A034087 -id:A034087 - cp's OEIS frontend

A034088 Numbers divisible by the sum of the cubes of their digits.

1, 10, 100, 110, 111, 153, 200, 221, 370, 371, 407, 500, 702, 1000, 1010, 1011, 1040, 1100, 1101, 1110, 1120, 1210, 1215, 1232, 1323, 1530, 1728, 2000, 2030, 2080, 2110, 2210, 2240, 2331, 2352, 2376, 2464, 2580, 3212, 3213, 3304, 3456, 3520, 3700, 3710

Offset: 1

Erich Friedman

Ratio between original number and a(n) = 1 for n = 1, 6, 9, 10, 11, ... a(n)= 1, 153, 370, 371, 407,... - Carmine Suriano, May 04 2013

			12672 is a term since 1^3+2^3+6^3+7^3+2^3 = 576 divides 12672. - _Carmine Suriano_, May 04 2013

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1001 from Carmine Suriano)

Cf. A034087, A055012, A169665, A169666.

Select[Range[4000], Divisible[#, Plus @@ (IntegerDigits[#]^3)] &] (* Amiram Eldar, Jan 31 2021 *)

Offset corrected by Amiram Eldar, Jan 31 2021

A117562 Numbers n such that n is a multiple of the sum of decimal digits squared of n.

Original entry on oeis.org

0, 1, 10, 20, 50, 100, 110, 111, 120, 130, 133, 200, 210, 240, 267, 298, 310, 315, 360, 372, 376, 400, 420, 480, 500, 532, 550, 630, 803, 917, 973, 1000, 1010, 1011, 1020, 1030, 1071, 1100, 1101, 1110, 1134, 1148, 1200, 1211, 1222, 1290, 1300, 1302, 1316

Offset: 1

Zak Seidov, Apr 27 2006

The same as A034087 with 0 inserted as first term.

Cf. A034087.

Reap[Do[If[IntegerQ[n/Total[IntegerDigits[n]^2]],Sow[n]],{n,1,2000}]][[2,1]]
Join[{0},Select[Range[1500],Divisible[#,Total[IntegerDigits[#]^2]]&]] (* Harvey P. Dale, Feb 28 2023 *)

Edited by Robert Israel, Apr 14 2020

A169665 Numbers divisible by the sum of 4th powers of their digits.

Original entry on oeis.org

1, 10, 100, 102, 110, 111, 1000, 1010, 1011, 1020, 1100, 1101, 1110, 1121, 1122, 1634, 2000, 2322, 4104, 5000, 8208, 9474, 10000, 10010, 10011, 10100, 10101, 10110, 10200, 10412, 11000, 11001, 11010, 11100, 11210, 11220, 12502, 12521, 14758

Offset: 1

Michel Lagneau, Apr 05 2010

			12521 is a term since 1^4 + 2^4 + 5^4 + 2^4 + 1^4 = 659, and 12521 = 19*659;
89295 is a term since 8^4 + 9^4 + 2^4 + 9^4 + 5^4 = 17859, and 89295 = 5*17859.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Digit.

Cf. A034087, A034088, A055013, A169666.

A:= proc(n) add(d^4, d=convert(n, base, 10)) ; end proc: for n from 1 to 200000 do:if irem( n,A(n))=0 then printf(`%d, `,n):else fi:od:

Select[Range[15000], Divisible[#, Plus @@ (IntegerDigits[#]^4)] &] (* Amiram Eldar, Jan 31 2021 *)

Numbers k such that A055013(k) | k.

A169666 Numbers divisible by the sum of 5th powers of their digits.

Original entry on oeis.org

1, 10, 100, 110, 111, 1000, 1010, 1011, 1100, 1101, 1110, 1122, 1232, 2112, 2210, 4100, 4150, 4151, 4224, 10000, 10010, 10011, 10100, 10101, 10110, 11000, 11001, 11010, 11022, 11100, 11122, 11220, 12012, 12110, 12210, 12320, 14550, 20000, 21120, 21321, 22100

Offset: 1

Michel Lagneau, Apr 05 2010

			21321 is a term since 2^5 + 1^5 + 3^5 + 2^5 + 1^5 = 309 and 21321 = 69*309.
54748 is a term since 5^5 + 4^5 + 7^5 + 4^5 + 8^5 = 54748.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Digit.

