A050689 -id:A050689 - cp's OEIS frontend

A057531 Numbers whose sum of digits and number of divisors are equal.

1, 2, 11, 22, 36, 84, 101, 152, 156, 170, 202, 208, 225, 228, 288, 301, 372, 396, 441, 444, 468, 516, 525, 530, 602, 684, 710, 732, 804, 828, 882, 952, 972, 1003, 1016, 1034, 1070, 1072, 1106, 1111, 1164, 1236, 1304, 1308, 1425, 1472, 1476, 1521, 1524

Offset: 1

Asher Auel, Sep 03 2000

[A007953(n)/A000005(n) = c] AND [A000005(n)/A007953(n) = c], c an integer. - Ctibor O. Zizka, Jun 26 2009

			36 is a term as the sum of the digits of 36 is 3+6 = 9 and the number of divisors is 9 too.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 210 terms from Daniel Arribas)
Code Golf StackExchange, Find the nth number where the digit sum equals the number of factors, coding challenge started Nov 28 2022.

Cf. A000005, A007953, A057532, A050689, A070274, A070275, A063737, A067077.

Select[ Range[ 1000 ], DivisorSigma[ 0, # ]==Plus@@IntegerDigits[ # ]& ] (* Harvey P. Dale, Feb 19 2004 *)

A067077 Numbers whose product of distinct prime factors is equal to its sum of digits.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 24, 375, 392, 640, 2401, 4802, 4913, 6400, 7744, 17576, 42592, 64000, 106496, 234256, 295936, 468750, 546875, 628864, 640000, 877952, 1124864, 1966080, 2839714, 3687936, 4687500, 4816896, 4952198, 6400000, 6453888

Offset: 1

Joseph L. Pe, Feb 18 2002

The product of the distinct prime factors of n (the squarefree kernel of n) is also denoted by rad(n) = A007947(n). - Giovanni Resta, Apr 21 2017

			The prime factors of 375 are 3,5, which have product = 15, the sum of the digits of 375, so 375 is a term of the sequence.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 71 terms from Harry J. Smith)

Cf. A007947, A006753, A057531, A057532, A050689, A070274, A070275, A063737, A285494.

f[n_] := Times@@ (First/@ FactorInteger[n]); g[n_] := Plus @@ IntegerDigits[n]; Select[Range[10^5], f[#] == g[#] &] (* or *)
nd=12; up=10^nd; L={1}; Do[If[SquareFreeQ[su], ps = First /@ FactorInteger[su]; nps = Length@ ps; Clear[ric]; ric[n_, i_] := Block[{e = 0, m}, If[i > nps, If[Plus @@ IntegerDigits[su n] == su, Sow[su n]], While[ (m = n ps[[i]]^e ) su < up, ric[m, i+1]; e++]]]; z = Reap[ ric[1, 1]][[2]]; If[z != {}, L = Union[L, z[[1]]]]], {su, 2, 9 nd}]; L (* fast, terms < 10^12, Giovanni Resta, Apr 21 2017 *)
Select[Range[65*10^5],Times@@FactorInteger[#][[All,1]]==Total[ IntegerDigits[ #]]&] (* Harvey P. Dale, Dec 16 2018 *)

isok(k)={vecprod(factor(k)[,1]) == sumdigits(k)} \\ Harry J. Smith, May 06 2010

a(19)-a(35) from Donovan Johnson, Sep 29 2009

a(1)=1 prepended by Giovanni Resta, Apr 21 2017

A070275 Numbers k such that the sum of the digits of k equals the sum of the prime divisors of k.

Original entry on oeis.org

2, 3, 5, 7, 84, 160, 250, 336, 468, 735, 936, 975, 1344, 1375, 1408, 1600, 1694, 1872, 2352, 2401, 2500, 2625, 2808, 3744, 3920, 4116, 4913, 5145, 5616, 6084, 6318, 7296, 7497, 7695, 8424, 8624, 8664, 8704, 9126, 9639, 10240, 12168, 12636, 12675, 14896

Offset: 1

Benoit Cloitre, May 09 2002

If k=10^s*m is a term of the sequence where s > 0 and gcd(m,10)=1, then for each positive integer j, 10^j*m is in the sequence, because the sum of the digits of 10^j*k equals the sum of the digits of k and the sum of the distinct prime factors of 10^j*k equals the sum of the distinct prime factors of k. Also it is obvious that m isn't in the sequence. [Jahangeer Kholdi, Oct 07 2013]

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A008472, A057531, A057532, A050689, A070274, A063737, A067077, A285494.

