A132359 -id:A132359 - cp's OEIS frontend

A221651 Numbers divisible by their first digit squared (excluding those whose first digit is 1).

20, 24, 28, 36, 48, 50, 200, 204, 208, 212, 216, 220, 224, 228, 232, 236, 240, 244, 248, 252, 256, 260, 264, 268, 272, 276, 280, 284, 288, 292, 296, 306, 315, 324, 333, 342, 351, 360, 369, 378, 387, 396, 400, 416, 432, 448, 464, 480, 496, 500, 525, 550, 575

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, May 04 2013

Numbers where floor(n/10^floor(log(n)))^2 divides n.

			48 is divisible by 4^2.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000
Christian N. K. Anderson, Ulam Spiral for the first 10000 terms. [Draw the usual Ulam spiral, containing all the nonnegative integers, and color the numbers belonging to this sequence yellow]

Cf. A132359, A034709, A034837, A005349, A007602, A034838, A225297, A225722, A225683.

x=0; y=rep(0,1000); len=0
firstdig<-function(x) as.numeric(substr(as.character(x),1,1))
isint<-function(x) x==as.integer(x)
while(len<10000) if((fd=firstdig((x=x+1)))>1) if(isint(x/fd^2)) y[(len=len+1)]=x

A225297 Numbers divisible by their last digit cubed.

Original entry on oeis.org

1, 11, 21, 31, 32, 41, 51, 61, 64, 71, 72, 81, 91, 101, 111, 112, 121, 125, 131, 141, 151, 152, 161, 171, 181, 191, 192, 201, 211, 216, 221, 231, 232, 241, 243, 251, 261, 271, 272, 281, 291, 301, 311, 312, 321, 331, 341, 351, 352, 361, 371, 375, 381, 384, 391

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, May 04 2013

Numbers k such that (k mod 10)^3 | k.
All numbers ending in 1 are trivially included in this sequence.
The sequence is { 1+10k, 32 + 40k, 243 + 270k, 64 + 320k, 125 + 250k, 216 + 1080k, 3087 + 3430k, 2048 + 2560k, 729 + 7290k ; k = 0,1,2,...}. - M. F. Hasler, Jan 31 2016
The asymptotic density of this sequence is 2201597407/16003008000 = 0.137573... . - Amiram Eldar, Aug 08 2023

			a(16) = 112 is divisible by 2^3.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000

Cf. A132359, A034709, A034837, A005349, A007602, A034838.

Select[Range[400], (m = Mod[#, 10]) > 0 && Divisible[#, m^3] &] (* Amiram Eldar, Aug 08 2023 *)

is(n)=n%10&&n%(n%10)^3==0 \\ M. F. Hasler, Jan 31 2016

x=0; y=rep(0,100); len=0; isint<-function(x) x==as.integer(x); while(len<100) if((x=x+1)%%10>0) if(isint(x/(x%%10)^3)) y[(len=len+1)]=x

A225296 Numbers divisible by their first digit cubed (excluding those whose first digit is 1).

Original entry on oeis.org

24, 200, 208, 216, 224, 232, 240, 248, 256, 264, 272, 280, 288, 296, 324, 351, 378, 448, 500, 648, 2000, 2008, 2016, 2024, 2032, 2040, 2048, 2056, 2064, 2072, 2080, 2088, 2096, 2104, 2112, 2120, 2128, 2136, 2144, 2152, 2160, 2168, 2176, 2184, 2192, 2200, 2208

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, May 04 2013

Numbers where floor(n/10^floor(log(n)))^3 | n.

			448 is divisible by 4^3.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000
Christian N. K. Anderson, Ulam Spiral of first 10000 terms.

Cf. A132359, A034709, A034837, A005349, A007602, A034838.

dfdcQ[n_]:=Module[{fidn=IntegerDigits[n][[1]]},fidn!=1&&Divisible[ n,fidn^3]]; Select[Range[2500],dfdcQ] (* Harvey P. Dale, Jan 13 2019 *)

x=0; y=rep(0,1000); len=0
firstdig<-function(x) as.numeric(substr(as.character(x),1,1))
isint<-function(x) x==as.integer(x)
while(len<1000) if((fd=firstdig((x=x+1)))>1) if(isint(x/fd^3)) y[(len=len+1)]=x

A225722 Numbers divisible by their last digit cubed, excluding those whose last digit is 1.

