A244069 Consider a number of k digits n = d_(k)*10^(k-1) + d_(k-1)*10^(k-2) + … + d_(2)*10 + d_(1). Sequence lists the numbers n such that Sum_{i=1..k-1}{phi(Sum_{j=1..i}{d_(j)*10^(j-1)})} = Sum_{i=1..k-1}{sigma(Sum_{j=1..i}{d_(k-j+1)*10^(i-j)})} (see example below).

11, 21, 53, 75, 83, 95, 211, 506, 523, 708, 908, 932, 955, 1008, 5086, 6535, 7272, 7557, 9126, 20534, 31165, 51301, 52695, 71665, 73713, 85173, 90902, 93026, 93565, 210021, 313370, 330173, 406945, 423775, 521427, 633190, 728687, 850123, 926281

Offset

1,1

Comments

No single-digit terms are permitted. - Harvey P. Dale, Mar 08 2015

Links

Table of n, a(n) for n=1..39.

Example

If n = 423775, starting from the least significant digit, let us cut the number into the set {5, 75, 775, 3775, 23775}. We have:
sigma(5) = 6;
sigma(75) = 124;
sigma(775) = 992;
sigma(3775) = 4712;
sigma(23775) = 39432.
Then, starting from the most significant digit, let us cut the number into the set {4, 42, 423, 4237, 42377}. We have:
phi(4) = 2;
phi(42) = 12;
phi(423) = 276;
phi(4237) = 3996;
phi(42377) = 40980.
Finally, 6 + 124 + 992 + 4712 + 39432 = 2 + 12 + 276 + 3996 + 40980 = 45266.

Maple

with(numtheory); P:=proc(q) local a, b, k, n; for n from 10 to q do
a:=0; k:=1; while trunc(n/10^k)>0 do a:=a+phi(trunc(n/10^k)); k:=k+1; od;
b:=0; k:=1; while (n mod 10^k)<n do b:=b+sigma(n mod 10^k); k:=k+1; od;
if a=b then print(n); fi; od; end: P(10^9);

Mathematica

dsepQ[n_]:=Module[{idn=IntegerDigits[n], len}, len=Length[idn]-1; Total[ DivisorSigma[1, #]&/@(FromDigits/@Table[Take[idn, -k], {k, If[Last[idn] == 0, 2, 1], len}])]==Total[EulerPhi/@(FromDigits/@Table[Take[idn, i], {i, len}])]]; Select[Range[10, 10^6], dsepQ] (* Harvey P. Dale, Mar 08 2015 *)

Crossrefs

Cf. A000010, A000203, A240894-A240903, A241207, A241502, A241503, A244068.

Sequence in context: A127624 A097616 A146150 * A058489 A278780 A071158

Adjacent sequences: A244066 A244067 A244068 * A244070 A244071 A244072

Keyword

nonn,base

Author

Paolo P. Lava, Jun 19 2014

Extensions

Corrected (a(18) added) by Harvey P. Dale, Mar 08 2015

Status

approved