A344026 -id:A344026 - cp's OEIS frontend

A344182 a(n) = A344026(n) XOR A344028(n).

0, 0, 0, 6, 0, 0, 5, 8, 0, 0, 0, 26, 15, 26, 18, 40, 0, 0, 0, 22, 0, 0, 63, 56, 9, 14, 31, 34, 82, 124, 119, 64, 0, 0, 0, 50, 0, 0, 45, 88, 0, 0, 0, 98, 99, 38, 234, 88, 29, 114, 29, 202, 35, 34, 136, 160, 162, 444, 406, 130, 393, 430, 452, 224, 0, 0, 0, 42, 0, 0, 97, 120, 0, 0, 0, 46, 215, 222, 130, 136, 0, 0, 0, 202

Offset: 0

Antti Karttunen, May 16 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for sequences related to binary expansion of n

Cf. A003415, A003714 (positions of zeros), A005940, A069359, A344026, A344028.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A069359(n) = (n*sumdiv(n, d, isprime(d)/d)); \\ From A069359
A344182(n) = { my(u=A005940(1+n)); bitxor(A003415(u),A069359(u)); };

a(n) = A344026(n) XOR A344028(n) = A003415(A005940(1+n)) XOR A069359(A005940(1+n)).

A369065 Lexicographically earliest infinite sequence such that a(i) = a(j) => A344026(i) = A344026(j) for all i, j >= 0.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 5, 6, 2, 7, 8, 9, 10, 11, 12, 13, 2, 14, 10, 15, 6, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 2, 27, 19, 13, 9, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 2, 55, 9, 56, 31, 57, 23, 34, 58, 59, 60, 61, 62, 63, 50, 64, 15, 65, 66

Offset: 0

Antti Karttunen, Jan 16 2024

nonn

Restricted growth sequence transform of A344026, i.e., of the arithmetic derivative (A003415) as reordered by the Doudna sequence (A005940).

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A003415, A005940, A344026.

Cf. also A366802, A369067.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A344026(n) = A003415(A005940(1+n));
v369065 = rgs_transform(vector(1+up_to,n,A344026(n-1)));
A369065(n) = v369065[1+n];

A344027 Arithmetic derivative applied to prime shift array: Square array A(n,k) = A003415(A246278(n,k)), read by falling antidiagonals.

Original entry on oeis.org

1, 4, 1, 5, 6, 1, 12, 8, 10, 1, 7, 27, 12, 14, 1, 16, 10, 75, 18, 22, 1, 9, 39, 16, 147, 24, 26, 1, 32, 14, 95, 20, 363, 30, 34, 1, 21, 108, 18, 203, 28, 507, 36, 38, 1, 24, 55, 500, 24, 407, 32, 867, 42, 46, 1, 13, 51, 119, 1372, 30, 611, 40, 1083, 52, 58, 1, 44, 16, 135, 275, 5324, 36, 935, 48, 1587, 60, 62, 1

Offset: 1

Antti Karttunen, May 07 2021

For each column k, A343221(2*k) gives the least n (row number) where A(n,k) < A246278(n,k).

Each column is monotonic.

			The top left corner of the array:
    k = 1   2   3     4   5     6   7       8     9    10  11      12  13    14
   2k = 2   4   6     8  10    12  14      16    18    20  22      24  26    28
------+--------------------------------------------------------------------------
  n=1 | 1,  4,  5,   12,  7,   16,  9,     32,   21,   24, 13,     44, 15,   32,
    2 | 1,  6,  8,   27, 10,   39, 14,    108,   55,   51, 16,    162, 20,   75,
    3 | 1, 10, 12,   75, 16,   95, 18,    500,  119,  135, 22,    650, 24,  155,
    4 | 1, 14, 18,  147, 20,  203, 24,   1372,  275,  231, 26,   1960, 30,  287,
    5 | 1, 22, 24,  363, 28,  407, 30,   5324,  455,  495, 34,   6050, 40,  539,
    6 | 1, 26, 30,  507, 32,  611, 36,   8788,  731,  663, 42,  10816, 44,  767,
    7 | 1, 34, 36,  867, 40,  935, 46,  19652, 1007, 1071, 48,  21386, 54, 1275,
    8 | 1, 38, 42, 1083, 48, 1235, 50,  27436, 1403, 1463, 56,  31768, 60, 1539,
    9 | 1, 46, 52, 1587, 54, 1863, 60,  48668, 2175, 1955, 64,  58190, 66, 2231,
   10 | 1, 58, 60, 2523, 66, 2639, 70,  97556, 2759, 2987, 72, 102602, 76, 3219,
   11 | 1, 62, 68, 2883, 72, 3255, 74, 119164, 3663, 3503, 78, 136462, 84, 3627,
   12 | 1, 74, 78, 4107, 80, 4403, 84, 202612, 4715, 4551, 90, 219040, 96, 4847,
etc.

