A057531 -id:A057531 - cp's OEIS frontend

A057530 n is odd and divisible by number of divisors of n and sum of digits of n.

1, 9, 225, 441, 1521, 2025, 2601, 12321, 40401, 62001, 99225, 103041, 251001, 321489, 585225, 893025, 1022121, 1108809, 1212201, 1320201, 1946025, 2368521, 2480625, 2772225, 3101121, 3744225, 4473225, 4862025, 5517801, 6125625

Offset: 1

Asher Auel, Sep 03 2000

For most values (except 9,2025 and 99225) number of divisors of n = sum of digits of n, see A057531.
The above comment is wrong: for 16 out of the first 34 terms of the sequence, the number of divisors of n does not equal the sum of the digits of n. - Harvey P. Dale, Dec 31 2015
Since A000005(n) is odd, n must be a square. - Robert Israel, Oct 31 2019

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A005349, A033950, A057529.

[k:k in [1..6000001 by 2]| IsIntegral(k/NumberOfDivisors(k)) and IsIntegral(k/&+Intseq(k))]; // Marius A. Burtea, Oct 31 2019

filter:= proc(m) local n;
  n:= m^2;
  n mod numtheory:-tau(n) = 0 and n mod convert(convert(n,base,10),`+`) = 0
end proc:
map(`^`, select(filter, [seq(i,i=1..10000,2)]),2); # Robert Israel, Oct 31 2019

Select[Range[1,5*10^6,2],Divisible[#,DivisorSigma[0,#]] && Divisible[ #,Total[ IntegerDigits[#]]]&] (* Harvey P. Dale, Dec 31 2015 *)

More terms from Harvey P. Dale, Dec 31 2015

A075491 Sum of digits of n minus number of divisors of n.

Original entry on oeis.org

0, 0, 1, 1, 3, 2, 5, 4, 6, -3, 0, -3, 2, 1, 2, 2, 6, 3, 8, -4, -1, 0, 3, -2, 4, 4, 5, 4, 9, -5, 2, -1, 2, 3, 4, 0, 8, 7, 8, -4, 3, -2, 5, 2, 3, 6, 9, 2, 10, -1, 2, 1, 6, 1, 6, 3, 8, 9, 12, -6, 5, 4, 3, 3, 7, 4, 11, 8, 11, -1, 6, -3, 8, 7, 6, 7, 10, 7, 14, -2, 4, 6, 9, 0, 9, 10, 11, 8, 15, -3, 6, 5, 8, 9, 10, 3, 14, 11, 12, -8, 0, -5, 2, -3, -2, 3, 6, -3, 8, -6

Offset: 1

Labos Elemer, Sep 26 2002

			a[n]<0, see A075492, n=10, 20, 30, ... a[n]=0, see A057531, primes like 2, 11, 101 a[n]>0, see A075493, majority of primes and others

Cf. A007953, A000005, A057531, A075492, A075493.

sud[x_] := Apply[Plus, IntegerDigits[x]] Table[sud[w]-DivisorSigma[0, w], {w, 1, 128}]
f[n_]:=Total[IntegerDigits[n]]-DivisorSigma[0,n];Array[f,130] (* Harvey P. Dale, Aug 10 2011 *)

a(n)=A007953[n]-A000005[n]

A113761 Numbers k such that the number of divisors of k equals both the sum and the product of digits of k in base 10.

Original entry on oeis.org

1, 2, 22, 2114, 11222, 21122, 22211, 112116, 121116, 1111143, 1413111, 3411111, 11111128, 11111821, 11112118, 11121231, 11811112, 13111212, 18111112, 21111118, 21111181, 21121113, 23111121, 111112119, 111119211, 192111111

Offset: 1

Giovanni Resta, Jan 18 2006

Intersection of A074312 and A057531.

			2114 is a term since 2+1+1+4 = 2*1*1*4 = 8 and 2114 has 8 divisors, {1, 2, 7, 14, 151, 302, 1057, 2114}.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A034710, A057531, A074312.

L={};Do[d=IntegerDigits@n; p=Times@@d; If[p==Plus@@d && p==DivisorSigma[0, n], AppendTo[L, n];Print[n]], {n, 1000000}];L
lst = {}; fQ[n_] := (id = IntegerDigits@n; Plus @@ id == Times @@ id == DivisorSigma[0, n]); Do[ If[ fQ@n, AppendTo[lst, n]], {n, 2*10^8}]; lst

a(13)-a(26) from Robert G. Wilson v, Jan 19 2006

A263720 Palindromic numbers such that the sum of the digits equals the number of divisors.

Original entry on oeis.org

1, 2, 11, 22, 101, 202, 444, 525, 828, 1111, 2222, 4884, 5445, 5775, 12321, 13431, 18081, 21612, 24642, 26862, 31213, 44244, 44844, 51415, 52425, 56265, 62426, 80008, 86868, 89298, 99099, 135531, 162261, 198891, 217712, 237732, 301103, 343343, 480084, 486684, 512215, 521125

Offset: 1

Ilya Gutkovskiy, Oct 24 2015

Subsequence of A002113.
A000005(a(n)) = A007953(a(n)).
The only known palindromic primes whose sum of digits equals the numbers of divisors (primes of the form 10^k + 1) are 2,11,101.

			a(3) = 11, 11 is the palindromic number, digitsum(11) = 1 + 1 = 2, sigma_0(11) = 2.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Palindromic Number
Eric Weisstein's World of Mathematics, Divisor Function
Eric Weisstein's World of Mathematics, Digit Sum

Cf. A002113, A057531.

fQ[n_] := Block[{d = IntegerDigits@ n}, And[d == Reverse@ d, Total@ d == DivisorSigma[0, n]]]; Select[Range[2^19], fQ] (* Michael De Vlieger, Oct 27 2015 *)
Select[Range[600000],PalindromeQ[#]&&Total[IntegerDigits[#]] == DivisorSigma[ 0,#]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 28 2019 *)

lista(nn) = {for(n=1, nn, my(d = digits(n)); if ((Vecrev(d) == d) && (numdiv(n) == sumdigits(n)), print1(n, ", ")););} \\ Michel Marcus, Oct 25 2015

A280911 Numbers n such that sum of decimal digits of n equals number of prime divisors of n counted with multiplicity and sum of distinct decimal digits of n equals number of distinct primes dividing n.

