A285494 -id:A285494 - cp's OEIS frontend

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

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

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) }.

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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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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.

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