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A337037
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Numbers whose every unordered factorization has a distinct sum of factors.
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4
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1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 98, 99, 101
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OFFSET
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1,2
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COMMENTS
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The number 1 is in the sequence by convention.
All primes p are trivially in the sequence.
All semiprimes greater than 4 are in the sequence because they have only two unordered factorizations pq = p*q whose sums are distinct. They are distinct because the only solution to p*q = p+q is p=q=2.
If a number m is not in the sequence, then all multiples of m are not in the sequence. For example, multiples of 4 are not in the sequence because there always exist at least two factorizations 4*k = 2*2*k whose factors sum to the same value 4+k = 2+2+k.
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LINKS
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EXAMPLE
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All unordered factorization of 30 are 30 = 2*15 = 3*10 = 5*6 = 2*3*5. Corresponding sums of factors are distinct: 30, 17 = 15+2, 13 = 10+3, 11 = 6+5, 10 = 2+3+5. Therefore 30 is in the sequence.
All unordered factorization of 90 are 90 = 45*2 = 30*3 = 18*5 = 15*6 = 15*3*2 = 10*9 = 9*5*2 = 10*3*3 = 6*5*3 = 5*3*3*2. Corresponding sums of factors are not all distinct: 90, 57, 33, 23, 21, 20, 19, 16, 16, 14, 13 because the sum 16 = 10+3+3 = 9+5+2 appears twice. Therefore 90 is not in the sequence.
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PROG
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(PARI) factz(n, minn) = {my(v=[]); fordiv(n, d, if ((d>=minn) && (d<=sqrtint(n)), w = factz(n/d, d); for (i=1, #w, w[i] = concat([d], w[i]); ); v = concat(v, w); ); ); concat(v, [[n]]); }
factorz(n) = factz(n, 2);
isok(n) = my(vs = apply(x->vecsum(x), factorz(n))); #vs == #Set(vs); \\ Michel Marcus, Aug 13 2020
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CROSSREFS
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Cf. A001055 (number of unordered factorizations of n), A074206 (number of ordered factorizations of n).
Cf. A056472 (all factorizations of n), A069016 (number of distinct sums).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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