cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

A073907 Erroneous version of A085124.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 6, 7, 8, 9, 11, 12, 33, 24, 15, 36, 77, 24, 36, 0, 0, 168, 0, 378, 135, 48, 0, 36, 0, 0, 273, 0, 0, 168, 525, 0, 999, 476, 0, 0, 0, 288, 0, 0, 175, 1296, 0, 0, 0, 0, 0, 378, 0, 0, 495, 0, 0, 384, 3577, 0, 0, 0, 0, 1296, 0, 728, 0, 0, 0, 0, 0, 0, 1197, 448, 0, 0, 0, 0, 0, 0, 0, 1368, 0, 0
Offset: 1

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Author

Keywords

A073910 Smallest number m such that m and the product of digits of m are both divisible by 3n, or 0 if no such number exists.

Original entry on oeis.org

3, 6, 9, 168, 135, 36, 273, 168, 999, 0, 0, 1296, 0, 378, 495, 384, 0, 1296, 0, 0, 1197, 0, 0, 1368, 3525, 0, 2997, 672, 0, 0, 0, 384, 0, 0, 735, 1296, 0, 0, 0, 0, 0, 3276, 0, 0, 3915, 0, 0, 3168, 7497, 0, 0, 0, 0, 5994, 0, 7896, 0, 0, 0, 0, 0, 0, 7938, 2688, 0, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Amarnath Murthy, Aug 18 2002

Keywords

Comments

Here 0 is regarded as not divisible by any number.
a(n) = 0 if 10 divides 3n or n contains a prime divisor > 9. - Sascha Kurz, Aug 23 2002

Links

Crossrefs

Cf. A073906, A073908, A073909, A073911, A073912, A085124.

Programs

  • Maple
    f := 3:for i from 1 to 1000 do b := ifactors(f*i)[2]: if b[nops(b)][1]>9 or (f*i mod 10) =0 then a[i] := 0:else j := 0:while true do j := j+f*i:c := convert(j,base,10):d := product(c[k],k=1..nops(c)): if (d mod f*i)=0 and d>0 then a[i] := j:break:fi:od:fi:od:seq(a[k],k=1..1000);

Formula

a(n) = A085124(3*n). - R. J. Mathar, Jun 21 2018

Extensions

More terms from Sascha Kurz, Aug 23 2002

A073908 Smallest number m such that m and the product of digits of m are both divisible by 7n, or 0 if no such number exists.

Original entry on oeis.org

7, 378, 273, 476, 175, 378, 3577, 728, 1197, 0, 0, 672, 0, 7742, 735, 784, 0, 3276, 0, 0, 7497, 0, 0, 7896, 1575, 0, 7938, 69776, 0, 0, 0, 12768, 0, 0, 37975, 3276, 0, 0, 0, 0, 0, 71736, 0, 0, 9765, 0, 0, 8736, 47677, 0, 0, 0, 0, 7938, 0, 74872, 0, 0, 0, 0, 0, 0, 7497
Offset: 1

Views

Author

Amarnath Murthy, Aug 18 2002

Keywords

Comments

Here 0 is regarded as not divisible by any number.
a(n) = 0 if n is divisible by 10 or contains a prime divisor > 9. - Sascha Kurz, Aug 23 2002

Examples

			a(8) = 728 is divisible by 7*8 = 56 and also 7*2*8 = 112 = 2*56.

Crossrefs

Cf. A073906, A085124, A073909, A073910, A073911, A073912.

Programs

  • Maple
    f := 7:for i from 1 to 400 do b := ifactors(f*i)[2]: if b[nops(b)][1]>9 or (f*i mod 10) =0 then a[i] := 0:else j := 0:while true do j := j+f*i:c := convert(j,base,10): d := product(c[k],k=1..nops(c)): if (d mod f*i)=0 and d>0 then a[i] := j:break:fi: od:fi:od:seq(a[k],k=1..400);

Formula

a(n) = A085124(7*n). - R. J. Mathar, Jun 21 2018

Extensions

More terms from Sascha Kurz, Aug 23 2002

A073909 Smallest number m such that m and the product of digits of m are both divisible by 2n, or 0 if no such number exists.

Original entry on oeis.org

2, 4, 6, 8, 0, 168, 378, 48, 36, 0, 0, 168, 0, 476, 0, 288, 0, 1296, 0, 0, 378, 0, 0, 384, 0, 0, 1296, 728, 0, 0, 0, 448, 0, 0, 0, 1368, 0, 0, 0, 0, 0, 672, 0, 0, 0, 0, 0, 384, 7742, 0, 0, 0, 0, 1296, 0, 784, 0, 0, 0, 0, 0, 0, 3276, 2688, 0, 0, 0, 0, 0, 0, 0, 3168, 0, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Amarnath Murthy, Aug 18 2002

Keywords

Comments

Here 0 is regarded as not divisible by any number.
a(n) = 0 if n is divisible by 5 or contains a prime divisor > 9. - Sascha Kurz, Aug 23 2002

Crossrefs

Cf. A073906, A085124, A073908, A073910, A073911, A073912.

