A067170 Numbers n such that sum of the cubes of the distinct prime factors of n equals the sum of the cubes of the digits of n.

2, 3, 5, 7, 250, 735, 2500, 25000, 250000, 1858560, 2500000, 18585600, 25000000, 91990080, 185856000, 242121642, 250000000, 919900800, 1081088775, 1390120992, 1768635648, 1858560000, 2500000000, 5435938431, 7245987840, 9199008000, 9475854336, 17996666688, 18585600000, 24214634829, 25000000000

Offset: 1

Author

Joseph L. Pe, Feb 18 2002

Keywords

• nonn
• base

Comments

If 10*m is a term (e.g. m = 25, 185856, 9199008), then 10^k * m is a term for all k >= 1. Therefore this sequence is infinite. - Amiram Eldar, Sep 28 2019

The sum of cubes of digits of a k-digit number is at most 729*k. Therefore any term with at most k digits is p-smooth where p is the largest prime < (729*k)^(1/3). - David A. Corneth, Sep 28 2019

Examples

			The prime factors of 735 are 3,5,7, the sum of whose cubes = 495 = sum of the cubes of the digits of 735; so 735 is a term of the sequence.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..11790 (first 260 terms from Giovanni Resta, terms < 10^34)

Crossrefs

Cf. A005064, A006753, A055012, A067184.

Programs

Mathematica

f[n_] := Module[{a, l, t, r}, a = FactorInteger[n]; l = Length[a]; t = Table[a[[i]][[1]], {i, 1, l}]; r = Sum[(t[[i]])^3, {i, 1, l}]]; g[n_] := Module[{b, m, s}, b = IntegerDigits[n]; m = Length[b]; s = Sum[(b[[i]])^3, {i, 1, m}]]; Select[Range[2, 10^6], f[ # ] == g[ # ] &]

PARI

sd(n) = my(d=digits(n)); sum(k=1, #d, d[k]^3); \\ A055012
sp(n) = my(f=factor(n)); sum(k=1, #f~, f[k,1]^3); \\ A005064
isok(n) = sp(n) == sd(n); \\ Michel Marcus, Sep 28 2019

Extensions

a(10)-a(14) from Amiram Eldar, Sep 28 2019

a(15)-a(18) from Michel Marcus, Sep 28 2019

a(20)-a(29) from David A. Corneth, Sep 28 2019

Missing a(19) from Giovanni Resta, Sep 28 2019