Cf. A005349, A034087, A034088, A055014, A169665,

with(numtheory):for n from 1 to 200000 do:l:=evalf(floor(ilog10(n))+1): n0:=n:indic:=0:s5:=0:for m from 1 to l do:q:=n0:u:=irem(q,10):v:=iquo(q,10):n0:=v :s5:=s5+u^5: od:if irem(n,s5)=0 then print (n):else fi:od:

Select[Range[10^4], Divisible[#, Plus @@ (IntegerDigits[#]^5)] &] (* Amiram Eldar, Jan 31 2021 *)

is_A169666(n)=!(n%sum(i=1,#n=Vecsmall(Str(n)),(n[i]-48)^5))

A055014(a(n)) | a(n).

Corrected and edited by D. S. McNeil, Nov 20 2010

More terms from Amiram Eldar, Jan 31 2021

A379980 Numbers that are divisible by the square of the sum of the squares of their digits.

Original entry on oeis.org

1, 10, 100, 1000, 1100, 1200, 1300, 2000, 2023, 2100, 2400, 3100, 4332, 5000, 10000, 10100, 10200, 10300, 11000, 12000, 13000, 20000, 20100, 20230, 20400, 21000, 24000, 30100, 30324, 31000, 31311, 42000, 43011, 43320, 50000, 52022, 52215, 55000, 71824, 100000

Offset: 1

Amiram Eldar, Jan 07 2025

Called "Second-order Harshad numbers" by Pal and Gopalan (2023).

If k is a term, then 10*k is also a term.

			10 is a term since 10 is divisible by (1^2 + 0^2)^2 = 1.
1100 is a term since 1100 is divisible by (1^2 + 1^2 + 0^2 + 0^2)^2 = 4.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Pradip Kumar Pal and Kaushik Gopalan, Second Order Harshad Number, International Journal of Mathematical Education, Vol. 13, No. 1 (2023), pp. 25-26.

Cf. A003132, A005349, A072081, A180490 (binary analog).
Subsequence of A034087.
Subsequences: A379981, A379982.

Select[Range[10^5], Divisible[#, (Plus @@ (IntegerDigits[#]^2))^2] &]

isok(k) = !(k % vecsum(apply(x -> x^2, digits(k)))^2);

def ok(n): return n and n%sum(di**2 for di in map(int, str(n)))**2 == 0
print([k for k in range(100001) if ok(k)]) # Michael S. Branicky, Jan 10 2025

A007603 Power-sum numbers: let n = a_1 a_2 ... a_k be a k-digit number; n is a power-sum number if there are exponents e_1 ... e_m such that n = Sum_{i=1..m} Sum_{j=1..k} a_j^e_i.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 23, 24, 27, 30, 36, 40, 42, 45, 48, 50, 54, 60, 63, 70, 72, 80, 81, 84, 90, 100, 102, 104, 108, 110, 111, 112, 113, 114, 115, 116, 117, 120, 122, 126, 130, 131, 132, 133, 134, 135, 136, 140, 144, 150, 151, 152, 153, 154, 156, 160, 162, 170, 171, 172, 173, 174, 178, 180, 182

Offset: 1

N. J. A. Sloane, Robert G. Wilson v, Mira Bernstein

			21 = (2+1)+(2^3+1^3)+(2^3+1^3), with e_1, e_2, e_3 = 1, 3, 3.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Mike Keith, Power-sum numbers, J. Recreational Mathematics, Vol. 18, No. 4 (1986), pp. 275-278. (Annotated scanned copy)

Subsequences: A005349, A034087, A034088, A169665, A169666.

q[n_] := Module[{d = IntegerDigits[n], v = {}, k = 1, s, ans = False}, If[Max[d] == 1, ans = Divisible[n, Total[d]], While[(s = Total[d^k]) <= n, AppendTo[v, s]; If[Length[IntegerPartitions[n, All, v]] > 0, ans = True; Break[]]; k++]]; ans]; Select[Range[200], q] (* Amiram Eldar, Sep 04 2021 *)

Corrected and extended by Naohiro Nomoto, Mar 11 2001

A169662 Numbers divisible by the sum of their digits, and by the sum of their digits squared, by the sum of their digits cubed and by the sum of 4th powers of their digits.

