A297330 -id:A297330 - cp's OEIS frontend

A354212 Numbers k such that A297330(k)*k and k have the same digits but in a different order.

11688, 116688, 126888, 1166688, 1266888, 11666688, 12446778, 12666888, 116666688, 123456789, 124466778, 126666888

Offset: 1

J. M. Bergot and Robert Israel, May 19 2022

Contains all numbers of the forms 116...688, 12446...6778, and 126...6888 (with at least one 6).

All terms are divisible by 3.

			a(1) = 11688 is a term because A297330(11688) = 7 and 7*11688 = 81816 has the same digits as 11688 in a different order.

Cf. A297330.

filter:= proc(n) local L,m;
  L:= convert(n,base,10);
  m:= convert(map(abs,L[2..-1]-L[1..-2]),`+`);
  if m = 1 then return false fi;
  sort(L) = sort(convert(m*n,base,10))
end proc:
select(filter, [seq(i,i=3..10^7,3)]);

f(n) = my(d=digits(n)); sum(i=2, #d, if (d[i]d[i-1], d[i]-d[i-1])); \\ A297330
isok(k) = my(d=digits(k), dd = digits(k*f(k))); (d != dd) && vecsort(d) == vecsort(dd); \\ Michel Marcus, May 19 2022

from itertools import count, islice
def A354212_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        s = str(n)
        t = str(n*sum(abs(int(s[i])-int(s[i+1])) for i in range(len(s)-1)))
        if s != t and sorted(s) == sorted(t):
            yield n
A354212_list = list(islice(A354212_gen(),5)) # Chai Wah Wu, May 31 2022

A033264 Number of blocks of {1,0} in the binary expansion of n.

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 0, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 0, 1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 3, 2, 2, 2, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 0, 1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 3, 2, 2, 2, 2, 1, 2, 2, 3, 2, 3, 3, 3, 2, 2, 2, 3, 2, 2, 2, 2, 1, 1, 1, 2, 1, 2, 2, 2

Offset: 1

Clark Kimberling

Number of i such that d(i) < d(i-1), where Sum_{d(i)*2^i: i=0,1,....,m} is base 2 representation of n.

This is the base-2 down-variation sequence; see A297330. - Clark Kimberling, Jan 18 2017

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Jean-Paul Allouche and Jeffrey Shallit, Sums of digits and the Hurwitz zeta function, in: K. Nagasaka and E. Fouvry (eds.), Analytic Number Theory, Lecture Notes in Mathematics, Vol. 1434, Springer, Berlin, Heidelberg, 1990, pp. 19-30.
Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple ordinary generating functions, 2004.
Ralf Stephan, Table of generating functions.
Eric Weisstein's World of Mathematics, Digit Block.
Index entries for sequences related to binary expansion of n

Cf. A014081, A014082, A037800, A056974, A056975, A056976, A056977, A056978, A056979, A056980, A196521.
a(n) = A005811(n) - ceiling(A005811(n)/2) = A005811(n) - A069010(n).
Equals (A072219(n+1)-1)/2.
Cf. also A175047, A030308.
Essentially the same as A087116.

a033264 = f 0 . a030308_row where
   f c [] = c
   f c (0 : 1 : bs) = f (c + 1) bs
   f c (_ : bs) = f c bs
-- Reinhard Zumkeller, Feb 20 2014, Jun 17 2012

f:= proc(n) option remember; local k;
k:= n mod 4;
if k = 2 then procname((n-2)/4) + 1
elif k = 3 then procname((n-3)/4)
else procname((n-k)/2)
fi
end proc:
f(1):= 0: f(0):= q:
seq(f(i),i=1..100); # Robert Israel, Aug 31 2015

