A327931 -id:A327931 - cp's OEIS frontend

A351260 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j), A003557(i) = A003557(j) and A046523(i) = A046523(j), for all i, j >= 1.

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

Offset: 1

Antti Karttunen, Feb 06 2022

Restricted growth sequence transform of the triplet [A003415(n), A003557(n), A046523(n)].
For all i, j >= 1:
  A305800(i) = A305800(j) => a(i) = a(j),
  a(i) = a(j) => A294877(i) = A294877(j),
  a(i) = a(j) => A300249(i) = A300249(j),
  a(i) = a(j) => A344025(i) = A344025(j).

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

Cf. A003415, A003557, A046523.
Cf. also A305800, A294877, A300249, A344025.
Differs from A300235, A305895 and A327931 for the first time at n=105, where a(105) = 56, while A300235(105) = A305895(105) = A327931(105) = 75.
Differs from A300249 for the first time at n=425, where a(425) = 299, while A300249(425) = 198.

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) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A003557(n) = (n/factorback(factorint(n)[, 1]));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
Aux351260(n) = [A003415(n), A003557(n), A046523(n)];
v351260 = rgs_transform(vector(up_to,n,Aux351260(n)));
A351260(n) = v351260[n];

A353560 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j), A001065(i) = A001065(j) and A051953(i) = A051953(j), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Apr 29 2022

nonn

Restricted growth sequence transform of the triplet [A046523(n), A001065(n), A051953(n)].
For all i,j:
  A305800(i) = A305800(j) => a(i) = a(j),
  a(i) = a(j) => A300232(i) = A300232(j), [Combining A046523 and A051953]
  a(i) = a(j) => A300235(i) = A300235(j), [Combining A046523 and A001065]
  a(i) = a(j) => A305895(i) = A305895(j), [Combining A001065 and A051953]
  a(i) = a(j) => A353276(i) = A353276(j). [Needs all three components]

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A001065, A046523, A051953.
Cf. also A101296, A305800, A300232, A353276.
Differs from A300235 for the first time at n=153, where a(153) = 110, while A300235(153) = 106.
Differs from A305895 for the first time at n=3283, where a(3283) = 2502, while A305895(3283) = 1845.
Differs from A327931 for the first time at n=4433, where a(4433) = 2950, while A327931(4433) = 3393.
Differs from A300249 and from A351260 for the first time at n=105, where a(105) = 75, while A300249(105) = A351260(105) = 56.

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; };
A001065(n) = (sigma(n)-n);
A051953(n) = (n-eulerphi(n));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
Aux353560(n) = [A046523(n), A001065(n), A051953(n)];
v353560 = rgs_transform(vector(up_to,n,Aux353560(n)));
A353560(n) = v353560[n];

A373379 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j), A085731(i) = A085731(j) and A107463(i) = A107463(j), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 03 2024

nonn

Restricted growth sequence transform of the triple [A003415(n), A085731(n), A107463(n)].
For all i, j >= 1:
  A305800(i) = A305800(j) => a(i) = a(j),
  a(i) = a(j) => A369051(i) = A369051(j),
  a(i) = a(j) => A373363(i) = A373363(j),
  a(i) = a(j) => A373364(i) = A373364(j).
Starts to differ from A300235 at n=153. - R. J. Mathar, Jun 06 2024

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A001414, A003415, A085731, A107463, A305800, A369051, A373363, A373364.
Differs from A305895, A327931, and A353560 for the first time at n=1610, where a(1610) = 1112, while A305895(1610) = A327931(1610) = A353560(1610) = 1210.
Cf. also A373150, A373152, A373380.

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; };
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A085731(n) = gcd(A003415(n),n);
A001414(n) = ((n=factor(n))[, 1]~*n[, 2]);
A107463(n) = if(n<=1,n,if(isprime(n),1,A001414(n)));
Aux373379(n) = [A003415(n), A085731(n), A107463(n)];
v373379 = rgs_transform(vector(up_to, n, Aux373379(n)));
A373379(n) = v373379[n];

A327930 Product_{d|n, d>1} prime(A003415(d)), where A003415(x) gives the arithmetic derivative of x.

Original entry on oeis.org

1, 2, 2, 14, 2, 44, 2, 518, 26, 68, 2, 16324, 2, 92, 76, 67858, 2, 41756, 2, 42364, 116, 164, 2, 116569684, 58, 188, 2678, 84364, 2, 3609848, 2, 27753922, 172, 268, 148, 4353104756, 2, 292, 212, 528236716, 2, 10506584, 2, 256004, 164996, 388, 2, 9360895334252, 86, 388484, 284, 346108, 2, 1802063692, 212, 1495183172, 316

Offset: 1

Antti Karttunen, Sep 30 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A000040, A001221, A001222, A003415, A007814, A056239, A064989, A319356, A319684, A319685, A327931 (rgs-transform).

A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A327930(n) = { my(m=1); fordiv(n,d,if((d>1), m *= prime(A003415(d)))); (m); };

a(n) = Product_{d|n, d>1} A000040(A003415(d)).
For all n >= 2, a(n) = prime(A003415(n)) * A064989(A319356(n)).
A001221(a(n)) = A319685(n).
A001222(a(n)) = A032741(n).
A007814(a(n)) = A001221(n).
A056239(a(n)) = A319684(n).

cp's OEIS Frontend

A351260 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j), A003557(i) = A003557(j) and A046523(i) = A046523(j), for all i, j >= 1.

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A353560 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j), A001065(i) = A001065(j) and A051953(i) = A051953(j), for all i, j >= 1.

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A373379 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j), A085731(i) = A085731(j) and A107463(i) = A107463(j), for all i, j >= 1.

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A327930 Product_{d|n, d>1} prime(A003415(d)), where A003415(x) gives the arithmetic derivative of x.

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