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 1-4 of 4 results.

A297330 Total variation of base-10 digits of n; see Comments.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 8, 7, 6, 5, 4, 3, 2
Offset: 1

Views

Author

Clark Kimberling, Jan 17 2018

Keywords

Comments

Suppose that a number n has base-b digits b(m), b(m-1), ..., b(0). The base-b down-variation of n is the sum DV(n,b) of all d(i)-d(i-1) for which d(i) > d(i-1); the base-b up-variation of n is the sum UV(n,b) of all d(k-1)-d(k) for which d(k) < d(k-1). The total base-b variation of n is the sum TV(n,b) = DV(n,b) + UV(n,b). Guide to related sequences and partitions of the natural numbers:
***
Base b {DV(n,b)} {UV(n,b)} {TV(n,b)}
2 A033264 A037800 A037834
3 A037853 A037844 A037835
4 A037854 A037845 A037836
5 A037855 A037846 A037837
6 A037856 A037847 A037838
7 A037857 A037848 A037839
8 A037858 A037849 A037840
9 A037859 A037850 A037841
10 A037860 A037851 A297330
11 A297231 A297232 A297233
12 A297234 A297235 A297236
13 A297237 A297238 A297239
14 A297240 A297241 A297242
15 A297243 A297244 A297245
16 A297246 A297247 A297247
For each b, let u = {n : UV(n,b) < DV(n,b)}, e = {n : UV(n,b) = DV(n,b)}, and d = {n : UV(n,b) > DV(n,b)}. The sets u,e,d partition the natural numbers. A guide to the matching sequences for u, e, d follows:
***
Base b Sequence u Sequence e Sequence d
2 A005843 A005408 (none)
3 A297249 A297250 A297251
4 A297252 A297253 A297254
5 A297255 A297256 A297257
6 A297258 A297259 A297260
7 A297261 A297262 A297263
8 A297264 A297265 A297266
9 A297267 A297268 A297269
10 A297270 A297271 A297272
11 A297273 A297274 A297275
12 A297276 A297277 A297278
13 A297279 A297280 A297281
14 A297282 A297283 A297284
15 A297285 A297286 A297287
16 A297288 A297289 A297290
Not a duplicate of A151950: e.g., a(100)=1 but A151950(100)=11. - Robert Israel, Feb 06 2018

Examples

			13684632 has DV = 8-4 + 6-3 + 3-2 = 8 and has UV = 3-1 + 6-3 + 8-6 + 6-4 = 9, so that a(13684632) = DV + UV = 17.

Links

Crossrefs

Cf. A037851, A297330, A297271, A297272.

Programs

  • Maple
    f:= proc(n) local L,i; L:= convert(n,base,10);
add(abs(L[i+1]-L[i]),i=1..nops(L)-1) end proc:
map(f, [$1..100]); # Robert Israel, Feb 04 2018
# alternative
A297330 := proc(n)
    A037860(n)+A037851(n) ;
end proc: # R. J. Mathar, Sep 27 2021
  • Mathematica
    b = 10; z = 120; t = Table[Total@Flatten@Map[Abs@Differences@# &, Partition[ IntegerDigits[n, b], 2, 1]], {n, z}] (* after Michael De Vlieger, e.g. A037834 *)
  • Python
    def A297330(n):
    s = str(n)
    return sum(abs(int(s[i])-int(s[i+1])) for i in range(len(s)-1)) # Chai Wah Wu, May 31 2022

A297270 Numbers whose base-10 digits have greater down-variation than up-variation; see Comments.

Original entry on oeis.org

10, 20, 21, 30, 31, 32, 40, 41, 42, 43, 50, 51, 52, 53, 54, 60, 61, 62, 63, 64, 65, 70, 71, 72, 73, 74, 75, 76, 80, 81, 82, 83, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 96, 97, 98, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200, 201, 210, 211, 220, 221
Offset: 1

Views

Author

Clark Kimberling, Jan 16 2018

Keywords

Comments

Suppose that n has base-b digits b(m), b(m-1), ..., b(0). The base-b down-variation of n is the sum DV(n,b) of all d(i)-d(i-1) for which d(i) > d(i-1); the base-b up-variation of n is the sum UV(n,b) of all d(k-1)-d(k) for which d(k) < d(k-1). The total base-b variation of n is the sum TV(n,b) = DV(n,b) + UV(n,b). See the guide at A297330.
Differs first from A071590 at 1101, which is in A071590, but not in here because UV(1101) = DV(1101). - R. J. Mathar, Jan 23 2018

Examples

			6151413121 in base-10:  6,1,5,1,4,1,3,1,2,1, having DV = 15, UV = 10, so that 6151413121 is in the sequence.

Links

Crossrefs

Cf. A297330, A297271, A297272.

Programs

  • Mathematica
    g[n_, b_] := Map[Total, GatherBy[Differences[IntegerDigits[n, b]], Sign]];
x[n_, b_] := Select[g[n, b], # < 0 &]; y[n_, b_] := Select[g[n, b], # > 0 &];
b = 10; z = 2000; p = Table[x[n, b], {n, 1, z}]; q = Table[y[n, b], {n, 1, z}];
w = Sign[Flatten[p /. {} -> {0}] + Flatten[q /. {} -> {0}]];
Take[Flatten[Position[w, -1]], 120]   (* A297270 *)
Take[Flatten[Position[w, 0]], 120]    (* A297271 *)
Take[Flatten[Position[w, 1]], 120]    (* A297272 *)

A297272 Numbers whose base-10 digits have greater up-variation than down-variation; see Comments.

