A049102 -id:A049102 - cp's OEIS frontend

A049106 Ratio from A049102.

1, 2, 37, 14, 1, 1, 2, 2, 88, 926, 247, 104, 114, 4, 196, 754, 214, 9, 289, 571, 47, 32, 2058, 1696, 36, 2428, 68, 3, 2946, 3071, 96, 86, 123, 3, 103, 61, 113, 119, 138, 97, 797, 41153, 30867, 5672, 30892, 644, 1146, 31142, 289, 41893, 499, 896, 109, 33642

Offset: 1

Olivier Gérard

Cf. A049101, A049102, A049105.

function a049106(a,b: integer); var n,k,j,p,r,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; r := p*k; if r > 0 then if n mod r = 0 then write(n div r,","); end; end; end; end; a049106(0,1220000);

More terms from Klaus Brockhaus, Dec 13 2001

A049101 Numbers m such that m divides (product of digits of m) * (sum of digits of m).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 15, 18, 24, 45, 48, 135, 144, 288, 378, 476, 756, 864, 1575, 39366, 69984, 139968

Offset: 1

Olivier Gérard

Next term if it exists is greater than 4*10^7. - Michel ten Voorde
Sequence is finite and bounded above by 10^84, since if 10^k <= n < 10^(k+1) (product of digits of n)*(sum of digits of n) <= k*9^(k+2) which is less than 10^k for k >= 84. - Henry Bottomley, May 18 2000
Numbers with a zero digit are not permitted. - Harvey P. Dale, Jul 16 2011
No further terms to 2.5*10^9. - Robert G. Wilson v, Jul 17 2011
Sequence is complete. - Giovanni Resta, Mar 20 2013
If product of digits is performed on nonzero digits only, then 1088 is also in the sequence. - Giovanni Resta, Mar 22 2013

			139968 is in the sequence since it divides (1*3*9*9*6*8) * (1+3+9+9+6+8). - _Giovanni Resta_, Mar 20 2013

Giovanni Resta, Method used to compute the full sequence

Cf. A038369, A049102, A049105, A049106.

okQ[n_]:=Module[{idn=IntegerDigits[n]},!MemberQ[idn,0] && Divisible[ (Total[idn]*Times@@idn),n]] (* Harvey P. Dale, Jul 16 2011 *)
(* full sequence *) dig[nD_] := Block[{ric, sol = {}, check}, check[mu_, minN_] := Block[{di = DigitCount@minN, k = 1, r}, While[(r = mu/k) >= minN, If[IntegerQ[r] && DigitCount[r] == di, AppendTo[sol, r]]; k++]]; ric[n_, prod_, sum_, lastd_, cnt_] := Block[{t}, If[cnt == nD, check[prod*sum, n], Do[t = nD - cnt - 1; If[n*10^(t+1) <= d*prod*9^t*(sum + d + 9*t), ric[10*n + d, d*prod, d + sum, d, cnt + 1], Break[]], {d, 9, lastd, -1}]]]; ric[0, 1, 0, 1, 0]; Print["nDig=", nD, " sol=", sol = Sort@sol]; sol]; Flatten[dig /@ Range[84]] (* Giovanni Resta, Mar 20 2013 *)

A066308 a(n) = (sum of digits of n) * (product of digits of n).

Original entry on oeis.org

1, 4, 9, 16, 25, 36, 49, 64, 81, 0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 0, 6, 16, 30, 48, 70, 96, 126, 160, 198, 0, 12, 30, 54, 84, 120, 162, 210, 264, 324, 0, 20, 48, 84, 128, 180, 240, 308, 384, 468, 0, 30, 70, 120, 180, 250, 330, 420, 520, 630, 0, 42, 96, 162, 240, 330

Offset: 1

Labos Elemer, Dec 13 2001

a(n) can be greater than, less than, or equal to n; see Example section.

			For n = 12, a(12) = (1 + 2)*(1*2) = 3*2 = 6 < n;
for n = 19, a(19) = (1 + 9)*(1*9) = 90 > n;
for n = 135, a(135) =(1 + 3 + 5)*(1*3*5) = 135 = n.

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

Cf. A007953, A007954.
Cf. A038369 (fixed points), A034710, A061672.
Cf. A049101, A049102, A049103, A049104, A049105, A049106.

asum[x_] := Apply[Plus, IntegerDigits[x]] apro[x_] := Apply[Times, IntegerDigits[x]] a[n]=asum[n]*apro[n]
sdpd[n_]:=Module[{idn=IntegerDigits[n]},Total[idn]Times@@idn]; Array[ sdpd,70] (* Harvey P. Dale, Dec 31 2011 *)

a(n) = my(d = digits(n)); vecsum(d) * vecprod(d); \\ Michel Marcus, Feb 24 2017

Edited by Jon E. Schoenfield, Jul 09 2018

A049105 Ratio from A049101.

Original entry on oeis.org

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

Offset: 1

Olivier Gérard

Cf. A049101, A049102, A049106.

A330880 Numbers m such that m*p is divisible by m-p, where m > p > 0 and p = A007954(m) = the product of digits of m.

