A294898 -id:A294898 - cp's OEIS frontend

A297114 Möbius transform of A294898, where A294898 is deficiency minus binary weight.

0, 0, 0, 0, 2, -2, 3, 0, 3, -2, 7, -4, 9, -2, 0, 0, 14, -6, 15, -4, 4, -2, 18, -8, 14, -2, 7, -4, 24, -14, 25, 0, 9, -2, 14, -12, 33, -2, 9, -8, 37, -18, 38, -4, 3, -2, 41, -16, 35, -10, 12, -4, 48, -18, 24, -8, 15, -2, 53, -28, 55, -2, 6, 0, 33, -26, 63, -4, 21, -22, 66, -24, 69, -2, 6, -4, 44, -30, 73, -16, 28, -2

Offset: 1

Antti Karttunen, Dec 26 2017

sign

Cf. A000203, A005187, A051953, A083254, A294898, A297111, A297115, A297117, A317844, A324397.

Cf. A378986 (even bisection, -2*A176095).

Table[DivisorSum[n, MoebiusMu[n/#] (2 # - DigitCount[2 #, 2, 1] - DivisorSigma[1, #]) &], {n, 82}] (* Michael De Vlieger, Mar 11 2019 *)

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A297114(n) = sumdiv(n,d,moebius(n/d)*(A005187(d)-sigma(d)));

a(n) = Sum_{d|n} A008683(n/d)*A294898(d).
a(n) = A297111(n) - n.
a(n) = A297117(n) - A051953(n).
a(n) = A083254(n) - A297115(n).
a(2n) = A083254(2n) = A378986(n) = -2*A176095(n).
a(n) = A294898(n) - A317844(n).

A326130 a(n) = gcd(A000120(n), A294898(n)) = gcd(A000120(n), sigma(n)-A005187(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 09 2019

nonn

Cf. A000120, A000203, A005187, A033879, A326131.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326056, A326057, A326060, A326062, A326129, A326140, A326144.

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A294898(n) = (A005187(n)-sigma(n));
A326130(n) = gcd(hammingweight(n), A294898(n));

A326130(n) = gcd(hammingweight(n), sigma(n)-A005187(n));

a(n) = gcd(A000120(n), A294898(n)) = gcd(A000120(n), A000203(n)-A005187(n)).

A318311 Filter sequence combining A278222(n) and A294898(n).

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 5, 1, 6, 7, 8, 9, 10, 11, 12, 1, 13, 14, 15, 16, 17, 18, 19, 20, 21, 8, 22, 23, 24, 25, 26, 1, 27, 28, 29, 30, 31, 32, 19, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 1, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 10, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 62, 79

Offset: 1

Antti Karttunen, Aug 25 2018

nonn

Restricted growth sequence transform of ordered pair [A278222(n), A294898(n)].

For all i, j: a(i) = a(j) => A318310(i) = A318310(j) => A033879(i) = A033879(j).

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

Cf. A033879, A278222, A286622, A294898, A318310.

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; };
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A294898(n) = (A005187(n)-sigma(n));
A318311aux(n) = [A278222(n), A294898(n)]; \\ Needs also code from A286622.
v318311 = rgs_transform(vector(up_to,n,A318311aux(n)));
A318311(n) = v318311[n];

A324348 a(n) = A294898(A005940(1+n)), where A294898(k) = A005187(k) - A000203(k).

Original entry on oeis.org

0, 0, 0, 0, 2, -2, 3, 0, 3, 0, 2, -6, 16, -5, 10, 0, 7, 1, 7, -4, 19, -16, 8, -14, 38, 4, 22, -21, 88, -16, 38, 0, 9, 5, 16, -3, 33, -15, 16, -12, 54, -7, 14, -52, 96, -58, 26, -30, 104, 22, 62, -20, 142, -76, 43, -53, 280, 26, 119, -68, 464, -42, 116, 0, 14, 7, 18, 1, 44, -14, 38, -11, 65, -1, 38, -59, 174, -66, 52, -28, 113, 16, 72, -59, 191, -160, 0, -124

Offset: 0

Antti Karttunen, Feb 24 2019

sign

Positions of zeros is given by the sequence A156552(A295296(n)), n >= 1, sorted into ascending order: 0, 1, 2, 3, 7, 9, 15, 31, 63, 86, 127, 255, 511, 519, 1023, 2047, 4095, 8191, 16383, 32767, ...

Antti Karttunen, Table of n, a(n) for n = 0..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences related to sigma(n)

Cf. A000120, A000203, A005187, A005940, A294898, A324055.

Cf. also A156552, A295296, A323248.

A324348(n) = { my(m1=2, m2=1, p=2, mp=p*p); while(n, if(!(n%2), p=nextprime(1+p); mp = p*p, m1 *= p; if(3==(n%4), mp *= p, m2 *= (mp-1)/(p-1))); n>>=1); ((m1-m2)-hammingweight(m1)); };

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A294898(n) = (A005187(n) - sigma(n));
A324348(n) = A294898(A005940(1+n));

a(n) = A294898(A005940(1+n)).

a(n) = A324055(n) - A000120(A005940(1+n)).

