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A307980
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Numbers k whose number of divisors is the square of the number of decimal digits of k.
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1
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1, 10, 14, 15, 21, 22, 26, 27, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 100, 196, 225, 256, 441, 484, 676, 1000, 1026, 1032, 1064, 1110, 1122, 1128, 1144, 1155, 1160, 1190, 1218, 1230, 1240, 1242, 1254, 1272, 1288, 1290, 1302, 1326, 1330, 1365, 1408
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OFFSET
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1,2
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COMMENTS
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The terms with an odd number of digits are squares.
The terms with 2 digits are squarefree semiprimes (cf. A006881) Union {27}. The terms with 3 digits belong to A030627 (numbers with 9 divisors) and the ones with 4 digits belong to A030634 (numbers with 16 divisors).
The number of terms b(n) with n digits begins with: 1, 30, 7, 753, 3, 11409, 2, ... When there are an odd number of digits, the number of terms decreases from b(3) = 7, b(5) = 3, b(7) = 2. Is there a 2q+1 such that b(2q+1) = 0?
The sequence is infinite because 10^k is the term for each k. We have tau(10^k) = tau(2^k)*tau(5^k) = (k + 1)^2 and 10^k has k + 1 digits. - Marius A. Burtea, May 09 2019
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LINKS
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EXAMPLE
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65 is a term with 2 digits and 4 divisors: {1, 5, 13, 65}.
484 is a term with 3 digits and 9 divisors: {1, 2, 4, 11, 22, 44, 121, 242, 484}.
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PROG
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(Magma) [n:n in [1..1500]|NumberOfDivisors(n) eq (#Intseq(n))^2]; // Marius A. Burtea, May 09 2019
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CROSSREFS
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Cf. A095862 (number of decimal digits = number of divisors).
Cf. A006881 (squarefree semiprimes).
Cf. A030513 (numbers with 4 divisors), A030627 (numbers with 9 divisors), A030634 (numbers with 16 divisors).
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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