Original entry on oeis.org

32, 64, 72, 112, 125, 152, 192, 216, 232, 243, 272, 312, 352, 375, 384, 392, 432, 472, 512, 513, 552, 592, 625, 632, 672, 704, 712, 729, 752, 783, 792, 832, 872, 875, 912, 952, 992, 1024, 1032, 1053, 1072, 1112, 1125, 1152, 1192, 1232, 1272, 1296, 1312, 1323

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, May 13 2013

a(n) ~ n. For 69 < n < 10000, the formula 26.61*n - 2.76 provides an estimate of a(n) to within 1%.

The asymptotic density of this sequence is 601296607/16003008000 = 0.037573... . Therefore, contrary to the above comment, a(n) ~ c*n where c = 16003008000/601296607 = 26.614166... . - Amiram Eldar, Aug 08 2023

			a(5) = 125 is an example because its last digit is 5, and 5^3 = 125, and 125 is divisible by 125.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000

Subsequence of A225297.

Cf. A221651, A225296, A132359, A034709, A034837, A005349, A007602, A034838.

dldcQ[n_]:=Module[{ld=Last[IntegerDigits[n]]},ld>1&&Divisible[n,ld^3]]; Select[Range[1500],dldcQ] (* Harvey P. Dale, Aug 15 2014 *)

which(sapply(1:1000,function(x) x%%10>1 & (v=x/(x%%10)^3)==as.integer(v) ))

A225299 Numbers divisible by the square of each digit.

Original entry on oeis.org

1, 11, 12, 36, 111, 112, 128, 144, 212, 216, 224, 333, 432, 448, 612, 1111, 1112, 1116, 1212, 1296, 1332, 1424, 2112, 2144, 2212, 2224, 2232, 2916, 3132, 3312, 3636, 4112, 4144, 4224, 4288, 4464, 6336, 6624, 8128, 8448, 9396, 11111, 11112, 11133, 11172, 11212

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, May 04 2013

Includes all repunits.

			a(7) 128 is divisible by 1^2, by 2^2, and by 8^2.

Christian N. K. Anderson, Table of n, a(n) for n = 1..5000

Cf. A132359, A034709, A034837, A005349, A007602, A034838.

d[n_]:=IntegerDigits[n]; t={}; Do[If[!MemberQ[d[n],0] && Union[Mod[n,d[n]^2]] == {0}, AppendTo[t,n]], {n,11220}]; t (* Jayanta Basu, May 15 2013 *)
Select[Range[12000],DigitCount[#,10,0]==0&&And@@Divisible[ #,IntegerDigits[ #]^2]&] (* Harvey P. Dale, Jul 16 2018 *)

isint<-function(x) x==as.integer(x)
sqalldig<-function(x) as.numeric(strsplit(as.character(x),"")[[1]])^2
divby<-function(x) ifelse(length(grep(0,x))>0,F,all(isint(x/sqalldig(x))))
which(sapply(1:1000,divby))

A237275 Smallest k divisible by the n-th power of its last decimal digit > 1.

Original entry on oeis.org

2, 2, 12, 32, 32, 32, 192, 512, 512, 512, 3072, 8192, 8192, 8192, 49152, 131072, 131072, 131072, 786432, 2097152, 2097152, 2097152, 12582912, 33554432, 33554432, 33554432

Offset: 0

Michel Lagneau, Apr 22 2014

Conjecture: a(n) == 2 (mod 10).

			a(0) = 2 because 2 is divisible by 2^0 = 1.
a(1) = 2 because 2 is divisible by 2^1 = 2.
a(2) = 12 because 12 is divisible by 2^2 = 4.

Cf. A132359.

Do[k=1;While[!Total[Transpose[IntegerDigits[k][[-1]]>0&&Mod[k,IntegerDigits[k][[-1]]^n]==0&&!Mod[k,10]==1],k++]];Print[n," ",k-1],{n,0,25}]

a(n) = 3*2^n if n mod 4 = 2; 2^(n+2-((n+1) mod 4)) otherwise. - Jon E. Schoenfield, Sep 12 2017

cp's OEIS Frontend

A221651 Numbers divisible by their first digit squared (excluding those whose first digit is 1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

R

A225297 Numbers divisible by their last digit cubed.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

R

A225296 Numbers divisible by their first digit cubed (excluding those whose first digit is 1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

R

A225722 Numbers divisible by their last digit cubed, excluding those whose last digit is 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

R

A225299 Numbers divisible by the square of each digit.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

R

A237275 Smallest k divisible by the n-th power of its last decimal digit > 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula