A038364 -id:A038364 - cp's OEIS frontend

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

0, 1, 135, 144

Offset: 1

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)).

A062237 Numbers k which are (sum of digits of k) concatenated with (product of digits of k).

Original entry on oeis.org

0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 119, 1236, 19135, 19144, 261296, 3634992, 43139968

Offset: 1

Erich Friedman, Jun 30 2001

For a d-digit number with d >= 88, the sum and product of the digits together have fewer than d digits. So every element of this sequence has 87 or fewer digits, hence it is finite. - David W. Wilson, Apr 28 2005

Fixed points of the map A380873: concatenate sum and product of digits. - M. F. Hasler, Apr 01 2025

			1236 has sum of digits 12 and product of digits 36.

Cf. A007953 (sum of digs), A007954 (product of digs), A038364, A038369, A066282, A380873, A380872 (trajectories under map).

sdpdQ[n_]:=Module[{idn=IntegerDigits[n],s,p},s=Total[idn];p=Times@@idn;n==FromDigits[Join[IntegerDigits[s],IntegerDigits[p]]]]; Select[Range[44*10^6],sdpdQ] (* Harvey P. Dale, Nov 23 2024 *)

from math import prod
from sympy.utilities.iterables import multiset_permutations as mp
from itertools import count, islice, combinations_with_replacement as mc
def c(s):
    d = list(map(int, s))
    return sorted(s) == sorted(str(sum(d)) + str(prod(d)))
def ok(s):
    d = list(map(int, s))
    return s[0] != '0' and "".join(s) == str(sum(d)) + str(prod(d))
def nd(d): yield from ("".join(m) for m in mc("0123456789", d))
def b(): yield from (s for d in count(1) for s in nd(d) if c(s))
def a(): yield from (int("".join(p)) for s in b() for p in mp(s) if ok(p))
print(list(islice(a(), 16))) # Michael S. Branicky, Jun 30 2022

More terms from Harvey P. Dale, Jul 04 2001
More terms from David W. Wilson, Apr 28 2005; he reports on May 03 2005 that there are no further terms.
Offset corrected by Altug Alkan, Apr 10 2018

A066282 Numbers k such that k = (product of nonzero digits of k) * (sum of digits of k).

Original entry on oeis.org

0, 1, 135, 144, 1088

Offset: 1

Klaus Brockhaus, Dec 13 2001

Suppose a term k has d digits, then k > 10^(d-1), the product of nonzero digits <= 9^d, and the sum of digits <= 9*d. Since for d >= 85 we have 10^(d-1) > 9^d * (9*d), it follows that d <= 84. That is, the sequence is finite. I've further verified that there are no other terms, that is, the sequence is complete. - Max Alekseyev, Jul 29 2024

			(1+0+8+8) * (1*8*8) = 17*64 = 1088, so 1088 belongs to the sequence.

René-Louis Clerc, Nombres S+P, maxSP, minSP et |P-S|, hal-04507547 [math.nt], 2024. (In French)

Fixed points of A062331.

Cf. A023651, A038364, A038369, A062237.

function a066282(a,b: integer); var n,k,j,p,d: integer; s: string; begin for n := a to b do s := itoa(n); k := 0; p := 1; for j := 0 to length(s) - 1 do d := atoi(s[j..j]); k := k + d; if d > 0 then p := p*d; end; end; if n = p*k then write(n,","); end; end; end; a066282(0,25000).

Do[ d = Sort[ IntegerDigits[n]]; While[ First[d] == 0, d = Drop[d, 1]]; If[n == Apply[ Plus, d] Apply[ Times, d], Print[n]], {n, 0, 5*10^7} ]

a066282(a,b) = local(n,k,q,p,d); for(n=a,b,k=0; p=1; q=n; while(q>0,d=divrem(q,10); q=d[1]; k=k+d[2]; p=p*max(1,d[2])); if(n==p*k,print1(n,", ")))
a066282(0,25000)

Offset corrected by Mohammed Yaseen, Jul 21 2022

Keywords fini,full added by Max Alekseyev, Jul 29 2024

A023651 Numbers k such that (product of digits of k) * (sum of digits of k) = 2k.

Original entry on oeis.org

0, 2, 15, 24, 1575, 39366

Offset: 1

Jason Earls, Dec 11 2001

Except for k = 0, this sequence is a subsequence of A049101. - Jason Yuen, Feb 26 2024

Cf. A007953, A007954, A049101.

Cf. A038364, A038369, A062237, A066282.

Do[ If[ 2n == Apply[ Times, IntegerDigits[n]] Apply[ Plus, IntegerDigits[n]], Print[n]], {n, 0, 10^7} ]

isok(n) = if(n, factorback(digits(n)), 0) * sumdigits(n) == 2*n \\ Mohammed Yaseen, Jul 22 2022

from math import prod
def s(n): return sum(map(int, str(n)))
def p(n): return prod(map(int, str(n)))
for n in range(0, 10**6):
  if p(n)*s(n)==2*n:
    print(n) # Mohammed Yaseen, Jul 22 2022

Offset corrected by Arkadiusz Wesolowski, Oct 17 2012

A366832 Numbers k such that k = (product of nonzero digits) * (sum of digits) for the digits of k in base 9.

Original entry on oeis.org

1, 12, 1536, 172032, 430080, 4014080

Offset: 1

René-Louis Clerc, Jan 10 2024

There is a finite number of such numbers (Property 1' of Clerc).

			430080 = 724856_9, (7+2+4+8+5+6)*(7*2*4*8*5*6) = 32*13440 = 430080.

