A373151 -id:A373151 - cp's OEIS frontend

A373485 a(n) = gcd(A083345(n), A276085(n)), where A276085 is fully additive with a(p) = p#/p, and A083345 is the numerator of the fully additive function with a(p) = 1/p.

0, 1, 1, 1, 1, 1, 1, 3, 2, 7, 1, 4, 1, 1, 8, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 8, 1, 1, 1, 5, 2, 1, 12, 1, 1, 1, 8, 1, 1, 1, 1, 4, 1, 1, 1, 1, 2, 1, 4, 2, 1, 1, 8, 1, 2, 1, 1, 1, 1, 1, 17, 3, 6, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 4, 6, 1, 1, 1, 4, 1, 1, 1, 2, 1, 8, 1, 1, 1, 20, 4, 2, 1, 12, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Jun 09 2024

nonn

For all n >= 1, A373145(n) is a multiple of a(n).

For all i, j: A373151(i) = A373151(j) => a(i) = a(j) => A373483(i) = A373483(j).

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

Cf. A083345, A276085, A373145, A373151.

Cf. A369002 (positions of even terms), A369003 (of odd terms), A373483, A373484 (of multiples of 3).

A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, 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))); };
A373485(n) = gcd(A083345(n), A276085(n));

A373150 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(1) = 1, and for n>1, f(n) = [A003415(n), A085731(n), A373148(n)], 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, 56, 57, 2, 58, 59, 60, 2, 61, 62, 63, 64, 65, 2, 66, 67, 68, 69, 70, 71, 72, 2, 73, 74

Offset: 1

Antti Karttunen, May 27 2024

nonn

Restricted growth sequence transform of the function f defined as: f(1) = 1, and for n>1, f(n) = [A003415(n), A085731(n), A373148(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) => A373151(i) = A373151(j) => A373143(i) = A373143(j).

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

Cf. A003415, A002110, A085731, A276085, A373148.
Cf. A305800, A369051, A373143, A373151.
Differs from A369050 for the first time at n=91, where a(91)=67, while A369050(91)=37.
Differs from A300833 for the first time at n=133, where a(133)=133, while A300833(133)=50.

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);
A002110(n) = prod(i=1,n,prime(i));
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };
A373148(n) = (A276085(n)%A003415(n));
Aux373150(n) = if(1==n,1,[A003415(n), A085731(n), A373148(n)]);
v373150 = rgs_transform(vector(up_to, n, Aux373150(n)));
A373150(n) = v373150[n];

A374131 Lexicographically earliest infinite sequence such that for all i, j >= 1, a(i) = a(j) => f(i) = f(j), where f(1) = 1, and for n > 1, f(n) = [A083345(n), A374132(n), A374133(n)], where A083345 is the numerator of the fully additive function with a(p) = 1/p, and A374132 and A374133 are the 2- and 3-adic valuations of A276085, which is fully additive with a(p) = p#/p.

Original entry on oeis.org

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

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) => A035263(i) = A035263(j),
  a(i) = a(j) => A369001(i) = A369001(j),
  a(i) = a(j) => A369004(i) = A369004(j),
  a(i) = a(j) => A372573(i) = A372573(j),
  a(i) = a(j) => A373137(i) = A373137(j),
  a(i) = a(j) => A373258(i) = A373258(j),
  a(i) = a(j) => A373483(i) = A373483(j).

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

Cf. A083345, A374132, A374133.
Cf. A035263, A305900, A369001, A369004, A372573, A373137, A373258, A373483.
Cf. also A373151.

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; };
A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, 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))); };
Aux374131(n) = if(1==n, n, my(u=A276085(n)); [A083345(n), valuation(u, 2), valuation(u, 3)]);
v374131 = rgs_transform(vector(up_to, n, Aux374131(n)));
A374131(n) = v374131[n];

A373152 Lexicographically earliest infinite sequence such that a(i) = a(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, 2, 2, 4, 5, 6, 2, 7, 2, 2, 8, 9, 2, 10, 2, 11, 12, 2, 2, 13, 14, 2, 15, 16, 2, 2, 2, 17, 12, 2, 18, 19, 2, 2, 8, 13, 2, 2, 2, 7, 10, 2, 2, 20, 21, 22, 23, 11, 2, 24, 8, 13, 12, 2, 2, 3, 2, 2, 25, 26, 27, 2, 2, 11, 12, 2, 2, 28, 2, 2, 22, 29, 27, 2, 2, 20, 30, 2, 2, 3, 12, 2, 8, 13, 2, 10, 31, 7, 12, 2, 18, 32, 2, 33, 10, 34, 2, 2, 2, 13, 2

Offset: 1

Antti Karttunen, Jun 03 2024

nonn

Restricted growth sequence transform of the ordered pair [A085731(n), A373145(n)], i.e., the ordered pair [gcd(n, A003415(n)), gcd(A003415(n), A276085(n))].

For all i, j >= 1: A373150(i) = A373150(j) => a(i) = a(j).

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

Cf. A085731, A276085, A373145.

Cf. also A373150, A373151.

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))); };
Aux373152(n) = [gcd(n, A003415(n)), gcd(A003415(n), A276085(n))];
v373152 = rgs_transform(vector(up_to, n, Aux373152(n)));
A373152(n) = v373152[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];

A374480 Lexicographically earliest infinite sequence such that a(i) = a(j) => A083345(i) = A083345(j) and A343223(i) = A343223(j), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 06 2024

nonn

Restricted growth sequence transform of the ordered pair [A083345(n), A343223(n)].

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

Cf. A003415, A083345, A342926, A343223.

Cf. also A305800, A373151, A374131.

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; };
A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A342926(n) = (A003415(sigma(n))-n);
A343223(n) = gcd(A003415(n), A342926(n));
Aux374480(n) = [A083345(n), A343223(n)];
v374480 = rgs_transform(vector(up_to, n, Aux374480(n)));
A374480(n) = v374480[n];

cp's OEIS Frontend

A373485 a(n) = gcd(A083345(n), A276085(n)), where A276085 is fully additive with a(p) = p#/p, and A083345 is the numerator of the fully additive function with a(p) = 1/p.

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A373150 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(1) = 1, and for n>1, f(n) = [A003415(n), A085731(n), A373148(n)], for all i, j >= 1.

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A373152 Lexicographically earliest infinite sequence such that a(i) = a(j) => A085731(i) = A085731(j) and A373145(i) = A373145(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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A374480 Lexicographically earliest infinite sequence such that a(i) = a(j) => A083345(i) = A083345(j) and A343223(i) = A343223(j), for all i, j >= 1.

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