Original entry on oeis.org

12, 13, 14, 15, 16, 17, 18, 19, 23, 24, 25, 26, 27, 28, 29, 34, 35, 36, 37, 38, 39, 45, 46, 47, 48, 49, 56, 57, 58, 59, 67, 68, 69, 78, 79, 89, 102, 103, 104, 105, 106, 107, 108, 109, 112, 113, 114, 115, 116, 117, 118, 119, 122, 123, 124, 125, 126, 127, 128
Offset: 1

Views

Author

Clark Kimberling, Jan 16 2018

Keywords

Comments

Suppose that n has base-b digits b(m), b(m-1), ..., b(0). The base-b down-variation of n is the sum DV(n,b) of all d(i)-d(i-1) for which d(i) > d(i-1); the base-b up-variation of n is the sum UV(n,b) of all d(k-1)-d(k) for which d(k) < d(k-1). The total base-b variation of n is the sum TV(n,b) = DV(n,b) + UV(n,b). See the guide at A297330.
Differs from A071589 first at 1011 which is in A071589 but not in here because UV(1011) = DV(1011)=1. - R. J. Mathar, Jan 23 2018

Examples

			198765 in base-10:  1,9,8,7,6,5, having DV = 4, UV = 8, so that 198765 is in the sequence.

Links

Crossrefs

Cf. A297330, A297270, A297271.

Programs

  • Mathematica
    g[n_, b_] := Map[Total, GatherBy[Differences[IntegerDigits[n, b]], Sign]];
x[n_, b_] := Select[g[n, b], # < 0 &]; y[n_, b_] := Select[g[n, b], # > 0 &];
b = 10; z = 2000; p = Table[x[n, b], {n, 1, z}]; q = Table[y[n, b], {n, 1, z}];
w = Sign[Flatten[p /. {} -> {0}] + Flatten[q /. {} -> {0}]];
Take[Flatten[Position[w, -1]], 120]   (* A297270 *)
Take[Flatten[Position[w, 0]], 120]    (* A297271 *)
Take[Flatten[Position[w, 1]], 120]    (* A297272 *)

A342826 Numbers k such that d(1)^0 + d(2)^1 + ... + d(p)^(p-1) = d(1)^(p-1) + d(2)^(p-2) + ... + d(p)^0, where d(i), i=1..p, are the digits of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212, 222, 232, 242, 252, 262, 272, 282, 292, 303, 313, 323, 333, 343, 353, 363, 373, 383, 393, 404, 414, 424, 434, 444, 454, 464
Offset: 1

Views

Author

Carole Dubois, Mar 23 2021

Keywords

Comments

This sequence starts off like other palindromic sequences such as A178354, A002113, A110751, and A227858 but it differs starting at the 110th term: 109th: 1001, 110th: 1011, 111th: 1101, ..., 119th: 1863, etc.
Differs from A297271 (which e.g. has 1021, 1031, 1041,.. 1091 which are absent here). - R. J. Mathar, Sep 27 2021
Contains the palindromes, and palindromes where pairs of digits have been substituted by d(i)=0, d(p-i)=1 or d(i)=1, d(p-1)=0, and "genuine" numbers like 1863, 2450, 2804, 2814, 3681, 4081, 4182, 103221, 113221, 122301, 122311, 142023,.. - R. J. Mathar, Sep 27 2021

Examples

			1863 is in this sequence because 1^0 + 8^1 + 6^2 + 3^3 = 1^3 + 8^2 + 6^1 + 3^0 = 72.

Links

Crossrefs

Cf. A002113 (subset), A179309, A110751, A227858.

Programs

  • Maple
    isA342826 := proc(n)
    local dgs ;
    dgs := convert(n,base,10) ;
    if  add(op(i,dgs)^(i-1),i=1..nops(dgs)) = add(op(-i,dgs)^(i-1),i=1..nops(dgs)) then
        true;
    else
        false;
    end if;
end proc:
A342826 := proc(n)
    option remember ;
    if n =1 then
        1;
    else
        for a from procname(n-1)+ 1 do
            if isA342826(a) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Sep 27 2021
  • Mathematica
    Select[Range[500],Mod[#,10]!=0&&Total[IntegerDigits[#]^Range[0,IntegerLength[ #]-1]]==Total[IntegerDigits[#]^Range[IntegerLength[#]-1,0,-1]]&] (* Harvey P. Dale, Jan 18 2023 *)
  • Python
    def digpow(s): return sum(int(d)**i for i, d in enumerate(s))
def aupto(limit):
  alst = []
  for k in range(1, limit+1):
    s = str(k)
    if digpow(s) == digpow(s[::-1]): alst.append(k)
  return alst
print(aupto(464)) # Michael S. Branicky, Mar 23 2021
Showing 1-4 of 4 results.

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