Programs

  • Maple
    f := 2:for i from 1 to 400 do b := ifactors(f*i)[2]: if b[nops(b)][1]>9 or (f*i mod 10) =0 then a[i] := 0:else j := 0:while true do j := j+f*i:c := convert(j,base,10): d := product(c[k],k=1..nops(c)): if (d mod f*i)=0 and d>0 then a[i] := j:break:fi: od:fi:od:seq(a[k],k=1..400);

Formula

a(n) = A085124(2*n). - R. J. Mathar, Jun 21 2018

Extensions

More terms from Sascha Kurz, Aug 23 2002

A073911 Smallest number m such that m and the product of digits of m are both divisible by 5n, or 0 if no such number exists.

Original entry on oeis.org

5, 0, 135, 0, 525, 0, 175, 0, 495, 0, 0, 0, 0, 0, 3525, 0, 0, 0, 0, 0, 735, 0, 0, 0, 55125, 0, 3915, 0, 0, 0, 0, 0, 0, 0, 1575, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15975, 0, 0, 0, 37975, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9765, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 155625, 0, 0, 0, 0, 0, 31995, 0, 0
Offset: 1

Views

Author

Amarnath Murthy, Aug 18 2002

Keywords

Comments

Here 0 is regarded as not divisible by any number.
a(n) = 0 if n is divisible by 2 or contains a prime divisor > 9. - Sascha Kurz, Aug 23 2002

Crossrefs

Cf. A073906, A085124, A073908, A073909, A073910, A073912.

Programs

  • Maple
    f := 5:for i from 1 to 400 do b := ifactors(f*i)[2]: if b[nops(b)][1]>9 or (f*i mod 10) =0 then a[i] := 0:else j := 0:while true do j := j+f*i:c := convert(j,base,10): d := product(c[k],k=1..nops(c)): if (d mod f*i)=0 and d>0 then a[i] := j:break:fi: od:fi:od:seq(a[k],k=1..400);

Formula

a(n) = A085124(5*n). - R. J. Mathar, Jun 21 2018

Extensions

More terms from Sascha Kurz, Aug 23 2002

A073912 Smallest number m such that m and the product of digits of m are both divisible by 8n, or 0 if no such number exists.

Original entry on oeis.org

8, 48, 168, 288, 0, 384, 728, 448, 1368, 0, 0, 384, 0, 784, 0, 2688, 0, 3168, 0, 0, 7896, 0, 0, 2688, 0, 0, 4968, 12768, 0, 0, 0, 4864, 0, 0, 0, 4896, 0, 0, 0, 0, 0, 8736, 0, 0, 0, 0, 0, 2688, 74872, 0, 0, 0, 0, 22896, 0, 14784, 0, 0, 0, 0, 0, 0, 33768, 14848, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Amarnath Murthy, Aug 18 2002

Keywords

Comments

Here 0 is regarded as not divisible by any number.
a(n) = 0 if 5 divides n or n contains a prime divisor > 9. - Sascha Kurz, Aug 23 2002 [Corrected by Sean A. Irvine, Dec 23 2024]

Crossrefs

Cf. A073906, A085124, A073908, A073909, A073910, A073911.

Programs

  • Maple
    f := 8:for i from 1 to 400 do b := ifactors(f*i)[2]: if b[nops(b)][1]>9 or (f*i mod 10) =0 then a[i] := 0:else j := 0:while true do j := j+f*i:c := convert(j,base,10): d := product(c[k],k=1..nops(c)): if (d mod f*i)=0 and d>0 then a[i] := j:break:fi: od:fi:od:seq(a[k],k=1..400);

Formula

a(n) = A085124(8*n). - R. J. Mathar, Jun 21 2018

Extensions

More terms from Sascha Kurz, Aug 23 2002

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

Views

Author

Scott R. Shannon, May 11 2020

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

Cf. A334679, A334803, A007954, A049102, A085124.
Subsequence of A052382.

Programs

  • Mathematica
    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 *)
  • PARI
    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

Views

Author

Scott R. Shannon, May 08 2020

Keywords

Examples

			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.

Links

Crossrefs

Cf. A330880, A334803, A007954, A049102, A085124.
Subsequence of A052382.

Programs

  • Mathematica
    Select[Range[10^6], (p = Times @@ IntegerDigits@ #; p > 0 && Mod[# p, # + p] == 0) &] (* Giovanni Resta, May 08 2020 *)
  • PARI
    isok(m) = my(p=vecprod(digits(m))); p && !((m*p) % (m+p)); \\ Michel Marcus, May 08 2020

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

Views

Author

Scott R. Shannon, May 12 2020

Keywords

Comments

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

Examples

			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.

Crossrefs

Cf. A007954, A049102, A085124.
Subsequence of A052382. Intersection of A334679 and A330880.

Programs

  • PARI
    isok(m) = my(p=vecprod(digits(m))); p && (m-p) && !((m*p) % (m-p)) && !((m*p) % (m+p)); \\ Michel Marcus, May 12 2020
Showing 1-9 of 9 results.

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