Cf. A003415, A246278.
Cf. A068719 (row 1).
Cf. also A343221, A343222, A344026, A355927, A356155.

up_to = 91;
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A246278sq(row,col) = if(1==row,2*col, my(f = factor(2*col)); for(i=1, #f~, f[i,1] = prime(primepi(f[i,1])+(row-1))); factorback(f));
A344027sq(row,col) = A003415(A246278sq(row,col));
A344027list(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] = A344027sq(col,(a-(col-1))))); (v); };
v344027 = A344027list(up_to);
A344027(n) = v344027[n];

A366801 Arithmetic derivative without its inherited divisor applied to the Doudna sequence: a(n) = A342001(A005940(1+n)).

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 2, 3, 1, 7, 8, 8, 2, 7, 3, 4, 1, 9, 10, 12, 12, 31, 13, 11, 2, 9, 11, 10, 3, 9, 4, 5, 1, 13, 14, 16, 16, 41, 17, 17, 18, 59, 71, 46, 19, 41, 18, 14, 2, 11, 13, 14, 17, 37, 16, 13, 3, 11, 14, 12, 4, 11, 5, 6, 1, 15, 16, 24, 18, 61, 25, 23, 20, 87, 103, 62, 27, 55, 24, 22, 24, 113, 131, 94, 167, 247

Offset: 0

Antti Karttunen, Oct 24 2023

Antti Karttunen, Table of n, a(n) for n = 0..16384

Cf. A003415, A005940, A342001, A344026, A366802 (rgs-transform).

Cf. also A342002.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A342001(n) = (A003415(n) / A003557(n));
A366801(n) = A342001(A005940(1+n));

A344028 a(n) = A069359(A005940(1+n)).

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 3, 4, 1, 7, 8, 10, 5, 15, 9, 8, 1, 9, 10, 14, 12, 31, 24, 20, 7, 35, 40, 30, 25, 45, 27, 16, 1, 13, 14, 18, 16, 41, 30, 28, 18, 59, 71, 62, 60, 93, 72, 40, 11, 63, 70, 70, 84, 155, 120, 60, 49, 175, 200, 90, 125, 135, 81, 32, 1, 15, 16, 26, 18, 61, 42, 36, 20, 87, 103, 82, 80, 123, 90, 56, 24, 113, 131

Offset: 0

Antti Karttunen, May 11 2021

Coincides with A344026 on Fibbinary numbers, A003714.

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for sequences related to binary expansion of n

Cf. A000079 (positions of ones), A003714, A005940, A069359, A344026, A344182.

A069359(n) = (n*sumdiv(n, d, isprime(d)/d)); \\ From A069359
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A344028(n) = A069359(A005940(1+n));

a(n) = A069359(A005940(1+n)).

A369456 a(n) = A083345(A005940(1+n)), where A083345(n) = (n'/gcd(n,n')), n' means the arithmetic derivative of n (A003415), and A005940 is the Doudna-sequence.

Original entry on oeis.org

0, 1, 1, 1, 1, 5, 2, 3, 1, 7, 8, 4, 2, 7, 1, 2, 1, 9, 10, 6, 12, 31, 13, 11, 2, 9, 11, 5, 3, 3, 4, 5, 1, 13, 14, 8, 16, 41, 17, 17, 18, 59, 71, 23, 19, 41, 6, 7, 2, 11, 13, 7, 17, 37, 16, 13, 3, 11, 14, 2, 4, 11, 5, 3, 1, 15, 16, 12, 18, 61, 25, 23, 20, 87, 103, 31, 27, 55, 8, 11, 24, 113, 131, 47, 167, 247, 106, 61

Offset: 0

Antti Karttunen, Jan 27 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16384

Cf. A005940, A083345, A369457 (rgs-transform).

Cf. also A344026, A366801.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
A369456(n) = A083345(A005940(1+n));

For all n > 0, a(n)|A366801(n)|A344026(n).

cp's OEIS Frontend

A344182 a(n) = A344026(n) XOR A344028(n).

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A369065 Lexicographically earliest infinite sequence such that a(i) = a(j) => A344026(i) = A344026(j) for all i, j >= 0.

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A344027 Arithmetic derivative applied to prime shift array: Square array A(n,k) = A003415(A246278(n,k)), read by falling antidiagonals.

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A366801 Arithmetic derivative without its inherited divisor applied to the Doudna sequence: a(n) = A342001(A005940(1+n)).

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A344028 a(n) = A069359(A005940(1+n)).

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A369456 a(n) = A083345(A005940(1+n)), where A083345(n) = (n'/gcd(n,n')), n' means the arithmetic derivative of n (A003415), and A005940 is the Doudna-sequence.

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