René-Louis Clerc, Quelques nombres de Niven-Harshad particuliers, 2023.
René-Louis Clerc, Nombres S+P, maxSP, minSP et |P-S|, hal-04507547 [math.nt], 2024. (In French)

Cf. A066282, A062331, A023651, A038364, A038369, A062237.

Select[Range[5*10^6],Total[IntegerDigits[#,9]]*Fold[Times,1,IntegerDigits[#,9]]==#&] (* James C. McMahon, Jan 30 2024 *)

isok(k, b) = my(d=select(x->(x>0), digits(k,b))); vecprod(d)*vecsum(d) == k;
 for (k=1, 10^7, if (isok(k, 9), print1(k, ", ")))

A367070 Numbers k such that k = (product of nonzero digits) * (sum of digits) for the digits of k in base 7.

Original entry on oeis.org

1, 16, 128, 250, 480, 864, 21600, 62208, 73728

Offset: 1

René-Louis Clerc, Jan 10 2024

There is a finite number of such numbers; we only calculated the terms in [1, 10^10] (Property 1' of Clerc).

			21600 = 116655_7, (1+1+6+6+5+5)*(1*1*6*6*5*5) = 24*900 = 21600.

René-Louis Clerc, Quelques nombres de Niven-Harshad particuliers, 2023.
René-Louis Clerc, Nombres S+P, maxSP, minSP et |P-S|, hal-04507547 [math.nt], 2024. (In French)

Cf. A066282, A062331, A023651, A038364, A038369, A062237.

Select[Range[7^7], #1 == Times @@ DeleteCases[#2, 0]*Total[#2] & @@ {#, IntegerDigits[#, 7]} &] (* Michael De Vlieger, Mar 25 2024 *)

isok(k, b) = my(d=select(x->(x>0), digits(k,b))); vecprod(d)*vecsum(d) == k;
for (k=1, 10^5, if (isok(k, 7), print1(k, ", ")))

A370251 Base-12 numbers k such that k = (product of nonzero digits of k) * (sum of digits of k) (written in base 10).

Original entry on oeis.org

1, 176, 231, 495, 7040

Offset: 1

René-Louis Clerc, Feb 13 2024

There are only finitely many such numbers (Property 1' of Clerc).

			231 = 173_12, (1*7*3)*(1+7+3) = 21*11 = 231.

René-Louis Clerc, Quelques nombres de Niven-Harshad particuliers, 2023.
René-Louis Clerc, Nombres S+P, maxSP, minSP et |P-S|, hal-04507547 [math.nt], 2024. (In French)

Cf. A366832, A367070, A066282, A062331, A023651, A038364, A038369, A062237.

Select[Range[5*10^4], Total[IntegerDigits[#, 12]]*Fold[Times, 1, Select[IntegerDigits[#, 12],#>0&]]==#&] (* James C. McMahon, Feb 14 2024 *)

isok(k, b) = my(d=select(x->(x>0), digits(k, b))); vecprod(d)*vecsum(d) == k;
for (k=0, 10^10, if (isok(k, 12), print1(k, ", ")))

A371337 Numbers k>0 such that k = (sum of digits of k^2) + (product of nonzero digits of k^2).

Original entry on oeis.org

127, 1467, 3052, 5860, 653230, 3483702, 43352128, 783820873, 8092385362, 622196951140, 1061882796441600145, 178949702436677222562

Offset: 1

René-Louis Clerc, Mar 19 2024

			1467^2 = 2152089, (2+1+5+2+8+9) + (2*1*5*2*8*9) = 27 + 1440 = 1467.

René-Louis Clerc, Nombres S+P, maxSP, minSP et |P-S|, pp. 1-15, 2024.

Cf. A038364, A061763.

SplusP(k,r) = my(d=select(x->(x>0),digits(k^r))); vecsum(d) + vecprod(d) == k;
        resuSplusP(p,r)=for(k=1,10^p,if(SplusP(k,r) ==1,print1(k,",")))

a(9)-a(12) from Chai Wah Wu, Apr 20 2024

A371338 Numbers k>0 such that k = |(product of nonzero digits of k^2) - (sum of digits of k^2)|.

Original entry on oeis.org

161, 198, 1701, 604755, 629810, 4354506, 100018736, 411505847, 14869757951891, 2239397044538572646, 40766979086355529727820, 6289762487609138872319999999757

Offset: 1

René-Louis Clerc, Mar 19 2024

Most often P-S is strictly positive but to always have an application of N* in N* we prefer to use |P-S| (cf. Clerc).

			1701^2 = 2893401, |(2*8*9*3*4*1) - (2+8+9+3+4+1)| = 1728 - 27 = 1701.

René-Louis Clerc, Nombres S+P, maxSP, minSP et |P-S|, pp. 1-15, 2024.

Cf. A038364, A061763.

SmP(k,r)=my(d=select(x->(x>0),digits(k^r))); abs(vecsum(d)- vecprod(d)) == k;
 resuSmP(p,r)={for(k=1,10^p,if(SmP(k,r)==1, print1(k,";")))}

a(9)-a(12) from Chai Wah Wu, Apr 20 2024

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

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A062237 Numbers k which are (sum of digits of k) concatenated with (product of digits of k).

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A066282 Numbers k such that k = (product of nonzero digits of k) * (sum of digits of k).

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A023651 Numbers k such that (product of digits of k) * (sum of digits of k) = 2k.

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A366832 Numbers k such that k = (product of nonzero digits) * (sum of digits) for the digits of k in base 9.

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A367070 Numbers k such that k = (product of nonzero digits) * (sum of digits) for the digits of k in base 7.

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A370251 Base-12 numbers k such that k = (product of nonzero digits of k) * (sum of digits of k) (written in base 10).

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