A360519 -id:A360519 - cp's OEIS frontend

A361118 a(n) = gcd(A360519(n), A360519(n+1)).

1, 2, 5, 7, 3, 4, 5, 11, 3, 2, 7, 11, 3, 5, 2, 11, 13, 3, 4, 7, 13, 5, 2, 17, 7, 9, 2, 13, 17, 3, 2, 19, 5, 3, 4, 11, 17, 5, 2, 23, 3, 19, 4, 13, 3, 5, 2, 29, 3, 31, 2, 7, 3, 37, 2, 17, 3, 41, 2, 5, 23, 7, 12, 5, 29, 7, 2, 3, 43, 5, 2, 3, 47, 5, 2, 3, 7, 19, 2

Offset: 1

Scott R. Shannon, Rémy Sigrist and N. J. A. Sloane, Mar 03 2023

nonn

			a(7) = gcd(A360519(7), A360519(8)) = gcd(20, 55) = 5.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program

Cf. A360519.

PARI
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a(n) > 1 for any n > 1.

A361103 a(n) = k such that A360519(k) = A361102(n), or -1 if A361102(n) never appears in A360519.

Original entry on oeis.org

1, 2, 3, 6, 11, 14, 10, 7, 5, 16, 19, 28, 20, 23, 9, 24, 4, 27, 32, 18, 15, 31, 36, 34, 40, 35, 39, 30, 44, 68, 8, 52, 42, 48, 64, 51, 26, 22, 72, 56, 41, 47, 76, 55, 46, 43, 12, 80, 60, 59, 63, 38, 84, 49, 88, 87, 21, 92, 50, 96, 33, 91, 67, 13, 71, 95, 100, 53, 104, 99, 75, 54, 112, 108

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Mar 02 2023

nonn

Conjectured to be a permutation of the natural numbers (and if so, -1 will never appear). See A361104 for the putative inverse permutation.

Scott R. Shannon, Table of n, a(n) for n = 0..10000

Cf. A360519, A361102, A336957, A361104.

A361107 Records in A360519.

Original entry on oeis.org

1, 6, 10, 35, 55, 77, 99, 143, 221, 235, 301, 329, 371, 391, 497, 511, 623, 1243, 1253, 1379, 1393, 1799, 1837, 1969, 2513, 2629, 3353, 3493, 3601, 3983, 6259, 8063, 10417, 12991, 13453, 16003, 17413, 21967, 23089, 27049, 32329, 33737, 40079, 60073, 70103, 73411, 79673, 105131, 116677, 117799, 119933, 124619, 128227, 130537, 149083

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 03 2023

nonn

Scott R. Shannon, Table of n, a(n) for n = 1..116. The record values for the first 1 million terms of A360519.

Cf. A360519, A361108.

A361109 After A360519(n) has been found, a(n) is the smallest member of C (A361102) that is missing from A360519.

Original entry on oeis.org

6, 10, 12, 12, 12, 14, 14, 14, 14, 14, 15, 15, 15, 22, 22, 24, 24, 24, 26, 26, 26, 26, 26, 26, 26, 26, 26, 38, 38, 38, 38, 44, 44, 44, 44, 46, 46, 46, 46, 52, 52, 52, 52, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54, 54

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 03 2023

nonn

			After we have calculated A360519(4) = 35, the smallest term of C that is missing from A360519 is 12, so a(4) = 12.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program
N. J. A. Sloane, Table showing A360519(1)-A360519(13), also the smallest missing number (smn, A361109 and A361110), binary vectors showing which terms are divisible by the primes 2, 3, 5, 7, 11; and phi, a decimal representation of those binary vectors (A361111). This sequence forms the third row of the table.

Cf. A360519, A361110.

PARI
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More terms from Rémy Sigrist, Mar 03 2023

A361111 The binary expansion of a(n) specifies which primes divide A360519(n).

Original entry on oeis.org

0, 3, 5, 12, 10, 3, 5, 20, 18, 3, 9, 24, 18, 6, 5, 17, 48, 34, 3, 9, 40, 36, 7, 65, 72, 10, 3, 33, 96, 66, 11, 129, 132, 6, 3, 17, 80, 68, 5, 257, 258, 130, 129, 33, 34, 6, 13, 513, 514, 1026, 1025, 9, 14, 2050, 2049, 65, 66, 4098, 4097, 5, 260, 264, 11, 7

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 03 2023

			A360519(6) = 12, which is divisible by 2, 3, but not 5, 7, 11, ... So we write down 1, 1, 0, 0, 0, .... Thus a(6) has binary expansion ...00011, and so a(6) = 3.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program
N. J. A. Sloane, Table showing A360519(1)-A360519(13), also the smallest missing number (smn, A361109 and A361110), binary vectors showing which terms are divisible by the primes 2, 3, 5, 7, 11; and phi, a decimal representation of those binary vectors (A361111). This sequence forms the bottom row of the table.

Cf. A087207, A360519.

PARI
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a(n) = A087207(A360519(n)). - Rémy Sigrist, Mar 03 2023

More terms from Rémy Sigrist, Mar 03 2023

A361119 a(n) is the least prime factor of A360519(n) with a(1) = 1.

Original entry on oeis.org

1, 2, 2, 5, 3, 2, 2, 5, 3, 2, 2, 7, 3, 3, 2, 2, 11, 3, 2, 2, 7, 5, 2, 2, 7, 3, 2, 2, 13, 3, 2, 2, 5, 3, 2, 2, 11, 5, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 5, 7, 2, 2, 5, 7, 2, 2, 3, 5, 2, 2, 3, 5, 2, 2, 3, 7, 2, 2, 3, 5, 2, 2, 7, 5

Offset: 1

Scott R. Shannon, Rémy Sigrist and N. J. A. Sloane, Mar 03 2023

nonn

See A361120 for the greatest prime factors.

			A360519(2) = 2*3 so a(2) = 2.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program

Cf. A020639, A360519, A361120.

PARI
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See Links section.
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a(n) = A020639(A360519(n)).

A361120 a(n) is the greatest prime factor of A360519(n) with a(1) = 1.

Original entry on oeis.org

1, 3, 5, 7, 7, 3, 5, 11, 11, 3, 7, 11, 11, 5, 5, 11, 13, 13, 3, 7, 13, 13, 5, 17, 17, 7, 3, 13, 17, 17, 7, 19, 19, 5, 3, 11, 17, 17, 5, 23, 23, 19, 19, 13, 13, 5, 7, 29, 29, 31, 31, 7, 7, 37, 37, 17, 17, 41, 41, 5, 23, 23, 7, 5, 29, 29, 7, 3, 43, 43, 5, 11, 47

Offset: 1

Scott R. Shannon, Rémy Sigrist and N. J. A. Sloane, Mar 03 2023

See A361119 for the least prime factors.

			A360519(2) = 2*3 so a(2) = 3.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program

Cf. A006530, A360519, A361119.

PARI
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a(n) = A006530(A360519(n)).

A361128 Let b = A360519; let Lg = gcd(b(n-1),b(n)), Rg = gcd(b(n),b(n+1)); let L(n) = prod_{primes p|Lg} p-part of b(n), R(n) = prod_{primes p|Rg} p-part of b(n), M(n) = b(n)/(L(n)*R(n)); sequence gives L(n).

Original entry on oeis.org

1, 2, 5, 7, 3, 4, 5, 11, 9, 2, 7, 11, 3, 5, 2, 11, 13, 3, 4, 7, 13, 5, 2, 17, 7, 9, 2, 13, 17, 3, 2, 19, 5, 3, 4, 11, 17, 25, 2, 23, 3, 19, 4, 13, 3, 5, 2, 29, 3, 31, 8, 7, 3, 37, 4, 17, 3, 41, 16, 5, 23, 7, 12, 5, 29, 49, 2, 3, 43, 25, 2, 3, 47, 5, 8, 3, 7, 19, 2, 27, 5, 31

Offset: 2

Scott R. Shannon, Rémy Sigrist, and N. J. A. Sloane, Mar 09 2023

nonn

The p-part of a number k is the highest power of p that divides k. For example, the 2-part of 24 is 8, the 3-part is 3.
One can think of A360519 as a chain of circles, each circle linked to its neighbors to the left and the right. The n-th term b(n) = A360519(n) is a product L(n)*M(n)*R(n), where L(n) is the part of b(n) sharing primes with the term to the left, R(n) the part sharing primes with the term to the right, and M(n) is the rest of b(n).
By definition of A360519, the set of primes in L(n) is disjoint from the primes in R(n).

N. J. A. Sloane, Table of n, a(n) for n = 2..20000

Cf. A360519, A361118, A361129, A361130.

