A355833 -id:A355833 - cp's OEIS frontend

A342671 a(n) = gcd(sigma(n), A003961(n)), where A003961 is fully multiplicative with a(prime(k)) = prime(k+1), and sigma is the sum of divisors of n.

1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 21, 1, 3, 1, 15, 1, 3, 5, 1, 1, 3, 1, 9, 1, 3, 1, 1, 1, 3, 1, 9, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 1, 15, 1, 3, 5, 3, 1, 21, 1, 3, 1, 1, 7, 3, 1, 9, 1, 3, 1, 15, 1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 5, 9, 1, 3, 1, 3, 1, 3, 1, 9, 1, 3, 13, 7, 1, 3, 1, 3, 1

Offset: 1

Antti Karttunen, Mar 20 2021

nonn

Cf. A000203, A003961, A161942, A286385, A341529, A342672, A342673, A348992, A349161, A349162, A349163, A349164, A349165 (positions of 1's), A349166 (of terms > 1), A349167, A349756, A350071 [= a(n^2)], A355828 (Dirichlet inverse).
Cf. A349169, A349745, A355833, A355924 (applied onto prime shift array A246278).
Cf. also A336850, A354827/A354828, A355932.

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));

a(n) = gcd(A000203(n), A003961(n)).
a(n) = gcd(A000203(n), A286385(n)) = gcd(A003961(n), A286385(n)).
a(n) = A341529(n) / A342672(n).
From Antti Karttunen, Jul 21 2022: (Start)
a(n) = A003961(n) / A349161(n).
a(n) = A000203(n) / A349162(n).
a(n) = A161942(n) / A348992(n).
a(n) = A003961(A349163(n)) = A003961(n/A349164(n)).
(End)

A355924 Square array A(n,k) = A342671(A246278(n,k)), read by falling antidiagonals, where A342671(x) = gcd(sigma(x), A003961(x)).

Original entry on oeis.org

3, 1, 1, 3, 1, 1, 3, 1, 1, 1, 3, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 7, 1, 1, 17, 1, 1, 1, 21, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 19, 1, 1, 1, 1, 1, 37, 1, 1, 1

Offset: 1

Antti Karttunen, Jul 21 2022

			The top left corner of the array:
   n=  1  2  3   4  5  6  7   8  9  10 11  12  13  14 15 16 17 18 19  20 21
  2n=  2  4  6   8 10 12 14  16 18  20 22  24  26  28 30 32 34 36 38  40 42
-----+-----------------------------------------------------------------------
   1 | 3, 1, 3,  3, 3, 1, 3,  1, 3, 21, 3, 15,  3,  1, 3, 9, 3, 1, 3,  9, 3,
   2 | 1, 1, 1,  5, 1, 1, 1,  1, 1,  1, 1,  5,  1, 13, 1, 1, 5, 1, 1,  5, 1,
   3 | 1, 1, 1,  1, 1, 1, 7,  1, 1,  1, 1,  1,  1,  7, 1, 7, 1, 1, 1, 13, 7,
   4 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1,  1, 19, 1, 1, 1, 1, 1,  1, 1,
   5 | 1, 1, 1,  1, 1, 1, 1,  1, 1, 19, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
   6 | 1, 1, 1, 17, 1, 1, 1,  1, 1,  1, 1, 17,  1,  1, 1, 1, 1, 1, 1, 17, 1,
   7 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1, 19,  1, 1, 1, 1, 1, 1, 29, 1,
   8 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
   9 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
  10 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
  11 | 1, 1, 1, 37, 1, 1, 1,  1, 1,  1, 1, 37,  1,  1, 1, 1, 1, 1, 1, 37, 1,
  12 | 1, 1, 1,  1, 1, 1, 1, 41, 1,  1, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
  13 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
  14 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
  15 | 1, 1, 1,  1, 1, 1, 1,  1, 1, 61, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
  16 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
  17 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
  18 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
  19 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
  20 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,
  21 | 1, 1, 1,  1, 1, 1, 1,  1, 1,  1, 1,  1,  1,  1, 1, 1, 1, 1, 1,  1, 1,

Cf. A000203, A003961, A246278, A342671, A355833.

