A083254 -id:A083254 - cp's OEIS frontend

A318878 Sum of A083254(d) for all such divisors d of n for which A083254(d) > 0.

1, 1, 2, 1, 4, 2, 6, 1, 5, 4, 10, 2, 12, 6, 6, 1, 16, 5, 18, 4, 10, 10, 22, 2, 19, 12, 14, 6, 28, 6, 30, 1, 18, 16, 22, 5, 36, 18, 22, 4, 40, 10, 42, 10, 12, 22, 46, 2, 41, 19, 30, 12, 52, 14, 38, 6, 34, 28, 58, 6, 60, 30, 22, 1, 46, 18, 66, 16, 42, 22, 70, 5, 72, 36, 26, 18, 58, 22, 78, 4, 41, 40, 82, 10, 62, 42, 54, 10, 88, 12, 70, 22, 58, 46, 70, 2

Offset: 1

Antti Karttunen, Sep 05 2018

nonn

			n = 105 has divisors [1, 3, 5, 7, 15, 21, 35, 105]. When A083254 is applied to them, we obtain [1, 1, 3, 5, 1, 3, 13, -9]. Summing the positive numbers present, we get a(105) = 1+1+3+5+1+3+13 = 27.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000010, A033879, A083254, A318879.

Cf. also A318678, A318874, A318876.

A318878(n) = sumdiv(n,d,d=(2*eulerphi(d))-d; (d>0)*d);

a(n) = Sum_{d|n} [A083254(d) > 0]*A083254(d), where A083254(n) = 2*phi(n) - n, and [ ] are the Iverson brackets.

a(n) = A318879(n) + A033879(n).

A324052 a(n) = A083254(A005940(1+n)).

Original entry on oeis.org

1, 0, 1, 0, 3, -2, 3, 0, 5, -2, 1, -4, 15, -6, 9, 0, 9, -2, 3, -4, 13, -14, 3, -8, 35, -10, 5, -12, 75, -18, 27, 0, 11, -2, 7, -4, 25, -18, 9, -8, 43, -22, -9, -28, 65, -42, 9, -16, 99, -14, 21, -20, 91, -70, 15, -24, 245, -50, 25, -36, 375, -54, 81, 0, 15, -2, 9, -4, 31, -26, 21, -8, 53, -30, -5, -36, 125, -54, 27, -16, 97, -34, 9

Offset: 0

Antti Karttunen, Feb 18 2019

sign

Antti Karttunen, Table of n, a(n) for n = 0..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537

Cf. A005940, A083254, A054429, A290077, A324055, A324182.

Cf. also A324103.

A324052(n) = { my(m1=1,m2=2,p=2); while(n, if(!(n%2), p=nextprime(1+p), m1 *= p; m2 *= (p-(1==(n%4)))); n>>=1); (m2-m1); };

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A083254(n) = (2*eulerphi(n)-n);
A324052(n) = A083254(A005940(1+n));

a(n) = A083254(A005940(1+n)).
a(n) = 2*A290077(n) - A005940(1+n).
For n >= 1, a(n) = A324182(A054429(n)).
For n >= 1, a((2^n)-1) = 0.

A324103 a(1) = 0; for n > 1, a(n) = A083254(A156552(n)).

Original entry on oeis.org

0, 1, 0, 1, 0, 3, 0, 5, -2, 3, 0, 9, 0, 15, -2, 1, 0, 11, 0, 17, -6, 7, 0, 21, -4, 31, -2, 13, 0, 3, 0, 29, -2, 39, -4, 9, 0, 255, -26, 9, 0, 35, 0, 65, -2, 135, 0, 45, -8, 15, -34, 129, 0, 27, -12, 69, -90, 575, 0, 41, 0, 679, -2, 9, -4, 19, 0, 173, -2, 39, 0, 25, 0, 3583, -2, 301, -8, 83, 0, 77, -14, 2727, 0, 5, -52, 8703, -378, 9, 0, 3

Offset: 1

Antti Karttunen, Feb 18 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..4473

Cf. A000010, A083254, A156552, A323244, A324104, A324105.

Cf. also A324052, A324182.

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)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A083254(n) = (2*eulerphi(n)-n);
A324103(n) = if(1==n,0,A083254(A156552(n)));

a(1) = 0; for n > 1, a(n) = A083254(A156552(n)).

a(n) = 2*A324104(n) - A156552(n).

