A348041 -id:A348041 - cp's OEIS frontend

A347392 Numbers k such that nearest common ancestor of k and sigma(k) in Doudna tree (A347879) is the grandparent of k.

8, 9, 12, 13, 24, 35, 160, 455, 42550, 127650, 8041950, 22469750, 58506250, 67409250, 175518750, 394055550, 4246782750

Offset: 1

Antti Karttunen, Aug 30 2021

Note how 13 * 35 = 455.
If there exists any odd perfect numbers x, with sigma(x) = 2x, then 2*x would be a term of this sequence, as then sigma(2*x) = 6*x would be situated as a descendant under the other branch of the grandparent of 2*x (a parent of x), which is m = A064989(x), with m in A005101. Opn x itself would be a term of A336702. Furthermore, if such x is not a multiple of 3 (in which case m is odd and in A005231), then also 3x would be a term of this sequence as sigma(3*x) = 4*sigma(x) = 8*x would be situated as a grandchild of 2x, with 2x being a first cousin of 3x. Also, in that case, 6*x would be located in A336702 (particularly, in A027687) because then sigma(6*x) = 12*sigma(x) = 24*x = 4*(6*x).
.
                <--A003961--  m  ---(*2)--->
              .............../ \...............
             /                                 \
            /                                   \
           /                                     \
          x                                       2m
  etc..../ \......2x = sigma(x)          3x....../ \......4m
                  / \                     / \            / \
               etc.  \                 etc.  \        etc.  etc.
                      \                       \
                       4x          sigma(2x) = 6x
                      / \                     / \
                  etc    \                 etc.  \
                          \                       \
                          8x = sigma(3x)          12x
                             if m odd               \
                                                     \
                                                    24x = sigma(6x) if m odd.
.
Furthermore, if there were any hypothetical odd terms y in A005820 (triperfect numbers), then 2y would be a term of this sequence. See the diagram in A347391.
If it exists, a(18) > 2^33.

			455 is included in the sequence as sigma(455) = 672, and the nearest common ancestor of 455 and 672 in Doudna tree is 42, which is the grandparent of 455 [as 455 = A003961(A003961(42))] and the grand-grand-grand-parent of 672 [as 672 = (2^4)*42].

Positions of 2's in A347381.
Cf. A000203, A005101, A005231, A005820, A005940, A027687, A336702, A347391, A347393, A347394, A347879, A348041.
Cf. also A336834, A353365.

PARI
```
isA347392(n) = (2==A347381(n));
```

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A252463(n) = if(!(n%2),n/2,A064989(n));
isA347391(n) = if(1==n,0,my(m=A252463(n), s=sigma(n)); while(s>m, if(s==n, return(0)); s = A252463(s)); (s==m));
isA347391_or_A347392(n) = if(1==n,0,my(m=A252463(A252463(n)), s=sigma(n)); while(s>m, if(s==n, return(0)); s = A252463(s)); (s==m));
isA347392(n) = (isA347391_or_A347392(n) && !isA347391(n));

A341510 Symmetric square array A(n,k) = A005940(1+A156552(n)+A156552(k)), read by antidiagonals starting with A(1,1).

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 9, 9, 9, 9, 9, 7, 8, 10, 8, 8, 8, 8, 10, 8, 9, 7, 15, 7, 7, 7, 15, 7, 9, 10, 8, 10, 12, 10, 10, 12, 10, 8, 10, 11, 15, 7, 15, 25, 15, 25, 15, 7, 15, 11, 12, 14, 12, 10, 12, 18, 18, 12, 10, 12, 14, 12, 13, 25, 21, 25, 15, 25, 11, 25, 15, 25, 21, 25, 13

Offset: 1

Antti Karttunen, Feb 13 2021

Considered as a binary operation on the positive integers, A(x, y) returns the term of the Doudna-sequence from the position that is the sum of the positions of x and y in the same sequence. (This is based on giving the Doudna-sequence an offset of 0, rather than 1 as used in A005940.) - Peter Munn, Feb 14 2021

