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A387423 The length of binary expansion of n minus the length of the maximal common prefix of the binary expansions of n and sigma(n), where sigma is the sum of divisors function.

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

Offset: 1

Antti Karttunen, Sep 01 2025

Positions of 0's in this sequence is given by such numbers n that sigma(n) = 2^k * n + r, for some n >= 1, k >= 0, 0 <= r < 2^k. These would include also quasi-perfect numbers and their generalizations, numbers n such that sigma(n) = 2^k * n + 2^k - 1, for some n > 1, k > 0 (see comments in A332223), if such numbers exist. However, it is conjectured that there are no other zeros than those given by A336702.

Cf. A000203, A000523, A332223, A336700, A336701, A336702 (conjectured positions of 0's), A387422.

Cf. also A347381, A387413.

A387423[n_] := BitLength[n] - LengthWhile[Transpose[IntegerDigits[{n, DivisorSigma[1, n]}, 2][[All, ;; BitLength[n]]]], Equal @@ # &];
Array[A387423, 100] (* Paolo Xausa, Sep 03 2025 *)

A387423(n) = { my(a=binary(n), b=binary(sigma(n)), i=1); while(i<=#a,if(a[i]!=b[i],return(#a-(i-1))); i++); (0); };

from os.path import commonprefix
from sympy import divisor_sigma
def A387423(n): return n.bit_length()-len(commonprefix([bin(n)[2:],bin(divisor_sigma(n))[2:]])) # Chai Wah Wu, Sep 03 2025

a(n) = (1+A000523(n)) - A387422(n).

A387480 a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(k,n-k)^2.

Original entry on oeis.org

1, 3, 15, 99, 603, 3807, 24759, 162243, 1072683, 7147359, 47887767, 322330995, 2178055899, 14765637663, 100380161655, 684061007139, 4671543976587, 31962145170015, 219043736154711, 1503380943222867, 10332034575214779, 71092843087100319, 489712662842798007

Offset: 0

Seiichi Manyama, Aug 30 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A387481, A387482.

Cf. A083858, A108490.

[&+[3^k * 2^(n-k) * Binomial(k, n-k)^2: k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 01 2025

Table[Sum[3^k*2^(n-k)*Binomial[k,n-k]^2,{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Sep 01 2025 *)

a(n) = sum(k=0, n, 3^k*2^(n-k)*binomial(k, n-k)^2);

G.f.: 1/sqrt((1-3*x-6*x^2)^2 - 72*x^3).

A387481 a(n) = Sum_{k=0..floor(n/2)} 3^k * 2^(n-2k) binomial(k,n-2*k)^2.

Original entry on oeis.org

1, 0, 3, 6, 9, 72, 63, 486, 1053, 2808, 11907, 22518, 99225, 246888, 755487, 2554902, 6488829, 23112216, 63506835, 198653958, 623336553, 1781565192, 5807475711, 16898655942, 52699192029, 161995971384, 484990399395, 1525112887446, 4572778238649, 14184781485480, 43472894580063

Offset: 0

Seiichi Manyama, Aug 30 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A387480, A387482.

[(&+[3^k * 2^(n-2*k)* Binomial(k,n-2*k)^2: k in [0..Floor(n/2)]]): n in [0..40]]; // Vincenzo Librandi, Sep 01 2025

Table[Sum[3^k * 2^(n-2*k)*Binomial[k,n-2*k]^2,{k,0,Floor[n/2]}],{n,0,40}] (* Vincenzo Librandi, Sep 01 2025 *)

a(n) = sum(k=0, n\2, 3^k*2^(n-2*k)*binomial(k, n-2*k)^2);

G.f.: 1/sqrt((1-3*x^2-6*x^3)^2 - 72*x^5).

A387484 a(n) = Sum_{k=0..floor(n/3)} 2^(n-k) * binomial(k,n-3*k)^2.