Original entry on oeis.org

1, 10, 100, 110, 111, 1000, 1010, 1011, 1100, 1101, 1110, 2000, 5000, 10000, 10010, 10011, 10100, 10101, 10110, 11000, 11001, 11010, 11100, 20000, 50000, 55000, 100000, 100010, 100011, 100100, 100101, 100110, 101000, 101001, 101010, 101100

Offset: 1

Michel Lagneau, Apr 05 2010

The numbers such that all digits are nonzero are rare (see the subsequence A176194).

			1121211 is a term since 1^4 + 1^4 + 2^4 + 1^4 + 2^4 + 1^4 + 1^4 = 37 and 1121211 = 37*30303 ; 1^3 + 1^3 + 2^3 + 1^3 + 2^3 + 1^3 + 1^3 = 21 and 1121211 = 21*53391 ; 1^2 + 1^2 + 2^2 + 1^2 + 2^2 + 1^2 + 1^2 = 13 and 1121211 = 13* 86247 ; 1 + 1 + 2 + 1 + 2 + 1 + 1 = 9 and 1121211 = 9*124579.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A005349, A034087, A034088 and A169665.

Cf. A072081, A117562, A176194.

isA169662 := proc(n)
        dgs := convert(n,base,10) ;
        if (n mod ( add(d,d=dgs) ) = 0)  and (n mod (add(d^2,d=dgs) )) =0 and (n mod (add(d^3,d=dgs))) =0 and (n mod (add(d^4,d=dgs))) = 0 then
                true;
        else
                false;
        end if;
end proc:
for i from 1 to 110000 do
        if isA169662(i) then
                printf("%d,",i) ;
        end if;
end do: # R. J. Mathar, Nov 07 2011

q[n_] := And @@ Divisible[n, Plus @@@ Transpose @ Map[#^Range[4] &, IntegerDigits[n]]]; Select[Range[10^5], q] (* Amiram Eldar, Jan 31 2021 *)

{n : A007953(n)|n and A003132(n)|n and A055012(n)| n and A055013(n)| n}.

A176194 Numbers with no zero digits divisible by the sum of the k-th powers of their digits, for each k = 1,2,3,4.

Original entry on oeis.org

1, 111, 1121211, 11243232, 12132432, 12413232, 22331232, 23111352, 23411232, 24113232, 41223312, 42131232, 44662464, 111111111, 112452144, 114251424, 135964224, 211412544, 246134592, 313212312, 332131212, 382941675, 416283624, 442114512, 523173456, 671635575, 979652772

Offset: 1

Michel Lagneau, Apr 11 2010

For the numbers divisible by the sum of k-th powers of digits including 0, see A169662. The numbers such that the digits are > 0 are rare.

			For n = 246134592 we obtain :
2^4 + 4^4 + 6^4 + 1^4 + 3^4 + 4^4 + 5^4 + 9^4 + 2^4 = 9108, and 246134592 = 9108*27024 ;
2^3 + 4^3 + 6^3 + 1^3 + 3^3 + 4^3 + 5^3 + 9^3 + 2^3 = 1242, and 246134592 = 1242*198176 ;
2^2 + 4^2 + 6^2 + 1^2 + 3^2 + 4^2 + 5^2 + 9^2 + 2^2 = 192, and 246134592 = 192*1281951 ;
2 + 4 + 6 + 1 + 3 + 4 + 5 + 9 + 2 = 36, and 246134592 = 36*6837072.

Amiram Eldar, Table of n, a(n) for n = 1..464

Intersection of A052382 and A169662.

Cf. A034087, A072081, A117562, A034087, A072081, A117562, A034088, A034088.

with(numtheory):for n from 2 to 500000000 do:l:=evalf(floor(ilog10(n))+1):n0:=n:s1:=0:s2:=0:s3:=0:s4:=0:p:=1:for m from 1 to l do:q:=n0:u:=irem(q,10):v:=iquo(q,10):n0:=v :s1:=s1+u:p:=p*u:s2:=s2+u^2:s3:=s3+u^3:s4:=s4+u^4: od:if irem(n,s1)=0 and irem(n,s2)=0 and irem(n,s3)=0 and irem(n,s4)=0 and p<>0 then print(n):else fi:od:

A007953 (n)|n and A003132(n)|n and A055012 (n)| n and A055013 (n)| n and all digits < > 0.

a(1)-a(2) and more terms add by Amiram Eldar, Apr 20 2023

A169663 Numbers k divisible by the sum of the digits and the sum of the squares of digits of k (in base 10).