Cf. also A355925, A355926, A355927 for similarly constructed arrays.

up_to = 105;
A246278sq(row,col) = if(1==row,2*col, my(f = factor(2*col)); for(i=1, #f~, f[i,1] = prime(primepi(f[i,1])+(row-1))); factorback(f));
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));
A355924sq(row,col) = A342671(A246278sq(row,col));
A355924list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A355924sq(col,(a-(col-1))))); (v); };
v355924 = A355924list(up_to);
A355924(n) = v355924[n];

A(n,k) = A342671(A246278(n,k)).

A(n, k) = gcd(A246278(1+n,k), A355927(n, k)).

A355835 Lexicographically earliest infinite sequence such that a(i) = a(j) => A348717(i) = A348717(j) and A355442(i) = A355442(j) for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 20 2022

nonn

Restricted growth sequence transform of the ordered pair [A348717(n), A355442(n)].
For all i, j: a(i) = a(j) => A355836(i) = A355836(j).
Terms that occur in positions given by A355822 may occur only a finite number of times in this sequence. Most of these seem to be in the singular equivalence classes, i.e., have unique values, apart from exceptions like pairs {6, 15}, {273, 1729}, (see the examples and the array A355926). In a coarser variant A355836 multiple such finite equivalence classes may coalesce together into several infinite equivalence classes.

			a(6) = a(15) [= 5 as allotted by the rgs-transform] because 15 = A003961(6) [i.e., 15 is in the same column in prime shift array A246278 as 6 is], and because A355442(6) = A355442(15) = 5.
a(138) = a(435) [= 103 as allotted by the rgs-transform] because 435 = A003961(138), and A355442(138) = A355442(435) = 5.
a(273) = a(1729) [= 205 as allotted by the rgs-transform] because 1729 = A003961(A003961(273)) [i.e., 273 and 1729 are in the same column of A246278], and A355442(273) = A355442(1729) = 11.

Cf. A003961, A276086, A348717, A355442, A355821, A355822, A355926.

Cf. also A246278, A305801, A355833, A355834.

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; };
A348717(n) = if(1==n, 1, my(f = factor(n), k = primepi(f[1, 1])-1); for (i=1, #f~, f[i, 1] = prime(primepi(f[i, 1])-k)); factorback(f));
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A355442(n) = gcd(A003961(n), A276086(n));
Aux355835(n) = [A348717(n), A355442(n)];
v355835 = rgs_transform(vector(up_to,n,Aux355835(n)));
A355835(n) = v355835[n];

A355834 Lexicographically earliest infinite sequence such that a(i) = a(j) => A348717(i) = A348717(j) and A355931(i) = A355931(j) for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 20 2022

nonn

Restricted growth sequence transform of the ordered pair [A348717(n), A355931(n)], where A355931(n) = A000265(A009194(i)).

			a(450) = a(3675) [= 274 as allotted by rgs-transform] because A003961(450) = 3675, therefore 450 and 3675 are in the same column of the prime shift array A246278, and because A355931(450) = A355931(3675) = 3.
a(3185) = a(14399) [= 2020 as allotted by rgs-transform] because A003961(3185) = 14399 and A355931(3185) = A355931(14399) = 7.
a(5005) = a(17017) [= 3184 as allotted by rgs-transform] because A003961(5005) = 17017 and A355931(5005) = A355931(17017) = 7.

Cf. A000203, A000265, A009194, A246278, A348717, A355931.

Cf. also A300230, A305800, A305897, A355833.

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; };
A000265(n) = (n>>valuation(n,2));
A009194(n) = gcd(n, sigma(n));
A348717(n) = if(1==n, 1, my(f = factor(n), k = primepi(f[1, 1])-1); for (i=1, #f~, f[i, 1] = prime(primepi(f[i, 1])-k)); factorback(f));
Aux355834(n) = [A000265(A009194(n)), A348717(n)];
v355834 = rgs_transform(vector(up_to,n,Aux355834(n)));
A355834(n) = v355834[n];

cp's OEIS Frontend

A342671 a(n) = gcd(sigma(n), A003961(n)), where A003961 is fully multiplicative with a(prime(k)) = prime(k+1), and sigma is the sum of divisors of n.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A355924 Square array A(n,k) = A342671(A246278(n,k)), read by falling antidiagonals, where A342671(x) = gcd(sigma(x), A003961(x)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A355835 Lexicographically earliest infinite sequence such that a(i) = a(j) => A348717(i) = A348717(j) and A355442(i) = A355442(j) for all i, j >= 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A355834 Lexicographically earliest infinite sequence such that a(i) = a(j) => A348717(i) = A348717(j) and A355931(i) = A355931(j) for all i, j >= 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI