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A056240
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Smallest number whose prime divisors (taken with multiplicity) add to n.
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35
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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
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OFFSET
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2,1
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COMMENTS
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a(n) is the index of first occurrence of n in A001414.
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)
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LINKS
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FORMULA
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a(n) >= n with equality iff n = 4 or n is prime. - M. F. Hasler, Jan 19 2019
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EXAMPLE
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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) = 9973, giving a candidate 9973 * a(10000 - 9973) = 9973 * 92. pp(9973) = 9967, giving a candidate 9967 * a(10000 - 9967) = 9967 * 62. pp(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
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MATHEMATICA
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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]]];
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PROG
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(Haskell)
a056240 = (+ 1) . fromJust . (`elemIndex` a001414_list)
(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 < 7, return(n + 2*(n==6))); my(p = precprime(n), res); 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
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CROSSREFS
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First column of array A064364, n>=2.
See A000792 for the maximal numbers whose prime factors sums up to n.
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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