Original entry on oeis.org

1, 0, 0, 4, 8, 0, 16, 128, 64, 64, 1152, 2304, 768, 8192, 36864, 33792, 55296, 409600, 823296, 704512, 3719168, 13123584, 16351232, 33619968, 160890880, 329515008, 436731904, 1695809536, 5182586880, 7935623168, 18086887424, 67335356416, 141687783424

Offset: 0

Seiichi Manyama, Aug 30 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..2000

Cf. A387483, A387485.

[(&+[2^(n-k)* Binomial(k,n-3*k)^2: k in [0..Floor(n/3)]]): n in [0..40]]; // Vincenzo Librandi, Sep 01 2025

Table[Sum[2^(n-k)*Binomial[k,n-3*k]^2,{k,0,Floor[n/3]}],{n,0,40}] (* Vincenzo Librandi, Sep 01 2025 *)

a(n) = sum(k=0, n\3, 2^(n-k)*binomial(k, n-3*k)^2);

G.f.: 1/sqrt((1-4*x^3-8*x^4)^2 - 128*x^7).

A387485 a(n) = Sum_{k=0..floor(n/3)} 2^(n-2k) binomial(k,n-3*k)^2.

Original entry on oeis.org

1, 0, 0, 2, 4, 0, 4, 32, 16, 8, 144, 288, 80, 512, 2304, 2080, 1856, 12800, 25664, 17408, 58624, 204928, 242944, 299520, 1258752, 2541568, 2609152, 6824448, 20169728, 28344320, 41747456, 132358144, 268472320, 349177856, 807964672, 2116296704, 3336458240

Offset: 0

Seiichi Manyama, Aug 30 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..2000

Cf. A387483, A387484.

[(&+[2^(n-2*k)* Binomial(k,n-3*k)^2: k in [0..Floor(n/3)]]): n in [0..40]]; // Vincenzo Librandi, Sep 01 2025

Table[Sum[2^(n-2*k)*Binomial[k,n-3*k]^2,{k,0,Floor[n/3]}],{n,0,40}] (* Vincenzo Librandi, Sep 01 2025 *)

a(n) = sum(k=0, n\3, 2^(n-2*k)*binomial(k, n-3*k)^2);

G.f.: 1/sqrt((1-2*x^3-4*x^4)^2 - 32*x^7).

A379049 a(n) = prime(i)dp(n,i) + prime(i)dn(n,i) where dp(n,i) = 1 when the i-th trit of n is 1, dn(n,i) = 1 when the i-th trit of n is T, and dp(n,i) = dn(n,i) = 0 when the i-th trit of n is 0.

Original entry on oeis.org

2, 3, 5, 4, 7, 11, 8, 13, 7, 6, 11, 17, 16, 31, 37, 22, 29, 17, 12, 19, 31, 26, 47, 13, 10, 17, 9, 8, 15, 23, 22, 43, 41, 38, 73, 37, 36, 71, 107, 106, 211, 221, 116, 127, 81, 46, 57, 103, 68, 101, 53, 32, 43, 25, 18, 29, 47, 40, 73, 97, 76, 131, 69, 62, 117

Offset: 0

Lei Zhou, Dec 14 2024

The Balanced Ternary presentation of a number is a series of 1, 0, and T, where T represent -1. For example, 35 = 110T = 1 * 3^3 + 1* 3^2 + 0 * 3 - 1 = 27 + 9 + 0 - 1.

Conjecture: All positive integers greater than 1 appear in this sequence at least once.

			When n = 0, its BT presentation is 0, thus a(0) = 1 + 1 = 2;
When n = 1, its BT presentation is 1, the first prime is 2, thus a(1) = 2 + 1 = 3;
...
When n = 14, its BT presentation is 1TTT, thus prime 7 appears before the plus sign and primes 5, 3, and 2 appear in the term after the plus sign, a(14) = 7 + 5*3*2 = 37;
...
By the same rule, when n = 64, its BT presentation is 1T101, thus prime 11, 5, 2 appear before the plus sign and prime 7 appears in the term after the plus sign, a(64) = 11*5*2 + 7 = 117.

John Tyler Rascoe, Table of n, a(n) for n = 0..10000

Cf. A005812, A134022, A134023, A140267.