# Suppose bW is a list of the terms of A360519.
# Then f3(bW[n-1], bW[n], bW[n+1]); returns [L(n), M(n), R(n)] where:
with(numtheory);
f3:=proc(a,b,c)
local lefta,midb,rightc,i,p,pa,pc,ta,tb,tc,t1,t2;
ta:=a; tb:=b; tc:=c;
# left
t1:=igcd(a,b);
t2:=factorset(t1);
t2:=convert(t2,list);
lefta:=1;
for i from 1 to nops(t2) do
p:=t2[i];
while (tb mod p) = 0 do lefta:=lefta*p; tb:=tb/p; od;
od:
# right
t1:=igcd(b,c);
t2:=factorset(t1);
t2:=convert(t2,list);
rightc:=1;
for i from 1 to nops(t2) do
p:=t2[i];
while (tb mod p) = 0 do rightc:=rightc*p; tb:=tb/p; od;
od:
# middle
midb:=b/(lefta*rightc);
[lefta,midb,rightc];
end; # N. J. A. Sloane, Mar 09 2023

A361129 Let b = A360519; let Lg = gcd(b(n-1),b(n)), Rg = gcd(b(n),b(n+1)); let L(n) = prod_{primes p|Lg} p-part of b(n), R(n) = prod_{primes p|Rg} p-part of b(n), M(n) = b(n)/(L(n)*R(n)); sequence gives M(n).

Original entry on oeis.org

3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 11, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 5, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1

Offset: 2

Scott R. Shannon, Rémy Sigrist, and N. J. A. Sloane, Mar 09 2023

nonn

The p-part of a number k is the highest power of p that divides k. For example, the 2-part of 24 is 8, the 3-part is 3.
Since so many of the initial terms are 1, we show more than the usual number of terms in the DATA section.
Conjecture: All terms are odd, and every odd number eventually appears.

N. J. A. Sloane, Table of n, a(n) for n = 2..20000

Cf. A360519, A361118, A361128, A361130.

A361130 Let b = A360519; let Lg = gcd(b(n-1),b(n)), Rg = gcd(b(n),b(n+1)); let L(n) = prod_{primes p|Lg} p-part of b(n), R(n) = prod_{primes p|Rg} p-part of b(n), M(n) = b(n)/(L(n)*R(n)); sequence gives R(n).

Original entry on oeis.org

2, 5, 7, 3, 4, 5, 11, 3, 2, 7, 11, 9, 5, 8, 11, 13, 3, 8, 7, 13, 5, 2, 17, 7, 9, 4, 13, 17, 3, 2, 19, 5, 9, 16, 11, 17, 5, 2, 23, 3, 19, 4, 13, 9, 25, 2, 29, 3, 31, 2, 7, 3, 37, 2, 17, 9, 41, 2, 5, 23, 7, 12, 5, 29, 7, 2, 27, 43, 5, 4, 3, 47, 5, 2, 9, 49, 19, 8, 3, 5, 31, 4, 43, 7

Offset: 2

Scott R. Shannon, Rémy Sigrist, and N. J. A. Sloane, Mar 09 2023

nonn

The p-part of a number k is the highest power of p that divides k. For example, the 2-part of 24 is 8, the 3-part is 3.

N. J. A. Sloane, Table of n, a(n) for n = 2..20000

Cf. A360519, A361118, A361128, A361129.

cp's OEIS Frontend

A361118 a(n) = gcd(A360519(n), A360519(n+1)).

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A361103 a(n) = k such that A360519(k) = A361102(n), or -1 if A361102(n) never appears in A360519.

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A361107 Records in A360519.

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A361109 After A360519(n) has been found, a(n) is the smallest member of C (A361102) that is missing from A360519.

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PARI

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A361111 The binary expansion of a(n) specifies which primes divide A360519(n).

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PARI

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A361119 a(n) is the least prime factor of A360519(n) with a(1) = 1.

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PARI

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A361120 a(n) is the greatest prime factor of A360519(n) with a(1) = 1.

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A361128 Let b = A360519; let Lg = gcd(b(n-1),b(n)), Rg = gcd(b(n),b(n+1)); let L(n) = prod_{primes p|Lg} p-part of b(n), R(n) = prod_{primes p|Rg} p-part of b(n), M(n) = b(n)/(L(n)*R(n)); sequence gives L(n).

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Maple

A361129 Let b = A360519; let Lg = gcd(b(n-1),b(n)), Rg = gcd(b(n),b(n+1)); let L(n) = prod_{primes p|Lg} p-part of b(n), R(n) = prod_{primes p|Rg} p-part of b(n), M(n) = b(n)/(L(n)*R(n)); sequence gives M(n).

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