A351236 -id:A351236 - cp's OEIS frontend

A345000 a(n) = gcd(A003415(n), A003415(A276086(n))), where A003415(n) is the arithmetic derivative of n, and A276086(n) gives the prime product form of primorial base expansion of n.

0, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 16, 1, 3, 1, 2, 5, 1, 1, 4, 5, 5, 1, 2, 1, 1, 1, 10, 1, 1, 3, 12, 1, 1, 1, 2, 1, 1, 1, 4, 1, 5, 1, 2, 1, 5, 5, 4, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 12, 3, 1, 1, 2, 1, 1, 1, 12, 1, 1, 55, 10, 3, 1, 1, 16, 1, 1, 1, 2, 1, 5, 1, 140, 1, 3, 1, 16, 1, 49, 3, 2, 1, 7, 1, 28, 1, 7, 1, 2, 1

Offset: 0

Antti Karttunen, Jul 21 2021

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to primorial base

Cf. A003415, A276086, A327860, A347958 (inverse Möbius transform), A347959, A351083, A351085, A351086, A351235, A351236.
Cf. A166486 (a(n) mod 2, parity of terms, see comment in A327860).
Cf. also A324198, A327858.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A345000(n) = gcd(A003415(n), A003415(A276086(n)));

a(n) = gcd(A003415(n), A327860(n)) = gcd(A003415(n), A003415(A276086(n))).

A344025 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j) and A003557(i) = A003557(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, 21, 2, 42, 2, 43, 44, 45, 46, 47, 2, 48, 49, 50, 2, 51, 2, 52, 53, 54, 46, 55, 2, 56, 57, 58, 2, 59, 41, 60, 61, 62, 2, 63, 37, 64

Offset: 1

Antti Karttunen, May 07 2021

nonn

Restricted growth sequence transform of the ordered pair [A003415(n), A003557(n)], where A003415(n) is the arithmetic derivative of n, and A003557(n) is n divided by its largest squarefree divisor.
For all i, j:
  parent(i) = parent(j) => a(i) = a(j),
  a(i) = a(j) => A342001(i) = A342001(j),
  a(i) = a(j) => A369051(i) = A369051(j) => A085731(i) = A085731(j).
Where "parent" can be any of the sequences A351236, A351260, A353520, A353521, A369050, for example.

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

Cf. A003415, A003557, A085731, A342001, A369051.

Cf. also A305800, A351236, A351260, A353520, A353521, A369050, and also A353523.

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) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
Aux344025(n) = [A003415(n), A003557(n)];
v344025 = rgs_transform(vector(up_to, n, Aux344025(n)));
A344025(n) = v344025[n];

A351235 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j), A327858(i) = A327858(j) and A345000(i) = A345000(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, 7, 2, 15, 16, 17, 18, 8, 2, 19, 2, 20, 21, 7, 22, 23, 2, 24, 10, 25, 2, 19, 2, 26, 27, 28, 2, 29, 30, 31, 14, 32, 2, 33, 10, 25, 10, 7, 2, 34, 2, 9, 27, 35, 36, 19, 2, 13, 10, 19, 2, 37, 2, 9, 38, 39, 36, 19, 2, 40, 41, 7, 2, 34, 10, 17, 10, 42, 2

Offset: 1

Antti Karttunen, Feb 06 2022

Restricted growth sequence transform of the triplet [A046523(n), A327858(n), A345000(n)].
For all i, j >= 1:
  A305800(i) = A305800(j) => a(i) = a(j) => A351085(i) = A351085(j).

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

Cf. A003415, A046523, A276086, A327858, A345000, A351086.

Cf. also A101296, A305800, A329620, A351085, A351236.

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
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); };
A327858(n) = gcd(A003415(n),A276086(n));
A345000(n) = gcd(A003415(n),A003415(A276086(n)));
Aux351235(n) = [A046523(n), A327858(n), A345000(n)];
v351235 = rgs_transform(vector(up_to,n,Aux351235(n)));
A351235(n) = v351235[n];

A369046 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j) and A069359(i) = A069359(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, 21, 2, 42, 2, 43, 44, 45, 46, 47, 2, 48, 49, 50, 2, 51, 2, 52, 53, 54, 46, 55, 2, 56, 57, 58, 2, 59, 41, 60, 61, 62, 2, 63, 37

Offset: 1

Antti Karttunen, Jan 18 2024

nonn

Restricted growth sequence transform of the ordered pair [A003415(n), A069359(n)].
For all i,j >= 1:
  A369050(i) = A369050(j) => a(i) = a(j), [Conjectured]
  a(i) = a(j) => A329039(i) = A329039(j) => A008966(i) = A008966(j).

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

Cf. A003415, A008966, A069359, A329039.
Cf. also A369050, A369051.
Differs from A351236 for the first time at n=91, where a(91) = 37, while A351236(91) = 64.
Differs from A344025 for the first time at n=140, where a(140) = 97, while A344025(140) = 92.

