A066308 -id:A066308 - cp's OEIS frontend

A256240 Numbers n such that repeatedly setting n := A066308(n) yields a constant nonzero n.

1, 89, 98, 135, 139, 144, 153, 193, 233, 315, 319, 323, 332, 351, 391, 414, 441, 513, 531, 913, 931, 1224, 1242, 1367, 1376, 1422, 1637, 1673, 1736, 1763, 2124, 2142, 2214, 2241, 2412, 2421, 3167, 3176, 3617, 3671, 3716, 3761, 4122, 4212

Offset: 1

David A. Corneth, Mar 20 2015

Eventually, these values of n become nonzero elements of A038369; 1, 135 or 144.

			89 is an element because (8 + 9) * 8 * 9 = 1224, then (1 + 2 + 2 + 4) * 1 * 2 * 2 * 4 = 144, then (1 + 4 + 4) * 1 * 4 * 4 = 144. Repetition so stop. 144 > 0 so 89 is an element.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms n = 1..869 from David A. Corneth).
Eric Weisstein's World of Mathematics, Sum-Product Number

Cf. A038369, A066308.

Select[Range[5000], FixedPoint[Total[#] Apply[Times, #] &@ IntegerDigits@ # &, #] > 0 &] (* Michael De Vlieger, Aug 16 2017 *)

\\test if n is an element.
is(n)=while(n!=SP(n),n=SP(n));n>0
\\Sum of digits times product of digits of n (A066308(n))
SP(n)={d=digits(n);prod(i=1,#d,d[i])*vecsum(d)}

A256242 Numbers having digits in nondecreasing order and repeatedly setting n := A066308(n) yields a constant nonzero n.

Original entry on oeis.org

1, 89, 135, 139, 144, 233, 1224, 1367, 11249, 12222, 111126, 111266, 111338, 112229, 112337, 1111119, 1111134, 1111137, 1111177, 1111333, 1111346, 11111117, 11111119, 11111223, 11112236, 111111119, 111111139, 111111299, 111112334, 1111111169, 1111122233, 11111111118, 11111111133, 11111111369, 111111111133

Offset: 1

David A. Corneth, Mar 20 2015

Intersection of A009994 and A256240. All digits differ from 0.

Permutations of all numbers of the elements in the table give the first 56622402 elements from A256240 (unsorted).

David A. Corneth, Table of n, a(n) for n = 1..166 (all elements below 10^100).

Cf. A009994, A256240.

isok(n) = {d = digits(n); if (vecsort(d,,2) == d, ok = 1; while (ok, newn = sum(k=1, #d, d[k])*prod(k=1,#d, d[k]); if (! newn, return (0)); if (newn == n, return (1)); n = newn; d = digits(n);););} \\ Michel Marcus, Mar 27 2015

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

Original entry on oeis.org

0, 1, 135, 144

Offset: 1

Felice Russo

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

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

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.

Cf. A007953, A007954, A066308.

Cf. A038364, A062237, A066282.

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

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

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

A369529 a(n) = gcd(A007953(n), A007954(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 6, 1, 4, 9, 2, 1, 12, 1, 2, 3, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1, 8, 1, 2, 1, 4, 1, 2, 1, 16, 1, 9, 1, 1, 3

Offset: 1

Ctibor O. Zizka, Jan 25 2024

			a(15) = gcd(A007953(15), A007954(15)) = gcd(6,5) = 1.

Cf. A007953, A007954, A066308.

a[n_] := GCD @@ Apply[{Plus @@ #, Times @@ #} &, {IntegerDigits[n]}]; Array[a, 100] (* Amiram Eldar, Jan 26 2024 *)

cp's OEIS Frontend

A256240 Numbers n such that repeatedly setting n := A066308(n) yields a constant nonzero n.

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A256242 Numbers having digits in nondecreasing order and repeatedly setting n := A066308(n) yields a constant nonzero n.

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

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A369529 a(n) = gcd(A007953(n), A007954(n)).

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