A038369 Numbers k such that k = (product of digits of k) * (sum of digits of k).

0, 1, 135, 144

Offset: 1

Author

Felice Russo

Keywords

• nice
• nonn
• fini
• base
• full

Comments

The list is complete. Proof: One shows that the number of digits is at most 84 and then it is only necessary to consider numbers of the forms 2^i*3^j*7^k and 3^i*5^j*7^k. - David W. Wilson, May 16 2003

Examples

			144 belongs to the sequence because 1*4*4=16, 1+4+4=9 -> 16*9=144
		

Links

• Alan Beardon, S.P numbers, The Mathematical Gazette, 83(496), 25-32 (1999).
• Alan Beardon, Sums and Products of Digits and SP Numbers, NRICH, University of Cambridge, 1998.
• Alan Beardon, Recent Developments on S.P. Numbers, NRICH, University of Cambridge, 1998-2011.
• E. Bussmann, S.P numbers in bases other than 10, The Mathematical Gazette, 85(503), 245-248 (2001).
• K. McLean, There are only three S.P numbers!, The Mathematical Gazette, 83(496), 32-38 (1999).
• S. Parameswaran, Numbers and their digits - a structural pattern, Note 81.24, The Mathematical Gazette, 81(491), 263-263 (1997).
• Eric Weisstein's World of Mathematics, Sum-Product Number.
• Eric Weisstein's World of Mathematics, Digit.

Crossrefs

Cf. A007953, A007954, A066308.

Cf. A038364, A062237, A066282.

Programs

Mathematica

pdsdQ[n_]:=Module[{idn=IntegerDigits[n]},(Total[idn]Times@@idn)==n]; Select[Range[0,150],pdsdQ]  (* Harvey P. Dale, Apr 23 2011 *)

PARI

is(n)=my(d=digits(n)); factorback(d)*vecsum(d)==n \\ Charles R Greathouse IV, Feb 06 2017

Formula

a(n) = A007953(a(n)) * A007954(a(n)).