A323898 Lexicographically earliest sequence such that a(i) = a(j) => A000120(i) = A000120(j) and A083254(i) = A083254(j), for all i, j >= 1.

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 6, 2, 4, 5, 7, 8, 9, 10, 11, 2, 12, 13, 14, 8, 15, 10, 16, 17, 18, 10, 19, 20, 21, 22, 23, 2, 24, 5, 25, 26, 27, 10, 19, 17, 28, 29, 30, 20, 31, 32, 33, 34, 27, 35, 36, 20, 37, 38, 39, 40, 41, 32, 42, 43, 44, 45, 46, 2, 47, 48, 49, 8, 50, 51, 52, 53, 54, 10, 55, 20, 56, 57, 58, 34, 59, 10, 60, 61, 56, 32, 39, 40, 62, 63, 64, 65, 66, 45, 67, 68, 69, 70, 16

Offset: 1

Antti Karttunen, Feb 09 2019

nonn

Restricted growth sequence transform of the ordered pair [A000120(n), A083254(n)].

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A000120, A083254.

Cf. also A318310, A323892, A323898.

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; };
A083254(n) = (2*eulerphi(n)-n);
A323898aux(n) = [hammingweight(n), A083254(n)];
v323898 = rgs_transform(vector(up_to,n,A323898aux(n)));
A323898(n) = v323898[n];

a(2^n) = 2 for all n >= 1.

A323912 Dirichlet inverse of A083254(n), where A083254(n) = 2*phi(n) - n.

Original entry on oeis.org

1, 0, -1, 0, -3, 2, -5, 0, -2, 2, -9, 4, -11, 2, 5, 0, -15, 2, -17, 4, 7, 2, -21, 8, -6, 2, -4, 4, -27, -2, -29, 0, 11, 2, 17, 8, -35, 2, 13, 8, -39, -6, -41, 4, 8, 2, -45, 16, -10, -2, 17, 4, -51, 0, 29, 8, 19, 2, -57, 4, -59, 2, 12, 0, 35, -14, -65, 4, 23, -10, -69, 24, -71, 2, 4, 4, 47, -18, -77, 16, -8, 2, -81, -4, 47, 2, 29, 8, -87, 4

Offset: 1

Antti Karttunen, Feb 12 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000
Jon Maiga, Computer-generated formulas for A323912, Sequence Machine.
Wikipedia, Dirichlet convolution.

Cf. A083254, A323913.

Sequences that appear in the convolution formulas: A002033, A023900, A046692, A055615, A067824, A074206, A101035, A130054, A174725, A191161, A253249, A323910 (Möbius transform), A328722, A330575.

up_to = 16384;
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA083254(n) = (2*eulerphi(n)-n);
v323912 = DirInverse(vector(up_to,n,A083254(n)));
A323912(n) = v323912[n];

A083254(n) = (2*eulerphi(n)-n);
memoA323912 = Map();
A323912(n) = if(1==n,1,my(v); if(mapisdefined(memoA323912,n,&v), v, v = -sumdiv(n,d,if(dA083254(n/d)*A323912(d),0)); mapput(memoA323912,n,v); (v))); \\ Antti Karttunen, Nov 22 2024

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA083254(n/d) * a(d).
From Antti Karttunen, Nov 22 2024: (Start)
Following convolution formulas were conjectured for this sequence by Sequence Machine, with each one giving the first 10000 terms correctly. The first one is certainly true, because A083254 is Möbius transform of A033879:
a(n) = Sum_{d|n} A323910(d).
a(n) = Sum_{d|n} A023900(d)*A074206(n/d) = Sum_{d|n} A002033(d-1)*A023900(n/d).
a(n) = Sum_{d|n} A055615(d)*A067824(n/d)
a(n) = Sum_{d|n} A046692(d)*A253249(n/d)
a(n) = Sum_{d|n} A130054(d)*A174725(n/d)
a(n) = Sum_{d|n} A101035(d)*A330575(n/d)
a(n) = Sum_{d|n} A191161(d)*A328722(n/d)
(End)

A324182 a(n) = A083254(A163511(n)), where A083254(n) = 2*phi(n) - n, the Möbius transform of the deficiency of n.

