k=2); while(!isok(k, n), k++); k; \\ Michel Marcus, Jun 21 2017

(PARI) a(n) = {if(n <= 5, return(n)); my(p = precprime(n), res = p * (n - p)); if(p == n, return(p), p = precprime(n - 2); res = p * a(n - p); while(res > (n - p) * p && p > 2, p = precprime(p - 1); res = min(res, a(n - p) * p)); res)} \\ David A. Corneth, Mar 23 2018

CROSSREFS

Cf. A001414, A064502, A000040, A288313.

First column of array A064364, n>=2.

See A000792 for the maximal numbers whose prime factors sums up to n.

Sequence in context: A074756 A240221 A075162 * A069968 A298882 A086931

Adjacent sequences:  A056237 A056238 A056239 * A056241 A056242 A056243

KEYWORD

nonn,easy

AUTHOR

Adam Kertesz, Aug 19 2000

EXTENSIONS

More terms from James A. Sellers, Aug 25 2000

STATUS

approved

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Last modified November 21 11:01 EST 2018. Contains 317447 sequences. (Running on oeis4.)