A278226 -id:A278226 - cp's OEIS frontend

A286626 Restricted growth sequence computed for primorial base related filter-sequence A278226.

1, 2, 2, 3, 4, 5, 2, 3, 3, 6, 5, 7, 4, 5, 5, 7, 8, 9, 10, 11, 11, 12, 13, 14, 15, 16, 16, 17, 18, 19, 2, 3, 3, 6, 5, 7, 3, 6, 6, 20, 7, 21, 5, 7, 7, 21, 9, 22, 11, 12, 12, 23, 14, 24, 16, 17, 17, 25, 19, 26, 4, 5, 5, 7, 8, 9, 5, 7, 7, 21, 9, 22, 8, 9, 9, 22, 27, 28, 13, 14, 14, 24, 29, 30, 18, 19, 19, 26, 31, 32, 10, 11, 11, 12, 13, 14, 11, 12, 12, 23, 14, 24

Offset: 0

Antti Karttunen, May 11 2017

When filtering sequences (by equivalence class partitioning), this sequence (with its modestly sized terms) can be used instead of A278226, because for all i, j it holds that: a(i) = a(j) <=> A278226(i) = A278226(j).

For example, for all i, j: a(i) = a(j) => A276150(i) = A276150(j).

Antti Karttunen, Table of n, a(n) for n = 0..30030
Index entries for sequences related to primorial base

Cf. A278226, A276150.

Cf. also A101296, A286603, A286605, A286610, A286619, A286621, A286622, A286378 for similarly constructed sequences.

b = MixedRadix[Reverse@ Prime@ Range@ 12]; f[n_] := Times @@ MapIndexed[Prime[#2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]] - Boole[n == 1]; With[{nn = 102}, Function[s, Table[Position[Keys@ s, k_ /; MemberQ[k, n]][[1, 1]], {n, nn}]]@ Map[#1 -> #2 & @@ # &, Transpose@ {Values@ #, Keys@ #}] &@ PositionIndex@ Table[Function[k, f[Times @@ Power @@@ # &@ Transpose@ {Prime@ Range@ Length@ k, Reverse@ k}]]@ IntegerDigits[n, b], {n, 0, nn}]] (* Michael De Vlieger, May 12 2017, Version 10.2 *)

rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A278226(n) = A046523(A276086(n));
write_to_bfile(0,rgs_transform(vector(30031,n,A278226(n-1))),"b286626.txt");

A373982 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278226(A328768(i)) = A278226(A328768(j)), for all i, j >= 0.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 2, 3, 3, 5, 2, 6, 2, 7, 8, 4, 2, 9, 2, 7, 10, 11, 2, 10, 3, 5, 12, 13, 2, 11, 2, 10, 14, 15, 16, 8, 2, 10, 17, 18, 2, 19, 2, 20, 21, 22, 2, 23, 3, 24, 10, 25, 2, 20, 26, 17, 27, 28, 2, 29, 2, 7, 24, 16, 29, 30, 2, 20, 22, 31, 2, 32, 2, 33, 19, 34, 35, 36, 2, 11, 4, 37, 2, 22, 29, 11, 25, 38, 2, 33, 39, 17, 10, 40, 41, 10, 2, 42, 43, 20

Offset: 0

Antti Karttunen, Jun 25 2024

nonn

Restricted growth sequence transform of A278226(A328768(n)).
For all i, j >= 1:
  A305800(i) = A305800(j) => A373983(i) = A373983(j) => a(i) = a(j).
For all i, j >= 0: a(i) = a(j) => A328771(i) = A328771(j).

Antti Karttunen, Table of n, a(n) for n = 0..100000
Index entries for sequences related to primorial numbers

Cf. A002110, A046523, A276085, A276086, A278226, A328768, A373983.