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) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A069359(n) = (n*sumdiv(n, d, isprime(d)/d));
Aux369046(n) = [A003415(n), A069359(n)];
v369046 = rgs_transform(vector(up_to, n, Aux369046(n)));
A369046(n) = v369046[n];

A369047 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j) and A345000(i) = A345000(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, 14, 17, 18, 19, 2, 20, 2, 21, 22, 23, 24, 25, 2, 26, 27, 28, 2, 29, 2, 30, 31, 32, 2, 33, 22, 34, 35, 36, 2, 37, 27, 38, 39, 20, 2, 38, 2, 40, 41, 42, 43, 44, 2, 45, 46, 47, 2, 48, 2, 31, 49, 21, 43, 50, 2, 51, 52, 53, 2, 54, 39, 34, 55, 56, 2, 57

Offset: 1

Antti Karttunen, Jan 21 2024

nonn

Restricted growth sequence transform of the ordered pair [A003415(n), A345000(n)], or equally, of the pair [A003415(n), A369038(n)], or equally, of the pair [A345000(n), A369038(n)].
For all i, j >= 1:
  A351236(i) = A351236(j) => a(i) = a(j),
  a(i) = a(j) => A369034(i) = A369034(j),
  a(i) = a(j) => A369036(i) = A369036(j).

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

Cf. A003415, A345000, A369034, A369036, A369038.

Cf. also A351236, A369051.

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) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A345000(n) = gcd(A003415(n),A003415(A276086(n)));
Aux369047(n) = [A003415(n), A345000(n)];
v369047 = rgs_transform(vector(up_to, n, Aux369047(n)));
A369047(n) = v369047[n];

A373268 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j), A085731(i) = A085731(j) and A373145(i) = A373145(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, 21, 2, 42, 2, 43, 44, 45, 46, 47, 2, 48, 49, 50, 2, 51, 2, 52, 53, 54, 46, 55, 2, 56, 57, 58, 2, 59, 41, 60, 61, 62, 2, 63, 64, 65, 66, 67, 68, 69, 2, 70, 71, 72, 2, 73, 2, 74, 55

Offset: 1

Antti Karttunen, Jun 09 2024

nonn

Restricted growth sequence transform of the triple [A003415(n), A085731(n), A373145(n)].
For all i, j >= 1:
  A373150(i) = A373150(j) => a(i) = a(j),
  a(i) = a(j) => A373151(i) = A373151(j) => A373485(i) = A373485(j),
  a(i) = a(j) => A373152(i) = A373152(j),
  a(i) = a(j) => A373486(i) = A373486(j).

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

Cf. A003415, A083345, A085731, A276085, A373145.
Cf. A373150, A373151, A373152, A373485, A373486.
Differs from A344025 and A369046 for the first time at n=91, where a(91) = 64, while A344025(91) = A369046(91) = 37.
Differs from A351236 for the first time at n=143, where a(143) = 100, while A351236(143) = 68.

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]));
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*prod(i=1,primepi(f[k, 1]-1),prime(i))); };
Aux373268(n) = { my(d=A003415(n)); [d, gcd(d,n), gcd(d, A276085(n))]; };
v373268 = rgs_transform(vector(up_to, n, Aux373268(n)));
A373268(n) = v373268[n];

A369448 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j), A327858(i) = A327858(j) and A359589(i) = A359589(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, 14, 17, 18, 11, 2, 19, 2, 20, 21, 22, 23, 24, 2, 25, 26, 27, 2, 28, 2, 29, 30, 31, 2, 32, 21, 33, 34, 35, 2, 36, 26, 37, 38, 19, 2, 37, 2, 39, 40, 41, 42, 43, 2, 44, 45, 46, 2, 47, 2, 30, 48, 49, 42, 50, 2, 51, 52, 53, 2, 54, 38, 33, 55

Offset: 1

Antti Karttunen, Jan 26 2024

nonn

Restricted growth sequence transform of the triplet [A003415(n), A327858(n), A359589(n)].

For all i, j: A305800(i) = A305800(j) => a(i) = a(j) => A366297(i) = A366297(j).

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

Cf. A003415, A276086, A305800, A327858, A359589, A359595, A366297.

Cf. also A351236, A369047.

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(dA003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A327858(n) = gcd(A003415(n), A276086(n));
v359589 = DirInverseCorrect(vector(up_to,n,A327858(n)-1));
A359589(n) = v359589[n];
Aux369448(n) = [A003415(n), A327858(n), A359589(n)];
v369448 = rgs_transform(vector(up_to, n, Aux369448(n)));
A369448(n) = v369448[n];

cp's OEIS Frontend

A345000 a(n) = gcd(A003415(n), A003415(A276086(n))), where A003415(n) is the arithmetic derivative of n, and A276086(n) gives the prime product form of primorial base expansion of n.

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A344025 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j) and A003557(i) = A003557(j), for all i, j >= 1.

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A351235 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j), A327858(i) = A327858(j) and A345000(i) = A345000(j) for all i, j >= 1.

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A369046 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j) and A069359(i) = A069359(j), for all i, j >= 1.

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A369047 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j) and A345000(i) = A345000(j), for all i, j >= 1.

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A373268 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j), A085731(i) = A085731(j) and A373145(i) = A373145(j), for all i, j >= 1.

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A369448 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j), A327858(i) = A327858(j) and A359589(i) = A359589(j), for all i, j >= 1.

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