A326131 Positive numbers n for which A000120(n) = k*A294898(n), with k < 0; numbers for which A326130(n) = sigma(n) - A005187(n).

Original entry on oeis.org

6, 28, 110, 496, 884, 8128, 18632, 85936, 116624, 15370304, 33550336, 73995392, 815634435, 3915380170, 5556840416, 6800695312, 8589869056

Offset: 1

Antti Karttunen, Jun 09 2019

No further terms below 2^31.
See also comments in A326133.
The quotients A000120(k)/(sigma(k)-A005187(k)) for these terms are: 1, 1, 5, 1, 3, 1, 5, 9, 2, 2, 1, 2, 2. Ones occur at the positions of perfect numbers.
a(18) > 10^11. - Amiram Eldar, Jan 03 2021

			110 is "1101110" in binary, thus A000120(110) = 5. Sigma(110) = 216, while A005187(110) = 215, thus as 5 = 5*(216-215), 110 is included in this sequence.

Intersection of A326132 and A326133, also of A326132 and A326138.
Cf. A000120, A000396 (subsequence), A005187, A294898, A295296, A295297, A295298, A295299, A326130.
Cf. also A325981, A326141.

q[n_] := Module[{bw = DigitCount[n, 2, 1], ab = DivisorSigma[1, n] - 2*n, sum}, (sum = ab + bw) > 0 && Divisible[bw, sum]]; Select[Range[10^5], q] (* Amiram Eldar, Jan 03 2021 *)

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
isA326131(n) = { my(t=sigma(n)-A005187(n)); (gcd(hammingweight(n), t) == t); };

a(14)-a(17) from Amiram Eldar, Jan 03 2021

A317844 Difference between A294898 and its Möbius transform (A297114).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -2, 0, 3, 2, 0, 0, 1, 0, 0, 3, 7, 0, -6, 2, 9, 3, 1, 0, -2, 0, 0, 7, 14, 5, -9, 0, 15, 9, -4, 0, 3, 0, 5, 5, 18, 0, -14, 3, 14, 14, 7, 0, 2, 9, -3, 15, 24, 0, -24, 0, 25, 10, 0, 11, 12, 0, 12, 18, 15, 0, -29, 0, 33, 16, 13, 10, 14, 0, -12, 10, 37, 0, -23, 16, 38, 24, 1, 0, -16, 12, 16, 25

Offset: 1

Antti Karttunen, Aug 09 2018

sign

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

Cf. A000203, A001065, A005187, A008683, A294898, A297114, A300244.

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A317844(n) = -sumdiv(n,d,(dA005187(d)-sigma(d)));

a(n) = -Sum_{d|n, dA008683(n/d)*A294898(d).
a(n) = A294898(n) - A297114(n).
a(n) = A300244(n) - A001065(n).

A323167 a(n) = A294898(A122111(n)).

Original entry on oeis.org

0, 0, 0, 0, 0, -2, 0, 2, 3, -6, 0, 0, 0, -14, -5, 3, 0, 2, 0, -4, -21, -30, 0, 1, 10, -62, 16, -12, 0, -16, 0, 7, -53, -126, -16, 7, 0, -254, -117, -3, 0, -52, 0, -28, 4, -510, 0, 5, 38, 8, -245, -60, 0, 19, -68, -11, -501, -1022, 0, -15, 0, -2046, -20, 9, -172, -124, 0, -124, -1013, -58, 0, 16, 0, -4094, 22, -252, -42, -268, 0, 1, 38

Offset: 1

Antti Karttunen, Jan 10 2019

sign

Cf. A000203, A005187, A294898, A122111, A322867, A323174, A323168.

Cf. also A295296, A323248.

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)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A294898(n) = (A005187(n)-sigma(n));
A323167(n) = A294898(A122111(n));
\\ Or alternatively as:
A322867(n) = hammingweight(A122111(n));
A323174(n) = { my(k=A122111(n)); ((2*k)-sigma(k)); }
A323167(n) = (A323174(n) - A322867(n));

a(n) = A294898(A122111(n)).

a(n) = A323174(n) - A322867(n).

A379008 Square array A(n, k) = A294898(A246278(n, k)), read by falling antidiagonals; Difference A005187(n)-A000203(n) applied to the prime shift array.

Original entry on oeis.org

0, 0, 0, -2, 3, 2, 0, 2, 16, 3, 0, 10, 19, 38, 7, -6, 7, 88, 54, 104, 9, 1, 8, 33, 280, 113, 151, 14, 0, 16, 96, 65, 1192, 184, 268, 15, -5, 38, 44, 389, 152, 2009, 282, 336, 18, -4, 22, 464, 88, 1279, 207, 4600, 388, 502, 24, 5, 16, 142, 1996, 174, 2445, 345, 6470, 608, 806, 25, -14, 18, 174, 623, 13170, 257, 4834, 497, 11605, 833, 924, 33

Offset: 1

Antti Karttunen, Dec 14 2024

Question: Are all columns increasing, and strictly increasing after the leftmost column?