Table[Count[Partition[IntegerDigits[n, 2], 2, 1], {1, 0}], {n, 102}] (* Michael De Vlieger, Aug 31 2015, after Robert G. Wilson v at A014081 *)
Table[SequenceCount[IntegerDigits[n,2],{1,0}],{n,110}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 26 2017 *)

a(n) = { hammingweight(bitand(n>>1, bitneg(n))) }; \\ Gheorghe Coserea, Aug 30 2015

def A033264(n): return ((n>>1)&~n).bit_count() # Chai Wah Wu, Jun 25 2025

G.f.: 1/(1-x) * Sum_(k>=0, t^2/(1+t)/(1+t^2), t=x^2^k). - Ralf Stephan, Sep 10 2003
a(n) = A069010(n) - (n mod 2). - Ralf Stephan, Sep 10 2003
a(4n) = a(4n+1) = a(2n), a(4n+2) = a(n)+1, a(4n+3) = a(n). - Ralf Stephan, Aug 20 2003
a(n) = A087116(n) for n > 0, since strings of 0's alternate with strings of 1's, which end in (1,0). - Jonathan Sondow, Jan 17 2016
Sum_{n>=1} a(n)/(n*(n+1)) = Pi/4 - log(2)/2 (A196521) (Allouche and Shallit, 1990). - Amiram Eldar, Jun 01 2021

A037800 Number of occurrences of 01 in the binary expansion of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 2, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 2, 1, 1, 1, 2, 2, 2, 2, 3, 2, 2, 1, 2, 2, 2, 1, 2, 1, 1, 0, 1, 1, 1, 1, 2, 1

Offset: 0

Clark Kimberling

Number of i such that d(i)>d(i-1), where Sum{d(i)*2^i: i=0,1,...,m} is base 2 representation of n.

This is the base-2 up-variation sequence; see A297330. - Clark Kimberling, Jan 18 2017

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Jean-Paul Allouche and Jeffrey Shallit, Sums of digits and the Hurwitz zeta function, in: K. Nagasaka and E. Fouvry (eds.), Analytic Number Theory, Lecture Notes in Mathematics, Vol. 1434, Springer, Berlin, Heidelberg, 1990, pp. 19-30.
Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple ordinary generating functions, 2004.
Ralf Stephan, Table of generating functions.
Eric Weisstein's World of Mathematics, Digit Block.
Index entries for sequences related to binary expansion of n

Cf. A014081, A014082, A033264, A056974, A056975, A056976, A056977, A056978, A056979, A056980.

Cf. A030308, A231902.

a037800 = f 0 . a030308_row where
   f c [_]          = c
   f c (1 : 0 : bs) = f (c + 1) bs
   f c (_ : bs)     = f c bs
-- Reinhard Zumkeller, Feb 20 2014

Table[SequenceCount[IntegerDigits[n,2],{0,1}],{n,0,120}] (* Harvey P. Dale, Aug 10 2023 *)

a(n) = { if(n == 0, 0, -1 + hammingweight(bitnegimply(n, n>>1))) };  \\ Gheorghe Coserea, Aug 31 2015

a(2n) = a(n), a(2n+1) = a(n) + [n is even]. - Ralf Stephan, Aug 21 2003
G.f.: 1/(1-x) * Sum_{k>=0} t^5/(1+t)/(1+t^2) where t=x^2^k. - Ralf Stephan, Sep 10 2003
a(n) = A069010(n) - 1, n>0. - Ralf Stephan, Sep 10 2003
Sum_{n>=1} a(n)/(n*(n+1)) = log(2)/2 + Pi/4 - 1 = A231902 - 1 (Allouche and Shallit, 1990). - Amiram Eldar, Jun 01 2021

A297250 Numbers whose base-3 digits having equal up-variation and down-variation; see Comments.

Original entry on oeis.org

1, 2, 4, 8, 10, 13, 16, 20, 23, 26, 28, 31, 34, 37, 40, 43, 46, 49, 52, 56, 59, 62, 65, 68, 71, 74, 77, 80, 82, 85, 88, 91, 94, 97, 100, 103, 106, 109, 112, 115, 118, 121, 124, 127, 130, 133, 136, 139, 142, 145, 148, 151, 154, 157, 160, 164, 167, 170, 173

Offset: 1

Clark Kimberling, Jan 15 2018

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.