			The top left 16x16 corner of the array:
   1,  2,  3,  4,  5,  6,  7,  8,  9, 10,  11,  12,  13,  14,  15, 16,
   2,  3,  4,  5,  6,  9, 10,  7,  8, 15,  14,  25,  22,  21,  12, 11,
   3,  4,  5,  6,  9,  8, 15, 10,  7, 12,  21,  18,  33,  20,  25, 14,
   4,  5,  6,  9,  8,  7, 12, 15, 10, 25,  20,  27,  28,  35,  18, 21,
   5,  6,  9,  8,  7, 10, 25, 12, 15, 18,  35,  16,  55,  30,  27, 20,
   6,  9,  8,  7, 10, 15, 18, 25, 12, 27,  30,  11,  42,  45,  16, 35,
   7, 10, 15, 12, 25, 18, 11, 16, 27, 14,  49,  20,  77,  50,  21, 24,
   8,  7, 10, 15, 12, 25, 16, 27, 18, 11,  24,  21,  40,  49,  14, 45,
   9,  8,  7, 10, 15, 12, 27, 18, 25, 16,  45,  14,  63,  24,  11, 30,
  10, 15, 12, 25, 18, 27, 14, 11, 16, 21,  50,  35,  70,  75,  20, 49,
  11, 14, 21, 20, 35, 30, 49, 24, 45, 50,  13,  36, 121,  22,  75, 32,
  12, 25, 18, 27, 16, 11, 20, 21, 14, 35,  36,  45,  60, 125,  30, 75,
  13, 22, 33, 28, 55, 42, 77, 40, 63, 70, 121,  60,  17,  98, 105, 48,
  14, 21, 20, 35, 30, 45, 50, 49, 24, 75,  22, 125,  98,  33,  36, 13,
  15, 12, 25, 18, 27, 16, 21, 14, 11, 20,  75,  30, 105,  36,  35, 50,
  16, 11, 14, 21, 20, 35, 24, 45, 30, 49,  32,  75,  48,  13,  50, 81,

Cf. A341511 (the lower triangular section).
Cf. A003961 (main diagonal), A329603 (skewed diagonal).
Cf. A297165 (row 2 and column 2, when started from its term a(1)).
Cf. also A003056, A268715, A341515, A341520, A348041.

up_to = 105;
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A156552(n) = { my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A341510sq(n,k) = A005940(1+A156552(n)+A156552(k));
A341510list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A341510sq(col,(a-(col-1))))); (v); };
v341510 = A341510list(up_to);
A341510(n) = v341510[n];

A(n, k) = A(k, n) = A005940(1 + A156552(n) + A156552(k)).
A(n, n) = A003961(n).
A(n, 2*n) = A(2*n, n) = A329603(n).
A(n, 2) = A(2, n) = A297165(n).

A348040 Square array A(n,k) = the length of the common prefix in binary expansions of A156552(n) and A156552(k), read by antidiagonals.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Sep 27 2021

			The top left 17x17 corner of the array:
  n/k | 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17
------+----------------------------------------------------
   1  | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
   2  | 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
   3  | 0, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 2,
   4  | 0, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1,
   5  | 0, 1, 2, 1, 3, 2, 3, 1, 1, 3, 3, 2, 3, 3, 2, 1, 3,
   6  | 0, 1, 2, 1, 2, 3, 2, 1, 1, 2, 2, 3, 2, 2, 3, 1, 2,
   7  | 0, 1, 2, 1, 3, 2, 4, 1, 1, 3, 4, 2, 4, 4, 2, 1, 4,
   8  | 0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 3, 1,
   9  | 0, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 2, 1,
  10  | 0, 1, 2, 1, 3, 2, 3, 1, 1, 4, 3, 2, 3, 3, 2, 1, 3,
  11  | 0, 1, 2, 1, 3, 2, 4, 1, 1, 3, 5, 2, 5, 4, 2, 1, 5,
  12  | 0, 1, 2, 1, 2, 3, 2, 1, 1, 2, 2, 4, 2, 2, 3, 1, 2,
  13  | 0, 1, 2, 1, 3, 2, 4, 1, 1, 3, 5, 2, 6, 4, 2, 1, 6,
  14  | 0, 1, 2, 1, 3, 2, 4, 1, 1, 3, 4, 2, 4, 5, 2, 1, 4,
  15  | 0, 1, 2, 1, 2, 3, 2, 1, 1, 2, 2, 3, 2, 2, 4, 1, 2,
  16  | 0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, 1,
  17  | 0, 1, 2, 1, 3, 2, 4, 1, 1, 3, 5, 2, 6, 4, 2, 1, 7,

Index entries for sequences computed from indices in prime factorization

Cf. A252464 (main diagonal).
Cf. A005940, A156552, A348041.
Cf. also A347380, A347381.

up_to = 105;
Abincompreflen(n, m) = { my(x=binary(n),y=binary(m),u=min(#x,#y)); for(i=1,u,if(x[i]!=y[i],return(i-1))); (u);};
A156552(n) = {my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A348040sq(x,y) = Abincompreflen(A156552(x), A156552(y));
A348040list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A348040sq(col,(a-(col-1))))); (v); };
v348040 = A348040list(up_to);
A348040(n) = v348040[n];

A347879 The nearest common ancestor of n and sigma(n) in the Doudna tree (A005940).

Original entry on oeis.org

1, 2, 2, 2, 3, 6, 2, 2, 2, 2, 3, 3, 7, 3, 6, 2, 2, 2, 5, 10, 2, 2, 3, 6, 2, 5, 2, 28, 3, 2, 2, 2, 3, 2, 6, 2, 19, 3, 7, 3, 5, 3, 11, 5, 3, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 3, 5, 3, 3, 3, 31, 3, 5, 2, 5, 2, 17, 5, 3, 2, 2, 2, 37, 17, 2, 3, 6, 5, 5, 5, 4, 5, 5, 5, 2, 7, 3, 3, 3, 3, 5, 5, 2, 2, 3, 3, 2, 2, 7, 2, 13, 2

Offset: 1

Antti Karttunen, Oct 13 2021

nonn

The fixed points of this sequence is given by the union of {2} and A336702.
The positions x such that a(x) = A252463(x) is given by the union of {1} and A347391.
The positions x such that a(x) = A252463(A252463(x)) is given by the union of {1} and A347392.

Cf. A000203, A005940, A252463, A332221, A336702, A347381, A347391, A347392, A348040, A348041.

A347879(n) = A348041sq(n,sigma(n)); \\ Needs also code from A348041.

a(n) = A348041(n, A000203(n)).

A356300 Square array read by antidiagonals. A(n,k) is the nearest common ancestor of n and k in the binary tree depicted in A253563.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 03 2022

Array is symmetric and is read by antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ... .

Also the nearest common ancestor of n and k in the tree depicted in A253565 (the mirror image of the A253563-tree).