Rest[Select[Range[20000],Total[Transpose[FactorInteger[#]][[1]]] == Total[ IntegerDigits[#]] &]] (* Harvey P. Dale, Dec 15 2010 *)

isok(n) = sumdigits(n) == vecsum(factor(n)[,1]); \\ Michel Marcus, May 27 2018

A070274 Numbers n such that sum of digits of n equals the squarefree part of n.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 12, 24, 100, 150, 200, 300, 320, 375, 500, 600, 640, 700, 704, 735, 832, 960, 1014, 1088, 1200, 1815, 2023, 2400, 2535, 2940, 3549, 3610, 3840, 4046, 4335, 4913, 5054, 5376, 5415, 5491, 6069, 6137, 6358, 6647, 7260, 7581, 7942, 8959, 9386

Offset: 1

Benoit Cloitre, May 09 2002

The squarefree part of x, core(x), is the smallest integer such that x*core(x) is a square.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A007913, A057531, A057532, A050689, A070275, A063737, A067077, A285494.

core[n_]:=Block[{f=FactorInteger[n], p, e}, {p, e} = Transpose[f]; Times@@ (p^Mod[e, 2])]; Select[Range[10^4], core[#] == Plus @@ IntegerDigits[#] &] (* Giovanni Resta, Apr 21 2017 *)

list(lim)=my(v=List()); forfactored(n=1,lim\1, if(core(n)==sumdigits(n[1]), listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Nov 05 2017

Data corrected by Giovanni Resta, Apr 21 2017

A285494 Numbers k such that digit sum of k = number of distinct prime factors of k.

Original entry on oeis.org

20, 30, 102, 120, 200, 300, 1002, 1200, 2000, 2001, 2002, 3000, 3010, 10001, 10002, 10011, 10030, 10120, 11001, 11020, 11110, 12000, 20000, 20001, 21010, 30000, 30030, 30100, 100001, 100030, 100101, 100130, 100210, 100300, 101001, 101101, 101200, 102102, 110001, 110200

Offset: 1

Jonathan Frech, Apr 19 2017

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A001221, A007953, A057531, A057532, A050689, A070274, A070275, A063737, A067077.

Select[Range[2,10000],Total[IntegerDigits[#]]==Length[FactorInteger[#]]&] (* or *)
nd = 10; s = 1; Sort@ Flatten@ Reap[ While[ Times @@ Prime[ Range@ s] < 10^nd, pa = IntegerPartitions[s, {nd}, Range[0, 9]]; Do[Sow@ Select[ FromDigits /@ Permutations[p], PrimeNu[#] == s &], {p, pa}]; s++]][[2, 1]] (* terms < 10^10, Giovanni Resta, Apr 21 2017 *)

isok(n) = sumdigits(n) == omega(n); \\ Michel Marcus, Apr 20 2017

More terms from Michel Marcus, Apr 20 2017

A050690 Sum of digits of zero-absent composite a(n) equals number of prime factors.

Original entry on oeis.org

12, 32, 1152, 11232, 13122, 14112, 21312, 111132, 112112, 3121152, 11231232, 11354112, 812122112, 1251213312, 2211121152, 2211213312, 5121114112, 26122125312, 56321114112, 62214111232, 431711322112, 3421411213312, 11111212122112, 11112113242112

Offset: 1

Patrick De Geest, Aug 15 1999

10^11 < a(21) <= 431711322112. a(22) <= 3421411213312. - Donovan Johnson, May 30 2010

Do all terms end in 2, i.e., is each term = 2 mod 10? - Harvey P. Dale, May 26 2024

			E.g., 21312 (no zero in the string) gives 2+1+3+1+2 = 9 prime factors, namely, 2*2*2*2*2*2*3*3*37.

Cf. A050689, A057531, A057532, A070274, A070275, A063737, A067077.

t={}; Do[If[FreeQ[x=IntegerDigits[n],0]&&PrimeOmega[n]==Total[x],AppendTo[t,n]],{n,2,3220000,2}]; t (* Jayanta Basu, May 30 2013 *)

a(15)-a(20) from Donovan Johnson, May 30 2010
a(21)-a(22) confirmed by Giovanni Resta, Jun 02 2013
a(23)-a(24) from Giovanni Resta, Apr 23 2017

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] &]

A383971 Triprimes with sum of digits 3.