Cf. also A329045.

up_to = 100000;
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; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A278226(n) = A046523(A276086(n));
A002110(n) = prod(i=1,n,prime(i));
A328768(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*A002110(primepi(f[i,1])-1)/f[i, 1]));
v373982 = rgs_transform(vector(1+up_to, n, A278226(A328768(n-1))));
A373982(n) = v373982[1+n];

A328472 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278226(A328461(i)) = A278226(A328461(j)) for all i, j.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 4, 1, 2, 5, 4, 1, 4, 6, 7, 1, 2, 8, 4, 2, 4, 9, 7, 2, 4, 10, 7, 11, 7, 12, 13, 1, 2, 14, 4, 15, 4, 16, 7, 17, 4, 18, 7, 19, 7, 20, 13, 3, 4, 21, 7, 22, 7, 23, 13, 24, 7, 25, 13, 22, 13, 26, 27, 1, 2, 28, 4, 29, 4, 30, 7, 20, 4, 31, 7, 32, 7, 33, 13, 34, 4, 35, 7, 36, 7, 37, 13, 38, 7, 39, 13, 36, 13, 40, 27, 22, 4

Offset: 1

Antti Karttunen, Oct 18 2019

nonn

Restricted growth sequence transform of function f(n) = A046523(A276086(A276156(n)/A002110(A007814(n)))).

Cf. A002110, A007814, A046523, A276156, A276086, A278226, A328461, A328471.

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; };
A002110(n) = prod(i=1,n,prime(i));
A276156(n) = { my(p=2,pr=1,s=0); while(n,if(n%2,s += pr); n >>= 1; pr *= p; p = nextprime(1+p)); (s); };
A328461(n) = (A276156(n)/A002110(valuation(n,2)));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A278226(n) = A046523(A276086(n));
v328472 = rgs_transform(vector(up_to, n, A278226(A328461(n))));
A328472(n) = v328472[n];

A373983 Lexicographically earliest infinite sequence such that a(i) = a(j) = A246277(A324886(i)) = A246277(A324886(j)) and A278226(A328768(i)) = A278226(A328768(j)), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 25 2024

Restricted growth sequence transform of the ordered pair [A246277(A276086(A108951(n))), A046523(A276086(A328768(n)))].
For all i, j >= 1:
  A305800(i) = A305800(j) => a(i) = a(j),
  a(i) = a(j) => A329345(i) = A329345(j) => A329045(i) = A329045(j),
  a(i) = a(j) => A373982(i) = A373982(j) => A328771(i) = A328771(j).
It is hard to say for sure which graphical features in the scatter plot have their provenance in A373982, and which ones in A329345.

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences related to primorial numbers

Cf. A002110, A046523, A108951, A246277, A276086, A278226, A324886, A328768.
Cf. A305800, A329045, A329345, A328771, A373982.
Cf. also A329620.

up_to = 100000;
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; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };
A108951(n) = { my(f=factor(n)); prod(i=1, #f~,  prod(i=1, primepi(f[i, 1]), prime(i))^f[i, 2]); };
A246277(n) = if(1==n, 0, my(f = factor(n), k = primepi(f[1,1])-1); for (i=1, #f~, f[i,1] = prime(primepi(f[i,1])-k)); factorback(f)/2);
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A002110(n) = prod(i=1,n,prime(i));
A328768(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*A002110(primepi(f[i,1])-1)/f[i, 1]));
Aux373983(n) = [A246277(A276086(A108951(n))), A046523(A276086(A328768(n)))];
v373983 = rgs_transform(vector(up_to, n, Aux373983(n)));
A373983(n) = v373983[n];

A286382 Compound filter: a(n) = P(A257993(n), A278226(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

2, 5, 16, 12, 67, 9, 16, 23, 436, 80, 1771, 18, 67, 80, 1771, 668, 16111, 48, 277, 302, 7141, 2630, 64621, 156, 1129, 1178, 28681, 10442, 258841, 14, 16, 23, 436, 80, 1771, 31, 436, 467, 21946, 1832, 87991, 94, 1771, 1832, 87991, 16292, 793171, 328, 7141, 7262, 352381, 64982, 3173941, 1228, 28681, 28922, 1410361, 259562, 12698281, 25, 67, 80, 1771, 668, 16111

Offset: 1

Antti Karttunen, May 08 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..11550
Eric Weisstein's World of Mathematics, Pairing Function
Index entries for sequences related to primorial base

Cf. A000027, A257993, A278226, A286142.

Differs from A286381 for the first time at n=24, where a(24) = 156 while A286381(24) = 14.

(define (A286382 n) (* (/ 1 2) (+ (expt (+ (A257993 n) (A278226 n)) 2) (- (A257993 n)) (- (* 3 (A278226 n))) 2)))

a(n) = (1/2)*(2 + ((A257993(n)+A278226(n))^2) - A257993(n) - 3*A278226(n)).

A374032 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278226(A374031(i)) = A278226(A374031(j)), for all i, j >= 1.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 3, 4, 2, 2, 4, 2, 2, 4, 4, 2, 2, 2, 4, 3, 2, 2, 2, 4, 2, 2, 4, 2, 2, 2, 5, 4, 2, 4, 2, 2, 2, 4, 6, 2, 2, 2, 4, 2, 2, 2, 2, 4, 2, 4, 4, 2, 2, 3, 3, 7, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 4, 4, 2, 2, 2, 2, 2, 2, 4, 8, 2, 2, 2, 3, 2, 2, 2, 4, 2, 4, 3, 2, 2, 9, 4, 3, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 3, 6

Offset: 1

Antti Karttunen, Jun 27 2024

nonn

Restricted growth sequence transform of A278226(A374031(n)).
For all i, j >= 1:
  A305800(i) = A305800(j) => a(i) = a(j) => A374034(i) = A374034(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences related to primorial numbers

Cf. A002110, A046523, A276085, A276086, A278226, A328768, A374031.
Cf. A305800, A374034.
Cf. also A373982, A374033.

up_to = 100000;
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; };
A002110(n) = prod(i=1,n,prime(i));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A278226(n) = A046523(A276086(n));
A328768(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*A002110(primepi(f[i,1])-1)/f[i, 1]));
A374031(n) = gcd(A276085(n), A328768(n));
v374032 = rgs_transform(vector(up_to, n, A278226(A374031(n))));
A374032(n) = v374032[n];

A374033 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278226(A373985(i)) = A278226(A373985(j)), for all i, j >= 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 3, 3, 1, 1, 1, 1, 3, 2, 1, 2, 2, 2, 4, 1, 1, 1, 1, 1, 2, 2, 2, 5, 1, 2, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 3, 1, 1, 2, 6, 7, 3, 2, 1, 8, 1, 2, 1, 2, 6, 1, 1, 1, 2, 1, 1, 4, 1, 2, 1, 1, 6, 1, 1, 1, 7, 2, 1, 9, 2, 2, 3, 2, 1, 2, 6, 1, 3, 2, 2, 5, 1, 1, 10, 5, 1, 1, 1, 2, 1

Offset: 1

Antti Karttunen, Jun 27 2024

nonn

Restricted growth sequence transform of A278226(A373985(n)).
For all i, j >= 1:
  A305800(i) = A305800(j) => a(i) = a(j) => A373989(i) = A373989(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences related to primorial numbers

Cf. A002110, A046523, A108951, A276086, A278226, A373158, A373985.
Cf. A305800, A373989.
Cf. also A373982, A373983, A374032.

up_to = 100000;
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; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A278226(n) = A046523(A276086(n));
A373985(n) = { my(f=factor(n),m=1,s=0); for(i=1, #f~, my(x=prod(i=1,primepi(f[i, 1]),prime(i))); s += f[i, 2]*x; m *= x^f[i, 2]); gcd(m,s); };
v374033 = rgs_transform(vector(up_to, n, A278226(A373985(n))));
A374033(n) = v374033[n];

A374211 Lexicographically earliest infinite sequence such that for all i, j >= 1, a(i) = a(j) => f(i) = f(j), with f(1) = 1, and for n > 1, f(n) = [A278226(A328768(n)), A374212(n), A374213(n)], where A328768 is the first primorial based variant of the arithmetic derivative, and A374212 and A374213 are its 2- and 3-adic valuations.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 30 2024

nonn

Restricted growth sequence transform of the function f given in the definition.
For all i, j >= 1:
  A305900(i) = A305900(j) => a(i) = a(j),
  a(i) = a(j) => A152822(i) = A152822(j),
  a(i) = a(j) => A373982(i) = A373982(j) => A328771(i) = A328771(j),
  a(i) = a(j) => A373991(i) = A373991(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences related to primorial numbers

Cf. A002110, A278226, A328768, A374212, A374213.
Cf. A305900, A152822, A328771, A373982, A373991.
Cf. also A374131.

up_to = 100000;
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; };
A002110(n) = prod(i=1,n,prime(i));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A278226(n) = A046523(A276086(n));
A328768(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*A002110(primepi(f[i,1])-1)/f[i, 1]));
Aux374211(n) = if(1==n, n, my(u=A328768(n)); [A278226(u), valuation(u, 2), valuation(u, 3)]);
v374211 = rgs_transform(vector(up_to, n, Aux374211(n)));
A374211(n) = v374211[n];

A378230 Positions of 0's in A378226, where A278226 is XOR-Moebius transform of A318457, and A318457(n) = n XOR (sigma(n)-n).

Original entry on oeis.org

6, 14, 28, 42, 50, 62, 72, 114, 124, 150, 186, 248, 254, 376, 402, 412, 426, 434, 474, 496, 498, 508, 762, 784, 786, 796, 868, 938, 978, 994, 1002, 1016, 1302, 1528, 1568, 1578, 1626, 1778, 1834, 1852, 1888, 1948, 1988, 2032, 3056, 3064, 3094, 3282, 3350, 3556, 3568, 3644, 3682, 3794, 3858, 3868, 3882, 3954, 4064

Offset: 1

Antti Karttunen, Nov 26 2024

nonn

Positions of 0's in A378226.
Cf. A001065, A318457.
Subsequences: A000396 (at least the even terms).

A318457(n) = bitxor(n,sigma(n)-n);
A378226(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, A318457(d)))); (v); }
isA378230(n) = (0==A378226(n));

A378441 Fixed points of A378226, where A278226 is XOR-Moebius transform of A318457, and A318457(n) = n XOR (sigma(n)-n).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 11, 13, 16, 17, 19, 23, 27, 29, 31, 32, 37, 41, 43, 47, 53, 59, 61, 64, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 243, 251, 256, 257, 263, 269, 271, 277, 281

Offset: 1

Antti Karttunen, Nov 26 2024

nonn

Cf. A000040 (subsequence), A001065, A003987, A318457, A378226.

A318457(n) = bitxor(n,sigma(n)-n);
A378226(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, A318457(d)))); (v); }
isA378441(n) = (A378226(n)==n);

{k such that A378226(k) is equal to k}.

cp's OEIS Frontend

A286626 Restricted growth sequence computed for primorial base related filter-sequence A278226.

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A373982 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278226(A328768(i)) = A278226(A328768(j)), for all i, j >= 0.

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A328472 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278226(A328461(i)) = A278226(A328461(j)) for all i, j.

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A373983 Lexicographically earliest infinite sequence such that a(i) = a(j) = A246277(A324886(i)) = A246277(A324886(j)) and A278226(A328768(i)) = A278226(A328768(j)), for all i, j >= 1.

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A286382 Compound filter: a(n) = P(A257993(n), A278226(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A374032 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278226(A374031(i)) = A278226(A374031(j)), for all i, j >= 1.

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A374033 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278226(A373985(i)) = A278226(A373985(j)), for all i, j >= 1.

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A378230 Positions of 0's in A378226, where A278226 is XOR-Moebius transform of A318457, and A318457(n) = n XOR (sigma(n)-n).

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