			The top left 21x21 corner of the array:
n/k  |  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21
-----+----------------------------------------------------------------------------
   1 |  1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,
   2 |  1, 2, 2, 2, 2, 2, 2, 2, 2,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2,
   3 |  1, 2, 3, 2, 3, 2, 3, 2, 3,  2,  3,  2,  3,  2,  3,  2,  3,  2,  3,  2,  3,
   4 |  1, 2, 2, 4, 2, 4, 2, 4, 2,  4,  2,  4,  2,  4,  2,  4,  2,  4,  2,  4,  2,
   5 |  1, 2, 3, 2, 5, 2, 5, 2, 3,  2,  5,  2,  5,  2,  3,  2,  5,  2,  5,  2,  3,
   6 |  1, 2, 2, 4, 2, 6, 2, 4, 2,  6,  2,  4,  2,  6,  2,  4,  2,  6,  2,  4,  2,
   7 |  1, 2, 3, 2, 5, 2, 7, 2, 3,  2,  7,  2,  7,  2,  3,  2,  7,  2,  7,  2,  3,
   8 |  1, 2, 2, 4, 2, 4, 2, 8, 2,  4,  2,  8,  2,  4,  2,  8,  2,  4,  2,  8,  2,
   9 |  1, 2, 3, 2, 3, 2, 3, 2, 9,  2,  3,  2,  3,  2,  9,  2,  3,  2,  3,  2,  9,
  10 |  1, 2, 2, 4, 2, 6, 2, 4, 2, 10,  2,  4,  2, 10,  2,  4,  2,  6,  2,  4,  2,
  11 |  1, 2, 3, 2, 5, 2, 7, 2, 3,  2, 11,  2, 11,  2,  3,  2, 11,  2, 11,  2,  3,
  12 |  1, 2, 2, 4, 2, 4, 2, 8, 2,  4,  2, 12,  2,  4,  2,  8,  2,  4,  2, 12,  2,
  13 |  1, 2, 3, 2, 5, 2, 7, 2, 3,  2, 11,  2, 13,  2,  3,  2, 13,  2, 13,  2,  3,
  14 |  1, 2, 2, 4, 2, 6, 2, 4, 2, 10,  2,  4,  2, 14,  2,  4,  2,  6,  2,  4,  2,
  15 |  1, 2, 3, 2, 3, 2, 3, 2, 9,  2,  3,  2,  3,  2, 15,  2,  3,  2,  3,  2, 15,
  16 |  1, 2, 2, 4, 2, 4, 2, 8, 2,  4,  2,  8,  2,  4,  2, 16,  2,  4,  2,  8,  2,
  17 |  1, 2, 3, 2, 5, 2, 7, 2, 3,  2, 11,  2, 13,  2,  3,  2, 17,  2, 17,  2,  3,
  18 |  1, 2, 2, 4, 2, 6, 2, 4, 2,  6,  2,  4,  2,  6,  2,  4,  2, 18,  2,  4,  2,
  19 |  1, 2, 3, 2, 5, 2, 7, 2, 3,  2, 11,  2, 13,  2,  3,  2, 17,  2, 19,  2,  3,
  20 |  1, 2, 2, 4, 2, 4, 2, 8, 2,  4,  2, 12,  2,  4,  2,  8,  2,  4,  2, 20,  2,
  21 |  1, 2, 3, 2, 3, 2, 3, 2, 9,  2,  3,  2,  3,  2, 15,  2,  3,  2,  3,  2, 21,
.
A(3,6) = A(6,3) = 2 because the nearest common ancestor of 3 and 6 in the tree described in A253563 (and in A253565) is 2.
A(4,6) = A(6,4) = 4 because 6 occurs as a descendant of 4 in A253563-tree, thus their nearest common ancestor is 4 itself.

Index entries for sequences computed from indices in prime factorization

Cf. A253553, A253563, A253565, A356301.

Cf. also A348041.

up_to = 105;
A253553(n) = if(n<=2,1,my(f=factor(n), k=#f~); if(f[k,2]>1,f[k,2]--,f[k,1] = precprime(f[k,1]-1)); factorback(f));
A356300sq(x,y) = if(1==x||1==y,1, my(lista=List([]), i, k=x, stemvec, stemlen, h=y); while(k>1, listput(lista,k); k = A253553(k)); stemvec = Vecrev(Vec(lista)); stemlen = #stemvec; while(1, if((i=vecsearch(stemvec,h))>0, return(stemvec[i])); h = A253553(h)));
A356300list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A356300sq(col,(a-(col-1))))); (v); };
v356300 = A356300list(up_to);
A356300(n) = v356300[n];