			173 in base-3:  2,0,1,0,2, having DV = 3, UV = 3, so that 173 is in the sequence.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A297249, A297251, A297330.

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 = 3; 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]   (* A297249 *)
Take[Flatten[Position[w, 0]], 120]    (* A297250 *)
Take[Flatten[Position[w, 1]], 120]    (* A297251 *)

A037851 a(n)=Sum{d(i-1)-d(i): d(i)

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

This is the base-10 up-variation sequence; see A297330.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A297330, A037860.

A037851 := proc(n)
    a := 0 ;
    dgs := convert(n,base,10);
    for i from 2 to nops(dgs) do
        if op(i,dgs)R. J. Mathar, Oct 19 2015

g[n_, b_] := Differences[IntegerDigits[n, b]]; b = 10; z = 120;
Table[-Total[Select[g[n, b], # < 0 &]], {n, 1, z}];  (*A037860*)
Table[Total[Select[g[n, b], # > 0 &]], {n, 1, z}];   (*A037851*)

Definition swapped with A037860. - R. J. Mathar, Oct 19 2015

Updated by Clark Kimberling, Jan 19 2018

A297271 Numbers whose base-10 digits have equal down-variation and up-variation; see Comments.

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, 474, 484

Offset: 1

Clark Kimberling, Jan 16 2018

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 after the zero from A002113 first at 1011, which is not a palindrome but has DV(1011,10) = UV(1011,10) =1. - R. J. Mathar, Jan 23 2018
Apart from 0, the initial terms coincide with those of A266140, but the two sequences are different. First disagreement: a(109) = 1001 and A266140(110) = 1111. - Georg Fischer, Oct 09 2018

			13601 in base-10:  1,3,6,0,1, having DV = 6, UV = 6, so that 13601 is in the sequence.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A002113, A266140, A297330, A297270, A297272.

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 *)

{k: A037851(k) = A037860(k)}. - R. J. Mathar, Sep 27 2021

A037846 a(n)=Sum{d(i-1)-d(i): d(i)

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 2, 3, 0, 0, 0, 1, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 0, 0, 1, 2, 3, 1, 1, 1, 2, 3, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 0, 1, 2, 3, 4, 0, 0, 1, 2, 3, 0, 0, 0, 1, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 0, 1, 2, 3, 4, 0, 0, 1, 2, 3, 0, 0, 0, 1, 2, 0

Offset: 1

Clark Kimberling

This is the base-5 up-variation sequence; see A297330. - Clark Kimberling, Jan 19 2017

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A297330.

A037846 := proc(n)
    a := 0 ;
    dgs := convert(n,base,5);
    for i from 2 to nops(dgs) do
        if op(i,dgs)R. J. Mathar, Oct 19 2015

g[n_, b_] := Differences[IntegerDigits[n, b]]; b = 5; z = 120;
Table[-Total[Select[g[n, b], # < 0 &]], {n, 1, z}];  (*A037855*)
Table[Total[Select[g[n, b], # > 0 &]], {n, 1, z}];   (*A037846*)

Definition swapped with A037855. - R. J. Mathar, Oct 19 2015

Updated by Clark Kimberling, Jan 19 2018

A037847 a(n)=Sum{d(i-1)-d(i): d(i)

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 0, 0, 0, 1, 2, 3, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 0, 0, 1, 2, 3, 4, 1, 1, 1, 2, 3, 4, 2, 2, 2, 2, 3, 4, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 0, 1, 2, 3, 4, 5, 0, 0, 1, 2, 3, 4, 0, 0, 0, 1, 2, 3, 1

Offset: 1

Clark Kimberling

This is the base-6 up-variation sequence; see A297330. - Clark Kimberling, Jan 18 2017

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A297330.