BTDigits[m_Integer, g_]:= Module[{n = m, d, sign, t = g}, If[n != 0, If[n > 0, sign = 1, sign = -1; n = -n]; d = Ceiling[Log[3, n]]; If[3^d - n <= ((3^d - 1)/2), d++]; While[Length[t] < d, PrependTo[t, 0]]; t[[Length[t] + 1 - d]] = sign; t = BTDigits[sign*(n - 3^(d - 1)), t]]; t];
res = {}; Do[BT = BTDigits[i, {0}]; BTl = Length[BT]; f = 1; b = 1; Do[If[BT[[j]] == 1, f = f*Prime[BTl - j + 1]]; If[BT[[j]] == -1, b = b*Prime[BTl - j + 1]], {j, 1, BTl}];  d = f + b; AppendTo[res, d], {i, 0, 64}]; res

from sympy import prime
def A140267(n): # see A140267
    return
def A379049(n):
    x,y,z = 1,1,str(A140267(n))[::-1]
    for i in range(len(z)):
        if z[i] == "1":
            x *= prime(i+1)
        if z[i] == "2":
            y *= prime(i+1)
    return x+y # John Tyler Rascoe, Feb 27 2025

A383904 a(n) is the number of complement pairs of primitive 2n-bead balanced binary necklaces.

Original entry on oeis.org

0, 0, 0, 1, 3, 11, 35, 118, 392, 1336, 4587, 15986, 56231, 199854, 716014, 2584742, 9390656, 34315811, 126039218, 465062362, 1723066193, 6407806833, 23910159818, 89493721076, 335912335304, 1264105728831, 4768446686910, 18027215660947, 68291877609003

Offset: 0

Tilman Piesk, Aug 07 2025

A022553(n) is the number of primitive 2n-bead balanced binary necklaces (corresponding to Lyndon words), and A000048 is the number of those that are self-complementary (i.e., can be rotated so that all beads change color). Their difference 2*a(n) is the number of those that are not self-complementary. a(n) is the number pairs of distinct complements.
Doubled entries: 0, 0, 0, 2, 6, 22, 70, 236, 784, 2672, 9174, 31972, 112462, 399708, 1432028, ...
Sequences counting 2n-bead balanced binary necklaces:
                       primitive  imprimitive
                     +-----------------------+---------+
  self-complementary |  A000048     A115118  | A000013 |
   complement pairs  |   this       A387130  | A386388 |
                     +-----------------------+---------+
                     |  A022553     A386946  | A003239 |
                     +-----------------------+---------+

			  n | A022553(n) A000048(n) | 2*a(n)    a(n)
  0 |         1          1  |     0       0
  1 |         1          1  |     0       0
  2 |         1          1  |     0       0
  3 |         3          1  |     2       1
  4 |         8          2  |     6       3
  5 |        25          3  |    22      11
  6 |        75          5  |    70      35
  7 |       245          9  |   236     118
  8 |       800         16  |   784     392
  9 |      2700         28  |  2672    1336
 10 |      9225         51  |  9174    4587
Examples for n=5 with necklaces of length 10:
The total number of necklaces is A003239(5) = 26.
Only A386946(5) = 1 of them is periodic, namely 0101010101.
The other A022553(5) = 25 are primitive.
A000048(5) = 3 among those are self-complementary:
 0000011111
 0001011101
 0010011011
The remaining 22 necklaces form a(5) = 11 complement pairs:
 0000101111 0000111101
 0000110111 0001111001
 0000111011 0001001111
 0001010111 0001110101
 0001011011 0010011101
 0001100111 0001110011
 0001101011 0010100111
 0001101101 0010010111
 0010101011 0011010101
 0010101101 0010110101
 0010110011 0011001101

Tilman Piesk, Table of n, a(n) for n = 0..1000

a(n) = (A022553(n) - A000048(n)) / 2.

A385476 Values of u in the quartets (1, u, v, w) of type 3; i.e., values of u for solutions to (1 - u) = v(v - w), in distinct positive integers, with v > 1, sorted by nondecreasing values of u; see Comments.