Original entry on oeis.org

30, 102, 1002, 1012, 1210, 2001, 2120, 3010, 10002, 10030, 20001, 20112, 20120, 100012, 100030, 101020, 102010, 110020, 110120, 120001, 121120, 200001, 200120, 211100, 221120, 230010, 300010, 320320, 400010, 400140, 1000002, 1000012, 1000140, 1000230, 1001020, 1003002, 1004010, 1010120, 1011300, 1013310, 1021100

Offset: 1

Ilya Gutkovskiy, Jan 10 2017

Numbers n such that A007953(n) = A001222(n) and A217928(n) = A001221(n).

			20112 is in the sequence because 20112 = 2^4*3*419  (6 prime factors, 3 distinct), 2 + 0 + 1 + 1 + 2 = 6 and 2 + 0 + 1 = 3.

Giovanni Resta, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Digit Sum
Eric Weisstein's World of Mathematics, Prime Factor
Eric Weisstein's World of Mathematics, Distinct Prime Factors

Cf. A001221, A001222, A007953, A050689, A050690, A057531, A217928.

Select[Range[1100000], Total[IntegerDigits[#1]] == PrimeOmega[#1] && Total[Union[IntegerDigits[#1]]] == PrimeNu[#1] &]

A075494 Squares whose sum of digits exceeds the number of divisors.

Original entry on oeis.org

4, 9, 16, 25, 49, 64, 81, 121, 169, 196, 256, 289, 361, 484, 529, 625, 676, 729, 784, 841, 961, 1089, 1156, 1225, 1369, 1444, 1681, 1849, 1936, 2116, 2209, 2401, 2809, 3025, 3249, 3364, 3481, 3721, 3844, 3969, 4096, 4225, 4489, 4624, 4761, 5041, 5329

Offset: 1

Labos Elemer, Sep 26 2002

			Sequence consists mainly of squares of primes and of special composites like 33, 34, 35, 38, 44, 46.

Cf. A007953, A000005, A057531, A075491-A075493.

sud[x_] := Apply[Plus, IntegerDigits[x]] Do[s=sud[n]-DivisorSigma[0, n]; If[Greater[s, 0]&&IntegerQ[Sqrt[n]], Print[n]], {n, 1, 10000}]
Select[Range[80]^2,Total[IntegerDigits[#]]>DivisorSigma[0,#]&] (* Harvey P. Dale, Feb 16 2020 *)

Numbers k^2 such that A007953(k^2) > A000005(k^2).

Edited by Jon E. Schoenfield, Sep 23 2018

A351842 Numbers whose sum of digits and number of proper divisors are equal.

Original entry on oeis.org

21, 32, 50, 70, 111, 162, 168, 201, 212, 232, 250, 308, 322, 380, 384, 405, 416, 430, 456, 546, 610, 650, 690, 740, 744, 812, 832, 870, 980, 1004, 1011, 1015, 1053, 1101, 1105, 1222, 1316, 1352, 1365, 1460, 1464, 1482, 1510, 1518, 1550, 1554, 1590, 1608, 1752

Offset: 1

Zdenek Cervenka, Feb 21 2022

			21 is a term since its digits sum to 2 + 1 = 3 and it has three proper divisors (1, 3, and 7).

Cf. A007953, A032741, A057531.

S := n -> add(convert(n, base, 10)):
PD := n -> nops(NumberTheory[Divisors](n)) - 1:
a := n -> select(x -> S(x) = PD(x), [seq(1..n)])

Select[Range[1, 1700], Total[IntegerDigits[#]] == Length[Divisors[#]] - 1 &]

isok(m) = sumdigits(m) == numdiv(m) - 1; \\ Michel Marcus, Feb 21 2022

list(nn) = forcomposite(n=1, nn, if (sumdigits(n) == (numdiv(n) - 1), print1(n, ", ")));
list(1700);

from sympy import divisor_count
def ok(n): return sum(map(int, str(n))) == divisor_count(n) - 1
print([k for k in range(1753) if ok(k)]) # Michael S. Branicky, Feb 21 2022

cp's OEIS Frontend

A057530 n is odd and divisible by number of divisors of n and sum of digits of n.

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A075491 Sum of digits of n minus number of divisors of n.

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A113761 Numbers k such that the number of divisors of k equals both the sum and the product of digits of k in base 10.

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A263720 Palindromic numbers such that the sum of the digits equals the number of divisors.

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A280911 Numbers n such that sum of decimal digits of n equals number of prime divisors of n counted with multiplicity and sum of distinct decimal digits of n equals number of distinct primes dividing n.

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A075494 Squares whose sum of digits exceeds the number of divisors.

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Mathematica

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A351842 Numbers whose sum of digits and number of proper divisors are equal.

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Maple

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Python