Original entry on oeis.org

24, 36, 45, 48, 144, 384, 624, 672, 798, 816, 3276, 3648, 4864, 5994, 7965, 18816, 56175, 83232, 98496, 177184, 199584, 275772, 344736, 377496, 784896, 879984, 1372896, 1378944, 1635795, 1886976, 2472736, 3364416, 4575375, 6595992, 9289728, 9377424, 28348416, 33247872, 36387792, 58677696

Offset: 1

Scott R. Shannon, May 11 2020

Every term m is the sum of two 7-smooth numbers. Proof: Since (m-p) | m*p, we have m*p = (m - p)*k for some k > 0. Suppose m is not the sum of two 7-smooth numbers. Then m - p is not 7-smooth and so there exists a prime q > 7 such that q | (m - p). Since q doesn't divide p and q | (m - p) but (m - p) | m*p, we have q | m. But since q | m and q | (m - p) we have q | (m - (m - p)) = p, a contradiction. Q.e.d. - David A. Corneth, Jun 15 2020

			24 is a term as p = 2*4 = 8 and 24*8 = 192 is divisible by 24 - 8 = 16.
3648 is a term as p = 3*6*4*8 = 576 and 3648*576 = 2101248 is divisible by 3648-576 = 3072.
1372896 is a term as p = 1*3*7*2*8*9*6 = 18144 and 1372896*18144 = 24909825024 is divisible by 1372896 - 18144 = 1354752.

David A. Corneth, Table of n, a(n) for n = 1..152 (terms <= 10^22; first 82 terms from Giovanni Resta)

Cf. A334679, A334803, A007954, A049102, A085124.

Subsequence of A052382.

npdQ[n_]:=Module[{p=Times@@IntegerDigits[n]},n>p>0&&Divisible[n*p,n-p]]; Select[Range[6*10^7],npdQ] (* Harvey P. Dale, Jun 14 2020 *)

isok(m) = my(p=vecprod(digits(m))); p && (m-p) && !((m*p) % (m-p)); \\ Michel Marcus, May 12 2020

A334679 Numbers k such that k*p is divisible by k+p, where p > 0 and p = A007954(k) = the product of digits of k.

Original entry on oeis.org

2, 4, 6, 8, 24, 36, 63, 456, 495, 3276, 6624, 7497, 8832, 19728, 23976, 127488, 167328, 273525, 274995, 297675, 576975, 661248, 797769, 853776, 1323648, 1378272, 1491264, 1886976, 3483648, 3679263, 3787749, 4644864, 6386688, 7886592, 7888896, 12841472, 15974784, 16224768

Offset: 1

Scott R. Shannon, May 08 2020

			8 is a term as p = 8 and 8*8 = 64 is divisible by 8+8 = 16.
3276 is a term as p = 3*2*7*6 = 252 and 3276*252 = 825552 is divisible by 3276+252 = 3528.
3787749 is a term as p = 3*7*8*7*7*4*9 = 296352 and 3787749*296352 = 1122506991648 is divisible by 3787749+296352 = 4084101.

Giovanni Resta, Table of n, a(n) for n = 1..95 (terms < 3.5*10^14)

Cf. A330880, A334803, A007954, A049102, A085124.

Subsequence of A052382.

Select[Range[10^6], (p = Times @@ IntegerDigits@ #; p > 0 && Mod[# p, # + p] == 0) &] (* Giovanni Resta, May 08 2020 *)

isok(m) = my(p=vecprod(digits(m))); p && !((m*p) % (m+p)); \\ Michel Marcus, May 08 2020

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

Original entry on oeis.org

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

A334803 Numbers k such that k*p is divisible by k+p and k-p, where k > p > 0 and p = A007954(k) = the product of digits of k.

Original entry on oeis.org

24, 36, 3276, 1886976

Offset: 1

Scott R. Shannon, May 12 2020

If a(5) exists it is at least 3*10^12.

a(5) > 1.5*10^14, if it exists. - Giovanni Resta, May 12 2020

			24 is a term as p = 2*4 = 8 and 24*8 = 192 is divisible by both 24-8 = 16 and 24+8 = 32.
36 is a term as p = 3*6 = 18 and 38*18 = 648 is divisible by both 36-18 = 18 and 36+18 = 54.
3276 is a term as p = 3*2*7*6 = 252 and 3276*252 = 825552 is divisible by both 3276-252 = 3024 and 3276+252 = 3528.
1886976 is a term as p = 1*8*8*6*9*7*6 = 145152 and 1886976*145152 = 273898340352 is divisible by both 1886976-145152 = 1741824 and 1886976+145152 = 2032128.

Cf. A007954, A049102, A085124.

Subsequence of A052382. Intersection of A334679 and A330880.

isok(m) = my(p=vecprod(digits(m))); p && (m-p) && !((m*p) % (m-p)) && !((m*p) % (m+p)); \\ Michel Marcus, May 12 2020

cp's OEIS Frontend

A049106 Ratio from A049102.

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A049101 Numbers m such that m divides (product of digits of m) * (sum of digits of m).

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A066308 a(n) = (sum of digits of n) * (product of digits of n).

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A049105 Ratio from A049101.

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A330880 Numbers m such that m*p is divisible by m-p, where m > p > 0 and p = A007954(m) = the product of digits of m.

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A334679 Numbers k such that k*p is divisible by k+p, where p > 0 and p = A007954(k) = the product of digits of k.

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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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A334803 Numbers k such that k*p is divisible by k+p and k-p, where k > p > 0 and p = A007954(k) = the product of digits of k.

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