A286603 -id:A286603 - cp's OEIS frontend

A351037 Lexicographically earliest infinite sequence such that a(i) = a(j) => A000593(i) = A000593(j), for all i, j >= 1, where A000593 is the sum of odd divisors function.

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

Offset: 1

Antti Karttunen, Jan 31 2022

Restricted growth sequence transform of A000593.

Question: To which set of n does the horizontal stripe at around a(n) = ~8000 correspond in the scatter plot of this sequence?

			a(21) = a(31) = 11 because A000593(21) = A000593(31) = 32, and 32 is the 11th distinct value obtained by A000593.

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

Cf. A000593.

Cf. also A286603, A347374, A351036, A351040, A351090.

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; };
v351037 = rgs_transform(vector(up_to, n, sigma(n>>valuation(n,2))));
A351037(n) = v351037[n];

A275987 Least k such that sigma(k) = sigma(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 6, 12, 13, 14, 14, 16, 10, 18, 19, 20, 21, 22, 14, 24, 16, 20, 27, 28, 29, 30, 21, 32, 33, 34, 33, 36, 37, 24, 28, 40, 20, 42, 43, 44, 45, 30, 33, 48, 49, 50, 30, 52, 34, 54, 30, 54, 57, 40, 24, 60, 61, 42, 63, 64, 44, 66, 67, 68, 42, 66, 30, 72, 73, 74, 48

Offset: 1

Altug Alkan, Aug 15 2016

nonn

			a(11) = 6 because sigma(6) = sigma(11) = 12.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
Index entries for sequences related to sigma(n).

Cf. A000203, A069822, A286603.

With[{nn = 76}, With[{s = Values@ PositionIndex@ Array[DivisorSigma[1, #] &, nn]}, Array[s[[FirstPosition[s, #][[1]], 1 ]] &, nn]]] (* Michael De Vlieger, Nov 16 2017 *)

a(n) = {my(k = 1); while(sigma(k) != sigma(n), k++); k; }

a(n) = invsigmaMin(sigma(n)); \\ Amiram Eldar, Dec 20 2024, using Max Alekseyev's invphi.gp

A369260 Lexicographically earliest infinite sequence such that a(i) = a(j) => A342671(i) = A342671(j) and A349162(i) = A349162(j), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 25 2024

nonn

Restricted growth sequence transform of the ordered pair [A342671(n), A349162(n)], or equally, of the pair [A000203(n), A342671(n)], or equally, of the pair [A000203(n), A349162(n)].
For all i, j >= 1:
  A369259(i) = A369259(j) => a(i) = a(j) => A286603(i) = A286603(j).

Cf. A000203, A003961, A342671, A348993, A349162.

Cf. also A286603, A291751, A369259, A369261.

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; };
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A342671(n) = gcd(sigma(n), A003961(n));
Aux369260(n) = { my(u=A342671(n)); [u, sigma(n)/u]; };
v369260 = rgs_transform(vector(up_to, n, Aux369260(n)));
A369260(n) = v369260[n];

A369261 Lexicographically earliest infinite sequence such that a(i) = a(j) => A324644(i) = A324644(j) and A369445(i) = A369445(j), for all i, j >= 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 15, 23, 16, 20, 24, 25, 26, 27, 21, 28, 29, 30, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 29, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 35, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 62, 53, 51, 70, 71, 72, 58, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 63

Offset: 1

Antti Karttunen, Jan 25 2024

nonn

Restricted growth sequence transform of the ordered pair [A324644(n), A369445(n)], or equally, of the pair [A000203(n), A324644(n)], or equally, of the pair [A000203(n), A369445(n)].

For all i, j >= 1: a(i) = a(j) => A286603(i) = A286603(j).

Cf. A000203, A276086, A324644, A369445.

Cf. also A286603, A291751, A369260 and A353523, A369047, A369051, A369062.

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; };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A324644(n) = gcd(sigma(n),A276086(n));
Aux369261(n) = { my(u=A324644(n)); [u, sigma(n)/u]; };
v369261 = rgs_transform(vector(up_to, n, Aux369261(n)));
A369261(n) = v369261[n];

A296088 Filter combining sigma(n) with the parity of n; restricted growth sequence transform of ((-1)^n)*A000203(n).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 07 2017

nonn

			For n = 21 and 31 the restricted growth sequence transform assigns the same value (we have a(21) = a(31) = 21) because both numbers are odd, and the sum of their divisors is equal as sigma(21) = sigma(31) = 32.
On the other hand, although sigma(14) = sigma(15) = 24, a(14) != a(15) because the other number is even and the other number is odd. Compare to A286603.

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

Cf. A000035, A000203, A286603, A291761, A296089.

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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
write_to_bfile(1,rgs_transform(vector(up_to,n,((-1)^n)*sigma(n))),"b296088.txt");

A366294 Lexicographically earliest infinite sequence such that a(i) = a(j) => A326042(i) = A326042(j) for all i, j >= 1, where A326042(n) = A064989(sigma(A003961(n))).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Oct 07 2023

nonn

Restricted growth sequence transform of A326042.

