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A172484
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Partial sums of extravagant numbers, also called prodigal numbers, or wasteful numbers.
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0
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4, 10, 18, 27, 39, 57, 77, 99, 123, 149, 177, 207, 240, 274, 310, 348, 387, 427, 469, 513, 558, 604, 652, 702, 753, 805, 859, 914, 970, 1027, 1085, 1145, 1207, 1270, 1335, 1401, 1469, 1538, 1608, 1680, 1754, 1829, 1905, 1982, 2060, 2140, 2222, 2306, 2391, 2477
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OFFSET
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1,1
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COMMENTS
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Every natural number, written in base 10, is either economical A046759 (also called frugal), or equidigital A046758, or extravagant (or prodigal or wasteful). An extravagant number is one for which the factorization requires more digits that the original number such as 30 = 2 * 3 * 5. The subsequence of economical partial sums of extravagant numbers begins: xxx, 18, 39, 57, 77, 99, 207, 240, 274, 310. The subsequence of equidigital partial sum of economical numbers begins: 10, 27, 123, 149, 177, 427, 469 (such as 1207 = 17 * 71). The subsequence of prime partial sums of economical numbers begins: xxx, 149, 859, 2477, 2833.
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LINKS
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FORMULA
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SUM[i=1..n] A046760(i) = Partial sum of {Write n as a product of primes raised to powers, let D(n) = number of digits in product, l(n) = number of digits in n; sequence gives n such that D(n)>l(n)}.
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EXAMPLE
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a(1) = A046760(1) = 4. a(2) = 4 + 6 = 10. a(67) = 4 + 6 + 8 + 9 + 12 + 18 + 20 + 22 + 24 + 26 + 28 + 30 + 33 + 34 + 36 + 38 + 39 + 40 + 42 + 44 + 45 + 46 + 48 + 50 + 51 + 52 + 54 + 55 + 56 + 57 + 58 + 60 + 62 + 63 + 65 + 66 + 68 + 69 + 70 + 72 + 74 + 75 + 76 + 77 + 78 + 80 + 82 + 84 + 85 + 86 + 87 + 88 + 90 + 91 + 92 + 93 + 94 + 95 + 96 + 98 + 99 + 100 + 102 + 104 + 108 + 110 + 114 = 4138 = 2 * 2069 which is thus an economical number, with 4 digits but 5 in its prime factorization.
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CROSSREFS
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KEYWORD
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base,easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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