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Term

Self Number

Numbers that cannot be written as m + digitsum(m) for any m (1, 3, 5, 7, 9, 20, 31, 42, 53, 64, 75, 86, 97, …).

1,012 numbers tagged.

A self number (or Colombian number) is one that no other number generates under the map \(m \mapsto m + \text{digitsum}(m)\). For instance, 21 is not self because 15 + (1+5) = 21; but nothing generates 20, so 20 is self.

Introduced by the Indian recreational mathematician D. R. Kaprekar (of Kaprekar-number and 6174 fame). In base 10 there are infinitely many self numbers, with a regular pattern in their distribution — roughly 10% of integers are self. The smallest self primes are 3, 5, 7, 31, 53, 97, 211, 233, 277.

2,007 2,022 2,033 2,044 2,055 2,066 2,077 2,088 2,099 2,213 2,347 2,437 2,459 2,503 2,549 2,593 2,617 2,683 2,729 2,819 2,953 3,023 3,067 3,089 3,313 3,359 3,449 3,517 3,539 3,583 3,673 3,719 3,853 3,919 3,943 4,057 4,079 4,259 4,327 4,349 4,483 4,663 4,909 4,933 4,999 5,113 5,227 5,519 5,563 5,653 5,743 5,923 6,011 6,037 6,217 6,329 6,373 6,553 6,619 6,709 6,733 6,823 6,911 7,001 7,027 7,229 7,699 7,723 7,789 7,879 7,901 8,039 8,219 8,263 8,353 8,419 8,443 8,599 8,623 8,689 8,779 9,029 9,319 9,343 9,433 9,521 9,679 9,769 9,791 9,859 9,949 10,111 10,133 10,177 10,223 10,267 10,289 10,313 10,357 10,513 10,559 10,627 10,739 11,213 11,257 11,279 11,369 11,437 11,503 11,527 11,549 11,593 11,617 11,863 11,953 12,113 12,157 12,269 12,517 12,539 12,583 12,763 12,853 12,919 13,009 13,147 13,259 13,327 13,417 13,619 13,933 13,999 14,159 14,249 14,293 14,519 14,563 14,653 14,923 15,149 15,193 15,217 15,329 15,373 15,619 15,643 15,733 15,823 15,889 16,001 16,139 16,183 16,229 16,273 16,319 16,363 16,453 16,519 16,633 16,699 16,811 16,879 16,901 17,419 17,443 17,509 17,599 17,623 17,891 17,959 17,981 18,119 18,253 18,433 18,521 18,523 18,679 18,701 18,859 19,219 19,309 19,333 19,421 19,423 19,489 19,759 19,961 20,123 20,347 20,369 20,549 20,593 20,639 20,773 21,023 21,067 21,089 21,157 21,179 21,247