Original entry on oeis.org

5, 7, 7, 9, 9, 10, 11, 11, 13, 13, 13, 13, 15, 15, 16, 16, 17, 17, 17, 19, 19, 19, 19, 21, 21, 21, 21, 22, 22, 23, 23, 25, 25, 25, 25, 25, 25, 26, 27, 27, 28, 28, 29, 29, 29, 29, 31, 31, 31, 31, 31, 31, 33, 33, 33, 33, 34, 34, 35, 35, 36, 36, 37, 37, 37, 37

Offset: 1

Clark Kimberling, Aug 16 2025

A 4-tuple (m, u, v, w) is a quartet of type 3 if m, u, v, w are distinct positive integers such that m < v and m*(m - u) = v*(v - w). Here, the values of u are arranged in nondecreasing order. When there is more than one solution for given m and u, the values of v are arranged in increasing order. Here, m = 1.

			First 20 quartets (1,u,v,w) of type 3:
   m    u    v    w
   1    5    2    4
   1    7    2    5
   1    7    3    5
   1    9    2    6
   1    9    4    6
   1   10    3    6
   1   11    2    7
   1   11    5    7
   1   13    2    8
   1   13    3    7
   1   13    4    7
   1   13    6    8
   1   15    2    9
   1   15    7    9
   1   16    3    5
   1   16    3    8
   1   17    2   10
   1   17    4    8
   1   17    8   10
   1   19    2   11
1(1-11) = 5(5-7), so (1, 11, 5, 7) is in the list.

Cf. A385182 (type 1), A386218 (type 2), A386631, A385246.

solnsM[m_Integer?Positive, u_Integer?Positive] :=
  Module[{n = m  (m - u), nn, sgn, ds, tups}, If[n == 0, Return[{}]];
   sgn = Sign[n]; nn = Abs[n];
   ds = Divisors[nn];
   If[sgn > 0, ds = Select[ds, # < nn/# &]];
   tups = ({m, u, nn/#, nn/# - sgn  #} & /@ ds);
   Select[tups, #[[3]] > 1 && #[[4]] > 0 && #[[2]] =!= #[[4]](*&&
     Length@DeleteDuplicates[#]==4*)&]];
(solns =
   Sort[Flatten[Map[solnsM[1, #] &, Range[2, 30]], 1]]) // ColumnForm
Map[#[[2]] &, solns] (*A385476*)
Map[#[[3]] &, solns] (*A163870*)
Map[#[[4]] &, solns] (*A385246*)
(* Peter J. C. Moses, Aug 22 2025 *)

A385852 Integers x such that there exist two integers 0

Original entry on oeis.org

79170, 150150, 158340, 161070, 232050, 237510, 300300, 316680, 322140, 395850, 450450, 464100, 468930, 474810, 475020, 483210, 554190, 570570, 600600, 622440, 633360, 641550, 644280, 696150, 712530, 750750, 791700, 805350, 937860, 949620, 950040, 963270, 966420

Offset: 1

S. I. Dimitrov, Aug 07 2025

The numbers x, y and z form a psi-amicable triple according to Dimitrov's definition.

			79170 is in the sequence since psi(79170) = psi(80850) = psi(81900) = 241920 = 79170 + 80850 + 81900. Other examples: (161070, 161070, 161700), (7063980, 7112490, 7112490).

S. I. Dimitrov, On psi-amicable numbers and their generalizations, arXiv:2508.02318 [math.NT], 2025.

Cf. A001615, A125490, A323329, A386901, A386933.

A386307 Ordered hypotenuses of Pythagorean triples that do not have the form (u^2 - v^2, 2uv, u^2 + v^2), where u and v are positive integers.