Original entry on oeis.org

1, 10, 20, 50, 100, 110, 111, 120, 133, 200, 210, 240, 315, 360, 372, 400, 420, 480, 500, 550, 630, 803, 1000, 1010, 1011, 1020, 1071, 1100, 1101, 1110, 1134, 1148, 1200, 1300, 1302, 1330, 1344, 1431, 1547, 2000, 2010, 2023, 2040, 2100, 2196, 2200, 2220

Offset: 1

Michel Lagneau, Apr 05 2010

			For k = 2196, 2^2 + 1^2 + 9^2 + 6^2 = 122, 2 + 1 + 9 + 6 = 18, and 2196 = 18*122 so it is divisible by both 18 and 122.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Digit.

Cf. A003132, A007953, A034087, A072081, A117562.

Intersection of A005349 and A034087.

with(numtheory):for n from 1 to 1000000 do:l:=evalf(floor(ilog10(n))+1):n0:=n:s1:=0:s2:=0:for m from 1 to l do:q:=n0:u:=irem(q,10):v:=iquo(q,10):n0:=v :s1:=s1+u:s2:=s2+u^2:od:if irem(n,s1)=0 and irem(n,s2)=0 then print(n):else fi:od:

Select[Range[2220], Divisible[#, Plus @@ (d = IntegerDigits[#])] && Divisible[#, Plus @@ (d^2)] &] (* Amiram Eldar, Mar 04 2023 *)

sd2(n) = my(d=digits(n)); sum(i=1, #d, d[i]^2);
isok(n) = !(n % sumdigits(n)) && !(n % sd2(n)); \\ Michel Marcus, Dec 21 2014

A007953(k)|k and A003132(k)|k.

A169664 Numbers k divisible respectively by the sum of digits, the sum of the squares and the sum of the cubes of digits in base 10 of k.

Original entry on oeis.org

1, 10, 100, 110, 111, 200, 500, 1000, 1010, 1011, 1100, 1101, 1110, 2000, 2352, 5000, 5500, 10000, 10010, 10011, 10100, 10101, 10110, 11000, 11001, 11010, 11100, 11112, 20000, 22000, 22200, 23520, 25032, 25110, 30100, 40000, 41013, 44160, 50000

Offset: 1

Michel Lagneau, Apr 05 2010

			For k = 174192, 1^3 + 7^3 + 4^3 + 1^3 + 9^3 + 2^3 = 1146, and 174192 = 152*1146; 1^2 + 7^2 + 4^2 + 1^2 + 9^2 + 2^2 = 152, and 174192 = 152*1146; 1 + 7 + 4 + 1 + 9 + 2 = 24, and 174192 = 24*7258.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Digit.

Cf. A003132, A007953, A034087, A055012, A072081, A117562.

with(numtheory):for n from 1 to 200000 do:l:=evalf(floor(ilog10(n))+1) : n0:=n:s1:=0:s2:=0: s3:=0:for m from 1 to l do:q:=n0:u:=irem(q,10):v:=iquo(q,10):n0:=v :s1:=s1+u:s2:=s2+u^2:s3:=s3+u^3:od:if irem(n,s1)=0 and irem(n,s2)=0 and irem(n,s3)=0 then print(n):else fi:od:

dsQ[n_]:=Module[{idn=IntegerDigits[n]}, Divisible[n,Total[idn]] && Divisible[n,Total[idn^2]] && Divisible[n,Total[idn^3]]]; Select[Range[50000],dsQ]  (* Harvey P. Dale, Feb 24 2011 *)

A007953(k)|k and A003132(k)|k and A055012(k)| k.

cp's OEIS Frontend

A034088 Numbers divisible by the sum of the cubes of their digits.

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A117562 Numbers n such that n is a multiple of the sum of decimal digits squared of n.

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A169665 Numbers divisible by the sum of 4th powers of their digits.

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A169666 Numbers divisible by the sum of 5th powers of their digits.

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A379980 Numbers that are divisible by the square of the sum of the squares of their digits.

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A007603 Power-sum numbers: let n = a_1 a_2 ... a_k be a k-digit number; n is a power-sum number if there are exponents e_1 ... e_m such that n = Sum_{i=1..m} Sum_{j=1..k} a_j^e_i.

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A169662 Numbers divisible by the sum of their digits, and by the sum of their digits squared, by the sum of their digits cubed and by the sum of 4th powers of their digits.

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