A348042 Square array A(n,k) = the nearest common ancestor of n, k and n*k in Doudna tree (A005940).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Sep 27 2021

Array is symmetric and is read by antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

			The top left 17x17 corner of the array:
  n/k |  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17
------+-------------------------------------------------------------
    1 |  1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1,  1,  1,  1,  1,  1,  1,
    2 |  1, 2, 2, 2, 2, 2, 2, 2, 2,  2,  2,  2,  2,  2,  2,  2,  2,
    3 |  1, 2, 2, 2, 3, 2, 3, 2, 2,  3,  3,  2,  3,  3,  3,  2,  3,
    4 |  1, 2, 2, 4, 2, 2, 2, 4, 4,  2,  2,  2,  2,  2,  2,  4,  2,
    5 |  1, 2, 3, 2, 2, 3, 3, 2, 2,  2,  5,  3,  5,  3,  2,  2,  5,
    6 |  1, 2, 2, 2, 3, 2, 3, 2, 2,  3,  3,  2,  3,  3,  6,  2,  3,
    7 |  1, 2, 3, 2, 3, 3, 2, 2, 2,  3,  3,  3,  5,  2,  3,  2,  7,
    8 |  1, 2, 2, 4, 2, 2, 2, 8, 4,  2,  2,  2,  2,  2,  2,  8,  2,
    9 |  1, 2, 2, 4, 2, 2, 2, 4, 4,  2,  2,  2,  2,  2,  2,  4,  2,
   10 |  1, 2, 3, 2, 2, 3, 3, 2, 2,  2,  5,  3,  5,  3,  2,  2,  5,
   11 |  1, 2, 3, 2, 5, 3, 3, 2, 2,  5,  2,  3,  3,  3,  3,  2,  5,
   12 |  1, 2, 2, 2, 3, 2, 3, 2, 2,  3,  3,  2,  3,  3,  6,  2,  3,
   13 |  1, 2, 3, 2, 5, 3, 5, 2, 2,  5,  3,  3,  2,  5,  3,  2,  3,
   14 |  1, 2, 3, 2, 3, 3, 2, 2, 2,  3,  3,  3,  5,  2,  3,  2,  7,
   15 |  1, 2, 3, 2, 2, 6, 3, 2, 2,  2,  3,  6,  3,  3,  2,  2,  3,
   16 |  1, 2, 2, 4, 2, 2, 2, 8, 4,  2,  2,  2,  2,  2,  2, 16,  2,
   17 |  1, 2, 3, 2, 5, 3, 7, 2, 2,  5,  5,  3,  3,  7,  3,  2,  2,

Index entries for sequences computed from indices in prime factorization

Cf. A005940, A156552, A348041, A348043, A348044 (main diagonal).

\\ Needs also code from A348041:
up_to = 105;
A348042sq(row,col) = A348041sq(row*col,A348041sq(row,col));
A348042list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A348042sq(col,(a-(col-1))))); (v); };
v348042 = A348042list(up_to);
A348042(n) = v348042[n];

A(n, k) = A(k, n).
A(n, k) = A348041(n*k, A348041(n, k)).
A(n, k) = A348041(n, A348043(k, n)) = A348041(k, A348043(n, k)).
For any two squares s=u^2 and t=v^2, A(s, t) is a square also.

A348043 Square array A(n,k) = the nearest common ancestor of n and n*k in Doudna tree (A005940).

