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Numbers such that the sum of their divisors, excluding 1 and the number itself, minus the sum of their digits equals the number.
2

%I #22 Jan 14 2020 07:23:34

%S 12,114256,6988996,8499988,8689996,8789788,8877988,8988868,8999956,

%T 9696988,9759988,9899596,9948988,9996868,9998884,9999892,15996988,

%U 16878988,17799796,17887996,17988796,17999884,18579988,18768988,18869788,18895996,18958996,18995788,19398988,19587988,19698868,19777996,19799668

%N Numbers such that the sum of their divisors, excluding 1 and the number itself, minus the sum of their digits equals the number.

%C After the second term, it seems that the digit sum is 55.

%C All terms after a(2) appear to be of the form 2^2 * 7 * p, where p is a prime. - _Scott R. Shannon_, Jan 09 2020

%C If there exists a third term not of the form 2^2*7*p, it is larger than 10^13. - _Giovanni Resta_, Jan 14 2020

%e a(3) = 6988996 as the sum of the divisors of 6988996, excluding 1 and 6988996, equals 6989051, the sum of its digits equals 55, and 6989051 - 55 = 6988996.

%t Select[Range[10^7], DivisorSigma[1, #] - Plus @@ IntegerDigits[#] == 2 # + 1 &] (* _Amiram Eldar_, Jan 08 2020 *)

%o (PARI) isok(n) = sigma(n) - n - 1 - sumdigits(n) == n; \\ _Michel Marcus_, Jan 09 2020

%Y Cf. A000203, A007953, A048050.

%Y Cf. A331037 (sum of divisors + digit sum = number).

%K nonn,base,less

%O 1,1

%A _Joseph E. Marrow_, Jan 08 2020

%E Terms a(7) and beyond from _Scott R. Shannon_, Jan 09 2020