Original entry on oeis.org

12, 30, 102, 1002, 2001, 10002, 10011, 11001, 20001, 100101, 101001, 110001, 200001, 1000002, 10001001, 10010001, 11000001, 20000001, 100000101, 1000000011, 1000001001, 1000010001, 1000100001, 1001000001, 1010000001, 10000000002, 10000000011, 10000010001, 11000000001, 100000000101, 100000001001

Offset: 1

Robert Israel, May 16 2025

Numbers that are the product of 3 primes, counted with multiplicity, and whose sum of decimal digits is 3.
Since all terms are divisible by 3, the only term ending with 0 is 30. All others are of the form 10^i + 10^j + 1 with 0 <= j <= i.
For each d from 2 to at least 71, there is at least one term with d digits.
Includes 10^k + 2 for k in A076850.
All terms except 12 are squarefree.
All even terms are Zumkeller numbers (A083207). - Ivan N. Ianakiev, May 18 2025

			a(4) = 1002 is a term because 1+0+0+2 = 3 and 1002 = 2 * 3 * 167 is the product of 3 primes, counted with multiplicity.

Robert Israel, Table of n, a(n) for n = 1..380

Intersection of any two of A014612, A050689, and A052217.

Cf. A001222, A007953, A076850, A083207.

istriprime:= proc(n) local F;
  F:= ifactors(n,easy)[2];
  if not hastype(F,symbol) then return convert(F[..,2],`+`)=3 fi;
  F:= remove(hastype,F,symbol);
  if nops(F) > 1 or (nops(F) = 1 and F[1,2] > 1) then return false fi;
  numtheory:-bigomega(n) = 3
end proc:
R:= 12, 30:
for d from 3 to 30 do
  V:= select(istriprime, [seq(seq(10^(d-1) + 10^j + 1,j=0..d-1)]);
  R:= R,op(V);
od:
R;

s={30};imax=11;Do[n=10^i+10^j+1;If[PrimeOmega[n]==3,AppendTo[s,n]],{i,0,imax},{j,0,i}];Sort[s] (* James C. McMahon, Jun 01 2025 *)

A376157 Numbers k such that the sum of the digits of k equals the sum of its prime factors plus the sum of the multiplicities of each prime factor.

Original entry on oeis.org

4, 25, 36, 54, 125, 192, 289, 297, 343, 392, 448, 676, 756, 1089, 1536, 1764, 1936, 2646, 2888, 3872, 4802, 4860, 6174, 6250, 6776, 6860, 7290, 7488, 7680, 8750, 8775, 9408, 9747, 10648, 14739, 15309, 16848, 18432, 18865, 21296, 22869, 25725, 29988, 33750, 33957

Offset: 1

Jordan Brooks, Sep 12 2024

			For k = 54, its prime factorization is 2^1*3^3: 5+4 = 2+1+3+3 = 9.
For k = 756, its prime factorization is 2^2*3^3*7^1: 7+5+6 = 2+2+3+3+7+1 = 18.

Similar to A006753, A019506, A050689, A063737, A070275, A285494.

Cf. A007953, A008474.

Select[Range[34000], DigitSum[#]==Total[Flatten[FactorInteger[#]]] &] (* Stefano Spezia, Sep 14 2024 *)

isok(k)={my(f=factor(k)); vecsum(f[,1]) + vecsum(f[,2]) == sumdigits(k)} \\ Andrew Howroyd, Sep 26 2024

from sympy.ntheory import factorint
c = 2
while c < 10000:
    charsum = 0
    for char in str(c):
        charsum += int(char)
    pf = factorint(c)
    cand = 0
    for p in pf.keys():
        cand += p
        cand += pf[p]
    if charsum == cand:
        print(c)
        print(pf)
    c += 1

{ k : A007953(k) = A008474(k) }.

cp's OEIS Frontend

A057531 Numbers whose sum of digits and number of divisors are equal.

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A067077 Numbers whose product of distinct prime factors is equal to its sum of digits.

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A070275 Numbers k such that the sum of the digits of k equals the sum of the prime divisors of k.

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A070274 Numbers n such that sum of digits of n equals the squarefree part of n.

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A285494 Numbers k such that digit sum of k = number of distinct prime factors of k.

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A050690 Sum of digits of zero-absent composite a(n) equals number of prime factors.

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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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A383971 Triprimes with sum of digits 3.

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