A038369 -id:A038369 - cp's OEIS frontend

A066042 Numbers k such that k divided by ((sum of digits of k) multiplied by (product of digits of k)) is prime.

12, 111, 216, 432, 41112, 81216, 186624, 248832, 311472, 316224, 341712, 422144, 714112, 1131111, 1131732, 1191915, 1211328, 1292112, 1418112, 2192832, 3112128, 4331232, 11127424, 11311272, 18122112, 21111192, 26726112, 28422144, 34338816

Offset: 1

Enoch Haga, Dec 13 2001

			a(2) = 111 because 1+1+1 = 3 and 1*1*1 = 1 and 3*1 = 3 and 111/3 = 37 and 37 is prime. [corrected by _Harry J. Smith_, Nov 08 2009]

David A. Corneth, Table of n, a(n) for n = 1..1744 (first 469 terms from Harry J. Smith and Chai Wah Wu)
David A. Corneth, a(n) = [product of digits of a(n)] * [sum of digits of a(n)] * [some prime]

Cf. A038369, A049102, A066146.

ndspQ[n_]:=Module[{idn=IntegerDigits[n]},FreeQ[idn,0]&&PrimeQ[n/(Total[ idn]Times@@idn)]]; Select[Range[35*10^6],ndspQ] (* Harvey P. Dale, Feb 09 2015 *)

isok(k) = { my(d=digits(k), q=vecsum(d)*vecprod(d)); q!= 0 && k%q==0 && isprime(k/q) }
{ for(k=0, 10^7, if(isok(k), print1(k, ", "))) } \\ Harry J. Smith, Nov 08 2009

Sum digits of n; take product of digits of n; multiply sum by product and divide into n. If prime, add to sequence.

Checked to over 10^8 (110508539) without finding another example.
Offset 1 from Harry J. Smith, Nov 08 2009
Should have found 34338816, 37121112, and 41174112 < 10^8. Term a(29) from Harry J. Smith, Nov 08 2009

A066309 Numbers k such that k > (product of digits of k) * (sum of digits of k).

Original entry on oeis.org

10, 11, 12, 13, 20, 21, 22, 30, 31, 32, 40, 41, 50, 51, 60, 61, 70, 71, 80, 81, 90, 91, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 130, 131, 132, 133, 134, 140, 141, 142

Offset: 1

Labos Elemer and Klaus Brockhaus, Dec 13 2001

			13 is in the sequence because (1*3)*(1+3) = 3*4 = 12 < 13.
125 is a term because (1*2*5)*(1+2+5) = 10*8 = 80 < 125.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A038369, A049101-A049106, A034710, A061672, A066306, A066307.

function a066312(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; p := p*d; end; if n > p*k then write(n,","); end; end; end; a066312(0,150);

asum[x_] := Apply[Plus, IntegerDigits[x]] apro[x_] := Apply[Times, IntegerDigits[x]] sz[x_] := asu[x]*apro[x] Do[s=sz[n]; If[Greater[n, s], Print[n]], {n, 1, 1000}]
okQ[n_]:=Module[{idn=IntegerDigits[n]},n> Total[idn]Times@@idn];Select[Range[150],okQ]  (* Harvey P. Dale, Mar 12 2011 *)

isok(k) = {my(d=digits(k)); k > vecprod(d) * vecsum(d)} \\ Harry J. Smith, Feb 10 2010

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

Original entry on oeis.org

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)}

A114457 Smallest k > 0 such that abs(S(k)P(k)-k) equals n, where S(k) is the sum and P(k) is the product of decimal digits of k or 0 if no such k exists.

Original entry on oeis.org

1, 13, 2, 219, 724, 1285, 3, 23, 7789816, 11, 10, 2891, 4, 127, 226, 15, 3248, 163, 52, 31, 5, 33, 262, 12857, 24, 325, 16, 243, 38428, 617, 6, 68177, 172, 0, 62, 2275, 272, 22577, 118, 17, 40, 43, 7, 1339, 136, 25, 154, 143, 128, 125599, 34, 5619, 352, 1483

