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Consider a non-palindromic 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}{sigma(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).
+10
5
324, 648, 756, 4448, 4961, 4983, 5849, 11124, 34453, 37609, 54575, 97888, 860858, 1089693, 3143632, 3192897, 3588047, 3768167, 5557853, 25485909, 32899939, 35699309, 58260393, 64564422, 120054389, 121554165, 356346023, 357507563, 755438130, 990227314
OFFSET
1,1
EXAMPLE
If n = 54575, starting from the least significant digit, let us cut the number into the set 5, 75, 575, 4575. We have:
sigma(5) = 6;
sigma(75) = 124;
sigma(575) = 744;
sigma(4575) = 7688.
Then, starting from the most significant digit, let us cut the number into the set 5, 54, 545, 5457. We have:
sigma(5) = 6;
sigma(54) = 120;
sigma(545) = 660;
sigma(5457) = 7776.
Finally, 6 + 124 + 744 + 7688 = 6 + 120 + 660 + 7776 = 8562.
MAPLE
for n from 1 do
if not isA002113(n) then
dgs := convert(n, base, 10) ;
ndgs := nops(dgs) ;
slo := 0 ;
shi := 0 ;
for sd from 1 to ndgs-1 do
lo := add( op(i, dgs)*10^(i-1), i=1..sd) ;
slo := slo + numtheory[sigma](lo) ;
hi := add( op(-i, dgs)*10^(sd-i), i=1..sd) ;
shi := shi + numtheory[sigma](hi) ;
end do:
if slo = shi then
print(n) ;
end if;
end if;
end do: # R. J. Mathar, Sep 09 2015
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Paolo P. Lava, Apr 24 2014
EXTENSIONS
a(16)-a(30) from Giovanni Resta, May 23 2016
STATUS
approved
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}{sigma(Sum_{j=1..i}{d_(j)*10^(j-1)})} = Sum_{i=1..k-1}{phi(Sum_{j=1..i}{d_(k-j+1)*10^(i-j)})} (see example below).
+10
3
11, 12, 35, 38, 57, 59, 152, 599, 2812, 3419, 3915, 6733, 11671, 16706, 16714, 16858, 25303, 26752, 128257, 171762, 238571, 265872, 345715, 375923, 486141, 496975, 573433, 1492832, 2324671, 2944061
OFFSET
1,1
EXAMPLE
If n = 345715, starting from the least significant digit, let us cut the number into the set {5, 15, 715, 5715, 45715}. We have:
phi(5) = 4;
phi(15) = 8;
phi(715) = 480;
phi(5715) = 3024;
phi(45715) = 35520.
Then, starting from the most significant digit, let us cut the number into the set {3, 34, 345, 3457, 34571}. We have:
sigma(3) = 4;
sigma(34) = 54;
sigma(345) = 576;
sigma(3457) = 3458;
sigma(34571) = 34944.
Finally, 4 + 8 + 480 + 3024 + 35520 = 4 + 54 + 576 + 3458 + 34944 = 39036.
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+sigma(trunc(n/10^k)); k:=k+1; od;
b:=0; k:=1; while (n mod 10^k)<n do b:=b+phi(n mod 10^k); k:=k+1; od;
if a=b then print(n); fi; od; end: P(10^9);
KEYWORD
nonn,base
AUTHOR
Paolo P. Lava, Jun 19 2014
STATUS
approved
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).
+10
3
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
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 *)
KEYWORD
nonn,base
AUTHOR
Paolo P. Lava, Jun 19 2014
EXTENSIONS
Corrected (a(18) added) by Harvey P. Dale, Mar 08 2015
STATUS
approved
Integers m such that m' = Sum_{i=1..k-1} (Sum_{j=1..i} d_(k-j+1)*10^(i-j))', where m' is the arithmetic derivative of m and the digits of m are given by d_(k)*10^(k-1) + d_(k-1)*10^(k-2) + ... + d_(2)*10 + d_(1).
+10
1
23, 29, 31, 37, 53, 59, 71, 73, 79, 113, 131, 137, 139, 173, 179, 191, 193, 197, 199, 6437, 8339, 14473, 60827, 95611, 107813, 321773, 495407, 1154383, 3001331, 5707209, 6208373, 12898591, 13481347, 18895997, 29223791, 39111253, 99058793, 175113571, 317624885
OFFSET
1,1
COMMENTS
From 23 to 199 only primes, then composites.
EXAMPLE
If m = 14473, let us cut the number into the set {1, 14, 144, 1447}. We have:
1' = 0;
14' = 9;
144' = 384;
1447' = 1.
Finally, 0 + 9 + 384 + 1 = 14473' = 394.
MAPLE
with(numtheory); P:=proc(q) local a, c, k, n, p;
for n from 10 to q do
a:=0; k:=1; while trunc(n/10^k)>0 do c:=trunc(n/10^k);
a:=a+c*add(op(2, p)/op(1, p), p=ifactors(c)[2]); k:=k+1; od;
if a=n*add(op(2, p)/op(1, p), p=ifactors(n)[2]) then print(n);
fi; od; end: P(10^10);
PROG
(PARI) ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
isok(k) = my(d=digits(k)); if (#d>1, sum(i=1, #d-1, ad(fromdigits(Vec(d, i)))) == ad(k)); \\ Michel Marcus, Dec 05 2025
(Python)
from sympy import factorint
from functools import cache
@cache
def ad(n): return 0 if n < 2 else sum(e*n//p for p, e in factorint(n).items())
def ok(n): return len(s:=str(n)) > 1 and ad(n) == sum(ad(int(s[:k])) for k in range(1, len(s)))
print([k for k in range(10**5) if ok(k)]) # Michael S. Branicky, Dec 05 2025
KEYWORD
nonn,base
AUTHOR
Paolo P. Lava, Jun 19 2014
EXTENSIONS
a(26)-a(37) and new name from Michel Marcus, Dec 05 2025
a(38)-a(39) from Michael S. Branicky, Dec 05 2025
STATUS
approved
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 n' = Sum_{i=1..k-1}{Sum_{j=1..i}{d_(j)*10^(j-1)}}', where n' is the arithmetic derivative of n (see example below).
+10
1
13, 17, 23, 37, 43, 53, 67, 73, 83, 97, 131, 211, 241, 271, 311, 331, 431, 461, 541, 571, 631, 641, 661, 761, 811, 911, 941, 971, 1601, 3701, 5101, 5701, 6101, 6701, 8101, 9601, 13001, 16138, 18497, 19001, 22879, 24001, 54001, 69001, 93001, 97001, 99361, 270001
OFFSET
1,1
COMMENTS
Mainly primes.
EXAMPLE
If n = 16138, let us cut the number into the set {8, 38, 138, 6138}. We have:
8' = 12;
38' = 21;
138' = 121;
6138' = 7917.
Finally, 12 + 21 + 121 + 7917 = 16138' = 8071.
MAPLE
with(numtheory); P:=proc(q) local a, c, k, n, p;
for n from 10 to q do
a:=0; k:=1; while (n mod 10^k)<n do c:=(n mod 10^k);
a:=a+c*add(op(2, p)/op(1, p), p=ifactors(c)[2]); k:=k+1; od;
if a=n*add(op(2, p)/op(1, p), p=ifactors(n)[2]) then print(n);
fi; od; end: P(10^10);
KEYWORD
nonn,base
AUTHOR
Paolo P. Lava, Jun 19 2014
STATUS
approved