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 2, 4, 5, 1, 2, 3, 2, 5, 6, 1, 2, 3, 4, 3, 6, 7, 1, 2, 2, 2, 5, 2, 7, 8, 1, 2, 3, 2, 2, 6, 5, 8, 9, 1, 2, 3, 2, 3, 6, 7, 2, 9, 10, 1, 2, 2, 4, 3, 2, 3, 8, 4, 10, 11, 1, 2, 3, 4, 5, 3, 5, 2, 9, 3, 11, 12, 1, 2, 3, 2, 3, 6, 2, 2, 2, 10, 7, 12, 13, 1, 2, 2, 2, 2, 2, 7, 2, 4, 2, 11, 2, 13, 14

Offset: 1

Antti Karttunen, Sep 27 2021

Array is read by falling antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

			The top left 17x17 corner of the array:
  n/k |   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17
------+----------------------------------------------------------------------
    1 |   1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,
    2 |   2,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2,
    3 |   3,  3,  2,  3,  3,  2,  3,  3,  2,  3,  3,  2,  3,  3,  3,  3,  3,
    4 |   4,  4,  2,  4,  2,  2,  2,  4,  4,  2,  2,  2,  2,  2,  2,  4,  2,
    5 |   5,  5,  3,  5,  2,  3,  3,  5,  3,  2,  5,  3,  5,  3,  2,  5,  5,
    6 |   6,  6,  2,  6,  6,  2,  3,  6,  2,  6,  3,  2,  3,  3,  6,  6,  3,
    7 |   7,  7,  5,  7,  3,  5,  2,  7,  5,  3,  3,  5,  5,  2,  3,  7,  7,
    8 |   8,  8,  2,  8,  2,  2,  2,  8,  4,  2,  2,  2,  2,  2,  2,  8,  2,
    9 |   9,  9,  4,  9,  2,  4,  2,  9,  4,  2,  2,  4,  2,  2,  2,  9,  2,
   10 |  10, 10,  3, 10,  2,  3,  3, 10,  3,  2, 10,  3,  5,  3,  2, 10,  5,
   11 |  11, 11,  7, 11,  5,  7,  3, 11,  7,  5,  2,  7,  3,  3,  5, 11,  5,
   12 |  12, 12,  2, 12,  6,  2,  3, 12,  2,  6,  3,  2,  3,  3, 12, 12,  3,
   13 |  13, 13, 11, 13,  7, 11,  5, 13, 11,  7,  3, 11,  2,  5,  7, 13,  3,
   14 |  14, 14,  5, 14,  3,  5,  2, 14,  5,  3,  3,  5,  5,  2,  3, 14, 14,
   15 |  15, 15,  6, 15,  2,  6, 15, 15,  6,  2,  3,  6,  3, 15,  2, 15,  3,
   16 |  16, 16,  2, 16,  2,  2,  2, 16,  4,  2,  2,  2,  2,  2,  2, 16,  2,
   17 |  17, 17, 13, 17, 11, 13,  7, 17, 13, 11,  5, 13,  3,  7, 11, 17,  2,

Index entries for sequences computed from indices in prime factorization

Cf. A005940, A156552, A348041, A348042, A348044 (main diagonal).

Cf. A000027 (all columns k that are powers of two: k = 2^e, for e >= 0).

\\ Needs also code from A348041:
A348043sq(x,y) = A348041sq(x,x*y);
A348043list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A348043sq(col,(a-(col-1))))); (v); };
v348043 = A348043list(up_to);
A348043(n) = v348043[n];

A(n, k) = A348041(n, n*k).

A348044 The nearest common ancestor of n and n^2 in the Doudna tree (A005940).

Original entry on oeis.org

1, 2, 2, 4, 2, 2, 2, 8, 4, 2, 2, 2, 2, 2, 2, 16, 2, 4, 2, 2, 2, 2, 2, 2, 4, 2, 8, 2, 2, 2, 2, 32, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 2, 2, 2, 8, 2, 2, 2, 2, 2, 2, 2, 2, 2, 64, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 4, 2, 2, 2, 2, 2, 16, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 4, 2, 2, 2, 2, 2

Offset: 1

Antti Karttunen, Sep 27 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Main diagonal of A348042 and A348043.
Cf. A102750 (the positions of 2's).
Cf. A005940, A156552, A348041.

A348044(n) = A348041sq(n,n*n); \\ Uses also code from A348041.

a(n) = A348041(n, n^2) = A348042(n, n) = A348043(n, n).