A037847 := proc(n)
    a := 0 ;
    dgs := convert(n,base,6);
    for i from 2 to nops(dgs) do
        if op(i,dgs)R. J. Mathar, Oct 19 2015

g[n_, b_] := Differences[IntegerDigits[n, b]]; b = 6; z = 120;
Table[-Total[Select[g[n, b], # < 0 &]], {n, 1, z}];  (*A037856*)
Table[Total[Select[g[n, b], # > 0 &]], {n, 1, z}];   (*A037847*)

Definition swapped with A037856. - R. J. Mathar, Oct 19 2015

Updated by Clark Kimberling, Jan 19 2018

A037857 Sum{d(i)-d(i-1): d(i)>d(i-1), i=1,...,m}, where Sum{d(i)*7^i: i=0,1,...,m} is base 7 representation of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 3, 2, 1, 0, 0, 0, 0, 4, 3, 2, 1, 0, 0, 0, 5, 4, 3, 2, 1, 0, 0, 6, 5, 4, 3, 2, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 3, 2, 1, 0, 0, 0, 0, 4, 3, 2, 1, 0, 0, 0, 5, 4, 3, 2, 1, 0, 0

Offset: 1

Clark Kimberling

This is the base-7 down-variation sequence; see A297330. - Clark Kimberling, Jan 18 2017

Clark Kimberling, Table of n, a(n) for n = 1..10000

A037857 := proc(n)
    a := 0 ;
    dgs := convert(n,base,7);
    for i from 2 to nops(dgs) do
        if op(i,dgs)>op(i-1,dgs) then
            a := a+op(i,dgs)-op(i-1,dgs) ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Oct 19 2015

g[n_, b_] := Differences[IntegerDigits[n, b]]; b = 7; z = 120;
Table[-Total[Select[g[n, b], # < 0 &]], {n, 1, z}];  (*A037857*)
Table[Total[Select[g[n, b], # > 0 &]], {n, 1, z}];   (*A037848*)

Definition swapped with A037848. - R. J. Mathar, Oct 19 2015

Updated by Clark Kimberling, Jan 19 2018

A037860 Sum{d(i)-d(i-1): d(i)>d(i-1), i=1,...,m}, where Sum{d(i)*10^i: i=0,1,...,m} is base 10 representation of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 3, 2, 1, 0, 0, 0, 0, 0, 0, 0, 4, 3, 2, 1, 0, 0, 0, 0, 0, 0, 5, 4, 3, 2, 1, 0, 0, 0, 0, 0, 6, 5, 4, 3, 2, 1, 0, 0, 0, 0, 7, 6, 5, 4, 3, 2, 1, 0, 0, 0, 8, 7, 6, 5, 4, 3, 2, 1, 0, 0, 9

Offset: 1

Clark Kimberling

This is the base-10 down-variation sequence; see A297330.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A297330, A037851.

A037860 := proc(n)
    a := 0 ;
    dgs := convert(n,base,10);
    for i from 2 to nops(dgs) do
        if op(i,dgs)>op(i-1,dgs) then
            a := a+op(i,dgs)-op(i-1,dgs) ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Oct 19 2015

g[n_, b_] := Differences[IntegerDigits[n, b]]; b = 10; z = 120;
Table[-Total[Select[g[n, b], # < 0 &]], {n, 1, z}];  (*A037860*)
Table[Total[Select[g[n, b], # > 0 &]], {n, 1, z}];   (*A037851*)

Definition swapped with A037851. - R. J. Mathar, Oct 19 2015

Updated by Clark Kimberling, Jan 19 2018

cp's OEIS Frontend

A354212 Numbers k such that A297330(k)*k and k have the same digits but in a different order.

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A033264 Number of blocks of {1,0} in the binary expansion of n.

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A037800 Number of occurrences of 01 in the binary expansion of n.

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A297250 Numbers whose base-3 digits having equal up-variation and down-variation; see Comments.

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A037851 a(n)=Sum{d(i-1)-d(i): d(i)

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A297271 Numbers whose base-10 digits have equal down-variation and up-variation; see Comments.

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A037846 a(n)=Sum{d(i-1)-d(i): d(i)

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