A145746 -id:A145746 - cp's OEIS frontend

A266816 Numbers whose arithmetic derivative is equal to the product of their digits.

4, 11, 25, 329, 3383, 4343, 5561, 6623, 12773, 17267, 21479, 57721, 129383, 136259, 142943, 172793, 256631, 292571, 364823, 413663, 413927, 619337, 653291, 1215659, 1218863, 1268951, 1276931, 1483751, 1655219, 1892327, 2952731, 4158719, 4973531, 5418671, 6377663

Offset: 1

Paolo P. Lava, Feb 10 2016

4 appears to be the only even number in the sequence.

			4’ = 4;
11’ = 1 = 1*1;
25’ = 10 = 2*5;
329’ = 54 = 3*2*9; etc.

Paolo P. Lava, Table of n, a(n) for n = 1..100

Cf. A003415, A007954, A058627, A145746.

with(numtheory):  P:=proc(q) local a,b,k,n; for n from 1 to q do a:=n; b:=1;
for k from 1 to ilog10(n)+1 do b:=b*(a mod 10); a:=trunc(a/10); od;
if n*add(op(2,a)/op(1,a),a=ifactors(n)[2])=b then print(n); fi; od; end: P(10^9);

Select[Range[3, 10^5], Times @@ IntegerDigits@ # == # Total[#2/#1 & @@@
FactorInteger@ Abs@ #] &] (* Michael De Vlieger, Feb 10 2016, after Michael Somos at A003415 *)

A279459 Numbers n such that sum of the proper divisors of n is the square of the sum of the digits of n.

Original entry on oeis.org

24, 153, 176, 794, 3071, 3431, 4607, 9671, 15599, 17711, 18167, 19511, 45671, 50927, 56471, 62807, 74639, 119279, 127559, 154199, 165791, 174719, 175871, 695399, 699359

Offset: 1

Ilya Gutkovskiy, Dec 12 2016

Subsequence of A073040.
Numbers n such that A001065(n) = A118881(n) or A000203(n) - n = (A007953(n))^2.
Every term in the sequence is composite (since the only proper divisor of a prime is 1). The sum of the proper divisors of a k-digit composite number n must exceed sqrt(n) >= sqrt(10^(k-1)), but the square of the sum of the digits of a k-digit number cannot exceed (9k)^2 = 81k^2. Since sqrt(10^(k-1)) > 81k^2 for all integers k > 8, every term in the sequence must be less than the smallest 9-digit number, 10^8. An exhaustive search through 10^8 shows that a(25)=699359 is the last term. - Jon E. Schoenfield, Dec 13 2016

			24 is in the sequence because 24 has 7 proper divisors {1,2,3,4,6,8,12}, 1 + 2 + 3 + 4 + 6 + 8 + 12 = 36 and (2 + 4)^2 = 36;
153 is in the sequence because 153 has 5 proper divisors {1,3,9,17,51}, 1 + 3 + 9 + 17 + 51 = 81 and (1 + 5 + 3)^2 = 81;
176 is in the sequence because 176 has 9 proper divisors {1,2,4,8,11,16,22,44,88}, 1 + 2 + 4 + 8 + 11 + 16 + 22 + 44 + 88 = 196 and (1 + 7 + 6)^2 = 196, etc.

Eric Weisstein's World of Mathematics, Digit Sum
Index entries for sequences related to sums of divisors

Cf. A073040, A001065, A118881, A145746.

Select[Range[1000000], DivisorSigma[1, #1] - #1 == Total[IntegerDigits[#1]]^2  &]

is(n) = sigma(n)-n==sumdigits(n)^2 \\ Felix Fröhlich, Dec 13 2016

cp's OEIS Frontend

A266816 Numbers whose arithmetic derivative is equal to the product of their digits.

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