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

Cf. A000203, A003961, A064989, A326042.

Cf. also A286603, A366295.

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; };
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A064989(n) = { my(f=factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A326042(n) = A064989(sigma(A003961(n)));
v366294  = rgs_transform(vector(up_to,n,A326042(n)));
A366294(n) = v366294[n];

A366295 Lexicographically earliest infinite sequence such that a(i) = a(j) => A349623(i) = A349623(j) for all i, j >= 1, where A349623 is the Dirichlet inverse of A064989(sigma(A003961(n))).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Oct 07 2023

nonn

Restricted growth sequence transform of A349623.

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

Cf. A000203, A003961, A064989, A326042, A349623.

Cf. also A286603, A366294.

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; };
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A064989(n) = { my(f=factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A326042(n) = A064989(sigma(A003961(n)));
v366295 = rgs_transform(DirInverseCorrect(vector(up_to,n,A326042(n))));
A366295(n) = v366295[n];

A369259 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003557(i) = A003557(j), A048250(i) = A048250(j) and A342671(i) = A342671(j), for all i, j >= 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 15, 23, 24, 25, 26, 27, 28, 29, 21, 30, 31, 32, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 29, 31, 43, 44, 45, 46, 47, 48, 49, 46, 50, 51, 52, 53, 54, 55, 39, 56, 57, 58, 59, 60, 61, 62, 59, 46, 63, 64, 65, 66, 67, 62, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 59

Offset: 1

Antti Karttunen, Jan 25 2024

nonn

Restricted growth sequence transform of the triplet [A003557(j), A048250(i), A342671(n)].
For all i, j >= 1:
  a(i) = a(j) => A323368(i) = A323368(j) => A291751(i) = A291751(j),
  a(i) = a(j) => A369260(i) = A369260(j) => A286603(i) = A286603(j).

Cf. A000203, A001615, A003557, A003961, A048250, A286603, A291751, A369260.

Differs from related A296089 and A323368 for the first time at n=79, where a(79) = 69, while A296089(79) = A323368(79) = 51.

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; };
A003557(n) = (n/factorback(factor(n)[, 1]));
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A048250(n) = if(n<1, 0, sumdiv(n, d, if(core(d)==d, d)));
A342671(n) = gcd(sigma(n), A003961(n));
Aux369259(n) = [A003557(n), A048250(n), A342671(n)];
v369259 = rgs_transform(vector(up_to, n, Aux369259(n)));
A369259(n) = v369259[n];

A374485 Lexicographically earliest infinite sequence such that a(i) = a(j) => A350388(i) = A350388(j) and A351569(i) = A351569(j), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 06 2024

nonn

Restricted growth sequence transform of the ordered pair [A350388(n), A351569(n)].

For all i, j >= 1: a(i) = a(j) => A000203(i) = A000203(j).

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

Cf. A000203, A350388, A350389, A351569.

Cf. also A286603, A291751.

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; };
A350388(n) = { my(m=1, f=factor(n)); for(k=1,#f~,if(0==(f[k,2]%2), m *= (f[k,1]^f[k,2]))); (m); };
A350389(n) = { my(m=1, f=factor(n)); for(k=1,#f~,if(1==(f[k,2]%2), m *= (f[k,1]^f[k,2]))); (m); };
A351569(n) = sigma(A350389(n));
Aux374485(n) = [A350388(n), A351569(n)];
v374485 = rgs_transform(vector(up_to, n, Aux374485(n)));
A374485(n) = v374485[n];

cp's OEIS Frontend

A351037 Lexicographically earliest infinite sequence such that a(i) = a(j) => A000593(i) = A000593(j), for all i, j >= 1, where A000593 is the sum of odd divisors function.

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A275987 Least k such that sigma(k) = sigma(n).

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Mathematica

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A369260 Lexicographically earliest infinite sequence such that a(i) = a(j) => A342671(i) = A342671(j) and A349162(i) = A349162(j), for all i, j >= 1.

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A369261 Lexicographically earliest infinite sequence such that a(i) = a(j) => A324644(i) = A324644(j) and A369445(i) = A369445(j), for all i, j >= 1.

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A296088 Filter combining sigma(n) with the parity of n; restricted growth sequence transform of ((-1)^n)*A000203(n).

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A366294 Lexicographically earliest infinite sequence such that a(i) = a(j) => A326042(i) = A326042(j) for all i, j >= 1, where A326042(n) = A064989(sigma(A003961(n))).

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A366295 Lexicographically earliest infinite sequence such that a(i) = a(j) => A349623(i) = A349623(j) for all i, j >= 1, where A349623 is the Dirichlet inverse of A064989(sigma(A003961(n))).

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A369259 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003557(i) = A003557(j), A048250(i) = A048250(j) and A342671(i) = A342671(j), for all i, j >= 1.

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A374485 Lexicographically earliest infinite sequence such that a(i) = a(j) => A350388(i) = A350388(j) and A351569(i) = A351569(j), for all i, j >= 1.

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