Original entry on oeis.org

15, 25, 30, 35, 39, 50, 51, 55, 60, 65, 65, 70, 75, 75, 78, 85, 85, 87, 91, 95, 100, 102, 105, 110, 111, 115, 119, 120, 123, 125, 130, 130, 135, 140, 143, 145, 145, 150, 150, 155, 156, 159, 165, 169, 170, 170, 174, 175, 175, 182, 183, 185, 185, 187, 190, 195, 195

Offset: 1

Felix Huber, Aug 13 2025

In the form (u^2 - v^2, 2*u*v, u^2 + v^2), u^2 + v^2) is the hypotenuse, max(u^2 - v^2, 2*u*v) is the long leg and min(u^2 - v^2, 2*u*v) is the short leg.
A101930(n) gives the total number of Pythagorean triples <= 10^n. The percentage of triangles in this sequence increases continuously:
               number of terms <= h     total number of
       h       in this sequence         hypotenuses <= h      percentage
      10                 0                    2                  0.0 %
     100                21                   52                 40.4 %
    1000               514                  881                 58.3 %
   10000              8629                12471                 69.2 %
  100000            122431               161436                 75.8 %

			The Pythagorean triple (9, 12, 15) does not have the form (u^2 - v^2, 2*u*v, u^2 + v^2), because 15 is not a sum of two nonzero squares. Therefore 15 is a term.

Felix Huber, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pythagorean Triple

Subsequence of A009000.

Cf. A101930, A366428, A380072, A386308, A386309, A386943.

A386307:=proc(N) # To get all hypotenuses <= N
    local i,l,m,u,v,r,x,y,z;
    l:={};
    m:={};
    for u from 2 to floor(sqrt(N-1)) do
        for v to min(u-1,floor(sqrt(N-u^2))) do
            x:=min(2*u*v,u^2-v^2);
            y:=max(2*u*v,u^2-v^2);
            z:=u^2+v^2;
            m:=m union {[z,y,x]};
            if gcd(u,v)=1 and is(u-v,odd) then
                l:=l union {seq([i*z,i*y,i*x],i=1..N/z)}
            fi
        od
    od;
    r:=l minus m;
    return seq(r[i,1],i=1..nops(r));
end proc;
A386307(1000);

a(n) = sqrt(A386308(n)^2 + A386309(n)^2).

{A009000(n)} = {a(n)} union {A020882(n)} union {A386943(n)}.

cp's OEIS Frontend

A387423 The length of binary expansion of n minus the length of the maximal common prefix of the binary expansions of n and sigma(n), where sigma is the sum of divisors function.

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A387480 a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(k,n-k)^2.

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A387481 a(n) = Sum_{k=0..floor(n/2)} 3^k * 2^(n-2*k) * binomial(k,n-2*k)^2.

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A387484 a(n) = Sum_{k=0..floor(n/3)} 2^(n-k) * binomial(k,n-3*k)^2.

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A387485 a(n) = Sum_{k=0..floor(n/3)} 2^(n-2*k) * binomial(k,n-3*k)^2.

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A379049 a(n) = prime(i)*dp(n,i) + prime(i)*dn(n,i) where dp(n,i) = 1 when the i-th trit of n is 1, dn(n,i) = 1 when the i-th trit of n is T, and dp(n,i) = dn(n,i) = 0 when the i-th trit of n is 0.

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A383904 a(n) is the number of complement pairs of primitive 2n-bead balanced binary necklaces.

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A385476 Values of u in the quartets (1, u, v, w) of type 3; i.e., values of u for solutions to (1 - u) = v(v - w), in distinct positive integers, with v > 1, sorted by nondecreasing values of u; see Comments.

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A387481 a(n) = Sum_{k=0..floor(n/2)} 3^k * 2^(n-2k) binomial(k,n-2*k)^2.

A387485 a(n) = Sum_{k=0..floor(n/3)} 2^(n-2k) binomial(k,n-3*k)^2.

A379049 a(n) = prime(i)dp(n,i) + prime(i)dn(n,i) where dp(n,i) = 1 when the i-th trit of n is 1, dn(n,i) = 1 when the i-th trit of n is T, and dp(n,i) = dn(n,i) = 0 when the i-th trit of n is 0.

A386307 Ordered hypotenuses of Pythagorean triples that do not have the form (u^2 - v^2, 2uv, u^2 + v^2), where u and v are positive integers.