Original entry on oeis.org

1, 0, 0, 1, 0, 3, -2, 3, 0, 9, -6, 15, -4, 1, -2, 5, 0, 27, -18, 75, -12, 5, -10, 35, -8, 3, -14, 13, -4, 3, -2, 9, 0, 81, -54, 375, -36, 25, -50, 245, -24, 15, -70, 91, -20, 21, -14, 99, -16, 9, -42, 65, -28, -9, -22, 43, -8, 9, -18, 25, -4, 7, -2, 11, 0, 243, -162, 1875, -108, 125, -250, 1715, -72, 75, -350, 637, -100, 147, -98, 1089

Offset: 0

Antti Karttunen, Feb 18 2019

Antti Karttunen, Table of n, a(n) for n = 0..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537

Cf. A054429, A083254, A163511, A324052, A324183, A324184, A324185 (compare the scatter plot), A366804 (rgs-transform).

Cf. also A324103.

A163511(n) = if(!n,1,my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A083254(n) = (2*eulerphi(n)-n);
A324182(n) = A083254(A163511(n));

a(n) = A083254(A163511(n)).

For n > 0, a(n) = A324052(A054429(n)).

A379011 Square array A(n, k) = 2phi(A246278(n, k)) - A246278(n, k), read by falling antidiagonals; A083254, (2phi(n)-n), applied to the prime shift array.

Original entry on oeis.org

0, 0, 1, -2, 3, 3, 0, 1, 15, 5, -2, 9, 13, 35, 9, -4, 3, 75, 43, 99, 11, -2, 3, 25, 245, 97, 143, 15, 0, 7, 65, 53, 1089, 163, 255, 17, -6, 27, 31, 301, 133, 1859, 253, 323, 21, -4, 5, 375, 73, 1067, 185, 4335, 355, 483, 27, -2, 9, 91, 1715, 151, 2119, 313, 6137, 565, 783, 29, -8, 9, 125, 473, 11979, 229, 4301, 457, 11109, 781, 899, 35

Offset: 1

Antti Karttunen, Dec 14 2024

Each column is strictly increasing.

			The top left corner of the array:
k=  |  1    2    3      4    5      6    7       8      9     10   11      12
2k= |  2    4    6      8   10     12   14      16     18     20   22      24
----+-------------------------------------------------------------------------
  1 |  0,   0,  -2,     0,  -2,    -4,  -2,      0,    -6,    -4,  -2,     -8,
  2 |  1,   3,   1,     9,   3,     3,   7,     27,     5,     9,   9,      9,
  3 |  3,  15,  13,    75,  25,    65,  31,    375,    91,   125,  43,    325,
  4 |  5,  35,  43,   245,  53,   301,  73,   1715,   473,   371,  83,   2107,
  5 |  9,  99,  97,  1089, 133,  1067, 151,  11979,  1261,  1463, 187,  11737,
  6 | 11, 143, 163,  1859, 185,  2119, 229,  24167,  2771,  2405, 295,  27547,
  7 | 15, 255, 253,  4335, 313,  4301, 403,  73695,  4807,  5321, 433,  73117,
  8 | 17, 323, 355,  6137, 457,  6745, 491, 116603,  8165,  8683, 593, 128155,
  9 | 21, 483, 565, 11109, 607, 12995, 733, 255507, 16385, 13961, 817, 298885,

Cf. A000010, A083254, A246278, A379010.
Cf. A040976 (column 1), A378986 (row 1).
Cf. also A378979.

up_to = 11325; \\ = binomial(150+1,2)
A083254(n) = (2*eulerphi(n)-n);
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));
A379011sq(row,col) = A083254(A246278sq(row,col));
A379011list(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] = A379011sq(col,(a-(col-1))))); (v); };
v379011 = A379011list(up_to);
A379011(n) = v379011[n];

A(n, k) = 2*A379010(n, k) - A246278(n, k).

A297153 Reversing binary representation of A083254(n), 2*phi(n) - n.

Original entry on oeis.org

1, 0, 1, 0, 7, 6, 13, 0, 7, 6, 25, 12, 31, 6, 1, 0, 19, 10, 49, 12, 7, 6, 61, 24, 19, 6, 25, 12, 47, 18, 37, 0, 11, 6, 21, 20, 103, 6, 25, 24, 107, 54, 121, 12, 7, 6, 117, 48, 103, 30, 21, 12, 87, 54, 41, 24, 19, 6, 73, 36, 79, 6, 25, 0, 35, 46, 193, 12, 55, 58, 205, 40, 203, 6, 13, 12, 127, 34, 213, 48, 47, 6, 241

Offset: 1

Antti Karttunen, Dec 26 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000010, A048724, A065620, A065621, A083254.

