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 A056240 Smallest number whose prime divisors (taken with multiplicity) add to n. 33
 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 FORMULA Trivial but essential: a(n) >= n. - David A. Corneth, Mar 23 2018 a(n) >= n with equality iff n is prime. - M. F. Hasler, Jan 19 2019 EXAMPLE a(8) = 15 = 3*5 because 15 is the smallest number whose prime divisors sum to 8. a(10000) = 586519: Let pp(n) be the largest prime < n and the candidate being the current value that might be a(10000). Then we see that pp(10000 - 1) = 9967, giving a candidate 9967 * a(10000 - 9967) = 9967 * 62. p(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. pp(9949) = 9941, giving a candidate 9941 * pp(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, edited by M. F. Hasler, Jan 19 2019 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-François 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 (PARI) A056240(n, p=n-1, m=oo)=if(n<6 || isprime(n), n, n==6, 8, until(p<3 || (n-p=precprime(p-1))*p >= m=min(m, A056240(n-p)*p), ); m) \\ M. F. Hasler, Jan 19 2019 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 September 25 17:14 EDT 2021. Contains 347659 sequences. (Running on oeis4.)