Offset: 0

Eric W. Weisstein, Nov 28 2005

a(33) > 2*10^9; then sequence continues 62, 2275, 272, 22577, 118, 17, 40, 43, 7, 1339, 136, 25, 154, 143, 128, 125599, 34, 5619, 352, 1483, 18, 145, 8, 15457, 173, 14963, 60, 1727, 517, 1197, 1787456, 235, 642, 53, 116, ... - Robert G. Wilson v, Dec 14 2005
a(33) > 2*10^16. - Floris M. Velleman, Dec 17 2014
a(33) = 0. Modification of David W. Wilson's proof for A038369 shows that if a(33) > 0, then a(33) has at most 84 digits. This allows an exhaustive search of numbers of the form 2^a*3^b*5^c*7^d which shows that no such number exists. Other values of n for which a(n) is currently unknown and may be equal to 0 (based on analysis of numbers with at most 20 digits) are: 69, 111, 127, 146, 168, 172, 233, 243, 249, 273, 279, 281, 316, 327, 372, 533, 557, 579, 587, 621, 623, 647, 649, 676, 683, 713, 721, 816, 819, 821, 827, 861, 872, 917, 926, 927, 928, 939, 983, 996, 999, ... - Chai Wah Wu, Nov 22 2015
a(69) = a(111) = 0. To compute a(111), numbers of at most 85 digits were checked. - Chai Wah Wu, Dec 04 2015

Chai Wah Wu, Table of n, a(n) for n = 0..126
Eric Weisstein's World of Mathematics, Sum-Product Number

Cf. A007953 (sum of digits), A007954 (product of digits), A038369.

a[n_] := Block[{k = 1}, While[id = IntegerDigits@k; Abs[(Plus @@ id)(Times @@ id) - k] != n, k++ ]; k];
Table[ a[n], {n, 0, 54}] (* Robert G. Wilson v, Dec 14 2005 *)

f(k) = my(d=digits(k)); abs(sum(j=1, #d, d[j])*prod(j=1,#d, d[j]) - k);
a(n) = {k = 1; while(f(k) != n, k++); k;} \\ Michel Marcus, Jan 02 2015

Added a(33), edited definition and verified a(34)-a(68) by Chai Wah Wu, Nov 22 2015

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, ", ")))

A114061 Numbers n such that n = (product of digits of n) * (sum of digits of n) in some base.

Original entry on oeis.org

0, 1, 6, 12, 16, 20, 30, 42, 48, 54, 56, 72, 90, 96, 110, 128, 132, 135, 144, 156, 160, 162, 176, 182, 210, 231, 240, 250, 272, 300, 306, 324, 336, 342, 380, 384, 420, 432, 448, 455, 462, 480, 495, 504, 506, 540, 552, 576, 600, 624, 650, 663, 686, 702, 720, 750

Offset: 1

Matthew Conroy, Feb 02 2006

This sequence is infinite since b^2+b is in the sequence for all b>1: in base b, b^2+b has digits {1,b} and (1*b)*(1+b)=b^2+b.

			12 is in the sequence since in base 9, 12 has digits {1,3} and (1*3)*(1+3)=12.

Cf. A038369.

A145745 Numbers n such that n = sigma(sum of digits of n)*sigma(product of digits of n).

Original entry on oeis.org

1, 12, 6266169, 6931848

Offset: 1

Farideh Firoozbakht, Oct 26 2008

No further terms up to 10^26. - Chai Wah Wu, Nov 28 2015

No other terms below 10^100. The sequence is likely finite and complete. - Max Alekseyev, Sep 05 2023

			12 = 4*3 = sigma(1+2)*sigma(1*2).
6931848 = 56*123783 = sigma(6+9+3+1+8+4+8)*sigma(6*9*3*1*8*4*8).

Cf. A038369.

Do[h=IntegerDigits[n]; s=Apply[Plus,h];p=Apply[Times,h];If[n== DivisorSigma[1,s]*DivisorSigma[1,p],Print[n]],{n,2000000000}]

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, ", ")))

cp's OEIS Frontend

A066042 Numbers k such that k divided by ((sum of digits of k) multiplied by (product of digits of k)) is prime.

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A066309 Numbers k such that k > (product of digits of k) * (sum of digits of k).

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

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A114457 Smallest k > 0 such that abs(S(k)P(k)-k) equals n, where S(k) is the sum and P(k) is the product of decimal digits of k or 0 if no such k exists.

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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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A114061 Numbers n such that n = (product of digits of n) * (sum of digits of n) in some base.

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A145745 Numbers n such that n = sigma(sum of digits of n)*sigma(product of digits of n).

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