Cf. also A297154, A295881.

(define (A297153 n) (let ((x (A083254 n))) (if (> x 0) (A065621 x) (A048724 (- x)))))

If A083254(n) > 0, then a(n) = A065621(A083254(n)), otherwise a(n) = A048724(-A083254(n)).

For all n >= 1, A065620(a(n)) = A083254(n)

A297154 Balanced ternary representation of A083254(n), 2*phi(n)-n.

Original entry on oeis.org

1, 0, 1, 0, 3, 7, 17, 0, 3, 7, 9, 8, 14, 7, 1, 0, 51, 21, 47, 8, 3, 7, 48, 19, 51, 7, 9, 8, 27, 67, 32, 0, 16, 7, 13, 24, 38, 7, 9, 19, 39, 63, 161, 8, 3, 7, 153, 68, 38, 20, 13, 8, 141, 63, 34, 19, 51, 7, 138, 56, 152, 7, 9, 0, 31, 55, 149, 8, 46, 71, 105, 57, 101, 7, 17, 8, 160, 60, 89, 68, 27, 7, 81, 72, 160, 7

Offset: 1

Antti Karttunen, Dec 26 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000010, A083254, A117966, A117967, A117968.

Cf. also A297153, A295882.

(define (A297154 n) (let ((x (A083254 n))) (if (>= x 0) (A117967 x) (A117968 (- x)))))

If A083254(n) >= 0, then a(n) = A117967(A083254(n)), otherwise a(n) = A117968(-A083254(n)).

For all n >= 1, A117966(a(n)) = A083254(n).

A318304 a(n) = A083254(n)/A003557(n) = (2*A173557(n) - A007947(n)).

Original entry on oeis.org

1, 0, 1, 0, 3, -2, 5, 0, 1, -2, 9, -2, 11, -2, 1, 0, 15, -2, 17, -2, 3, -2, 21, -2, 3, -2, 1, -2, 27, -14, 29, 0, 7, -2, 13, -2, 35, -2, 9, -2, 39, -18, 41, -2, 1, -2, 45, -2, 5, -2, 13, -2, 51, -2, 25, -2, 15, -2, 57, -14, 59, -2, 3, 0, 31, -26, 65, -2, 19, -22, 69, -2, 71, -2, 1, -2, 43, -30, 77, -2, 1, -2, 81, -18, 43, -2, 25

Offset: 1

Antti Karttunen, Aug 26 2018

sign

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000010, A003557, A007947, A083254, A173557, A318305.

Cf. A065463, A307868.

A173557(n) = my(f=factor(n)[, 1]); prod(k=1, #f, f[k]-1); \\ From A173557
A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
A318304(n) = (2*A173557(n) - A007947(n));

A083254(n) = (2*eulerphi(n)-n);
A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A318304(n) = (A083254(n)/A003557(n));

a(n) = A083254(n)/A003557(n) = 2*A173557(n) - A007947(n).
a(n) = A173557(n) - A318305(n).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 2 * A307868 - A065463 = 0.238919... . - Amiram Eldar, Dec 07 2023

cp's OEIS Frontend

A318878 Sum of A083254(d) for all such divisors d of n for which A083254(d) > 0.

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A324052 a(n) = A083254(A005940(1+n)).

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A324103 a(1) = 0; for n > 1, a(n) = A083254(A156552(n)).

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A323898 Lexicographically earliest sequence such that a(i) = a(j) => A000120(i) = A000120(j) and A083254(i) = A083254(j), for all i, j >= 1.

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A323912 Dirichlet inverse of A083254(n), where A083254(n) = 2*phi(n) - n.

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A324182 a(n) = A083254(A163511(n)), where A083254(n) = 2*phi(n) - n, the Möbius transform of the deficiency of n.

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A379011 Square array A(n, k) = 2*phi(A246278(n, k)) - A246278(n, k), read by falling antidiagonals; A083254, (2*phi(n)-n), applied to the prime shift array.

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A297153 Reversing binary representation of A083254(n), 2*phi(n) - n.

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A297154 Balanced ternary representation of A083254(n), 2*phi(n)-n.

A379011 Square array A(n, k) = 2phi(A246278(n, k)) - A246278(n, k), read by falling antidiagonals; A083254, (2phi(n)-n), applied to the prime shift array.