			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,   0,    -6,   1,      0,    -5,    -4,   5,    -14,
2   |  0,   3,   2,    10,   7,     8,  16,     38,    22,    16,  18,     26,
3   |  2,  16,  19,    88,  33,    96,  44,    464,   142,   174,  58,    495,
4   |  3,  38,  54,   280,  65,   389,  88,   1996,   623,   469, 103,   2737,
5   |  7, 104, 113,  1192, 152,  1279, 174,  13170,  1516,  1717, 211,  14102,
6   |  9, 151, 184,  2009, 207,  2445, 257,  26172,  3208,  2756, 328,  31850,
7   | 14, 268, 282,  4600, 345,  4834, 439,  78295,  5406,  5916, 473,  82285,
8   | 15, 336, 388,  6470, 497,  7455, 533, 123071,  9035,  9501, 638, 141745,
9   | 18, 502, 608, 11605, 653, 14081, 784, 267115, 17773, 15097, 870, 324077,
Here 0's occur also after the first row. For example column 30, which corresponds with numbers 60, 315, 1925, 7007, 26741, ..., begins as -52, 0, 868, 4428, 19958, etc. See also A295296.

Cf. A000203, A005187, A246278, A294898, A295296.
Cf. A080085 (column 1, incremented by one).
Cf. also array A378979, and A324348 (another permutation of A294898).

up_to = 11325; \\ = binomial(150+1,2)
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A294898(n) = (A005187(n)-sigma(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));
A379008sq(row,col) = A294898(A246278sq(row,col));
A379008list(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] = A379008sq(col,(a-(col-1))))); (v); };
v379008 = A379008list(up_to);
A379008(n) = v379008[n];

A318447 a(n) = Sum_{d|n, dA294898(d), where A294898(d) = A005187(d) - sigma(d).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -2, 0, 3, 2, 0, 0, 1, 0, 2, 3, 7, 0, -8, 2, 9, 3, 4, 0, 2, 0, 0, 7, 14, 5, -10, 0, 15, 9, -2, 0, 9, 0, 12, 7, 18, 0, -22, 3, 18, 14, 16, 0, 6, 9, 1, 15, 24, 0, -24, 0, 25, 13, 0, 11, 26, 0, 26, 18, 25, 0, -45, 0, 33, 20, 28, 10, 32, 0, -14, 13, 37, 0, -15, 16, 38, 24, 13, 0, -8, 12, 34, 25, 41, 17, -52, 0

Offset: 1

Antti Karttunen, Aug 27 2018

sign

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

Cf. A000203, A005187, A211779, A292257, A294898, A296074, A318445, A318448.

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A294898(n) = (A005187(n) - sigma(n));
A318447(n) = sumdiv(n,d,(dA294898(d));

a(n) = Sum_{d|n, dA294898(d).
a(n) = A318448(n) - A294898(n).
a(n) = A318445(n) - A211779(n).
a(n) = A296074(n) - A292257(n).

A318448 a(n) = Sum_{d|n} A294898(d), where A294898(d) = A005187(d) - sigma(d).

Original entry on oeis.org

0, 0, 0, 0, 2, -2, 3, 0, 3, 2, 7, -8, 9, 4, 4, 0, 14, -4, 15, -2, 10, 12, 18, -22, 18, 16, 13, 1, 24, -14, 25, 0, 23, 26, 24, -31, 33, 28, 27, -14, 37, -6, 38, 13, 15, 34, 41, -52, 41, 22, 40, 19, 48, -10, 42, -10, 45, 46, 53, -76, 55, 48, 29, 0, 55, 12, 63, 34, 57, 18, 66, -98, 69, 64, 42, 37, 64, 16, 73, -42, 51, 72, 78, -74, 74, 74, 73, 6

Offset: 1

Antti Karttunen, Aug 27 2018

sign

Inverse Möbius transform of A294898.

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

Cf. A000203, A005187, A007429, A093653, A294898, A296075, A318446, A318447.

Cf. also A297114, A317844.

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A294898(n) = (A005187(n) - sigma(n));
A318448(n) = sumdiv(n,d,A294898(d));

a(n) = Sum_{d|n} A294898(d).
a(n) = A318447(n) + A294898(n).
a(n) = A318446(n) - A007429(n).
a(n) = A296075(n) - A093653(n).

cp's OEIS Frontend

A297114 Möbius transform of A294898, where A294898 is deficiency minus binary weight.

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A326130 a(n) = gcd(A000120(n), A294898(n)) = gcd(A000120(n), sigma(n)-A005187(n)).

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A318311 Filter sequence combining A278222(n) and A294898(n).

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A324348 a(n) = A294898(A005940(1+n)), where A294898(k) = A005187(k) - A000203(k).

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A326131 Positive numbers n for which A000120(n) = k*A294898(n), with k < 0; numbers for which A326130(n) = sigma(n) - A005187(n).

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A317844 Difference between A294898 and its Möbius transform (A297114).

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A323167 a(n) = A294898(A122111(n)).

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A379008 Square array A(n, k) = A294898(A246278(n, k)), read by falling antidiagonals; Difference A005187(n)-A000203(n) applied to the prime shift array.

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