A356156 The nearest common ancestor of n and gcd(n, sigma(n)) in the Doudna tree (A005940).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 1, 2, 1, 12, 1, 2, 1, 28, 1, 6, 1, 1, 3, 2, 1, 1, 1, 2, 1, 10, 1, 3, 1, 2, 3, 2, 1, 2, 1, 1, 3, 2, 1, 2, 1, 2, 1, 2, 1, 6, 1, 2, 1, 1, 1, 3, 1, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 5, 1, 2, 3, 2, 1, 2, 5, 2, 1, 2, 5, 12, 1, 1, 3, 1, 1, 3, 1, 2, 3

Offset: 1

Antti Karttunen, Jul 30 2022

nonn

Cf. A000203, A007691 (fixed points), A009194, A348040, A348041.

Cf. also A347879, A356157, A356158, A356306.

Abincompreflen(n, m) = { my(x=binary(n),y=binary(m),u=min(#x,#y)); for(i=1,u,if(x[i]!=y[i],return(i-1))); (u);};
Abinprefix(n,k) = { my(digs=binary(n)); fromdigits(vector(k,i,digs[i]),2); };
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A156552(n) = {my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A348040sq(x,y) = Abincompreflen(A156552(x), A156552(y));
A348041sq(x,y) = A005940(1+Abinprefix(A156552(x),A348040sq(x,y)));
A356156(n) = A348041sq(n,gcd(n, sigma(n)));

a(n) = A348041(n, A009194(n)) = A348041(n, gcd(n, A000203(n))).

A356157 The nearest common ancestor of sigma(n) and gcd(n, sigma(n)) in the Doudna tree (A005940).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 2, 1, 2, 3, 1, 1, 3, 1, 2, 1, 2, 1, 6, 1, 2, 1, 28, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 3, 1, 6, 1, 2, 3, 2, 1, 2, 1, 1, 2, 2, 1, 6, 1, 2, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 2, 3, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 28, 1, 2, 3, 2, 1, 2, 7, 2, 1, 2, 3, 3, 1, 1, 3, 1, 1, 2, 1, 2, 3

Offset: 1

Antti Karttunen, Jul 30 2022

nonn

Cf. A000203, A009194, A336702 (fixed points), A348040, A348041.

Cf. also A347879, A356156, A356307.

Abincompreflen(n, m) = { my(x=binary(n),y=binary(m),u=min(#x,#y)); for(i=1,u,if(x[i]!=y[i],return(i-1))); (u);};
Abinprefix(n,k) = { my(digs=binary(n)); fromdigits(vector(k,i,digs[i]),2); };
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A156552(n) = {my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A348040sq(x,y) = Abincompreflen(A156552(x), A156552(y));
A348041sq(x,y) = A005940(1+Abinprefix(A156552(x),A348040sq(x,y)));
A356157(n) = A348041sq(sigma(n),gcd(n, sigma(n)));

cp's OEIS Frontend

A347392 Numbers k such that nearest common ancestor of k and sigma(k) in Doudna tree (A347879) is the grandparent of k.

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A341510 Symmetric square array A(n,k) = A005940(1+A156552(n)+A156552(k)), read by antidiagonals starting with A(1,1).

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A348040 Square array A(n,k) = the length of the common prefix in binary expansions of A156552(n) and A156552(k), read by antidiagonals.

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A347879 The nearest common ancestor of n and sigma(n) in the Doudna tree (A005940).

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A356300 Square array read by antidiagonals. A(n,k) is the nearest common ancestor of n and k in the binary tree depicted in A253563.

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A348042 Square array A(n,k) = the nearest common ancestor of n, k and n*k in Doudna tree (A005940).

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A348043 Square array A(n,k) = the nearest common ancestor of n and n*k in Doudna tree (A005940).

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A348044 The nearest common ancestor of n and n^2 in the Doudna tree (A005940).

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