A373364 -id:A373364

A373380

Lexicographically earliest infinite sequence such that a(i) = a(j) => A373145(i) = A373145(j), A373362(i) = A373362(j), and A373364(i) = A373364(j), for all i, j >= 1.

+20
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Restricted growth sequence transform of the triple [A373145(n), A373362(n), A373364(n)], i.e., the triple [gcd(x, y), gcd(x, z), gcd(y, z)], where x=A001414(n), y=A003415(n), z=A276085(n).

For all i, j >= 1:

A305800(i) = A305800(j) => a(i) = a(j),

a(i) = a(j) => A373367(i) = A373367(j).

LINKS

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

PROG

(PARI)

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; };

A001414(n) = ((n=factor(n))[, 1]~*n[, 2]);

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))); };

Aux373380(n) = { my(x=A001414(n), y=A003415(n), z=A276085(n)); [gcd(x, y), gcd(x, z), gcd(y, z)]; };

v373380 = rgs_transform(vector(up_to, n, Aux373380(n)));

A373380(n) = v373380[n];

CROSSREFS

Cf. A001414, A003415, A276085, A305800, A373145, A373362, A373364, A373367.

Cf. also A373150, A373379.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 03 2024

STATUS

approved

A373478

Numbers k such that A001414(k) and A003415(k) are both multiples of 3, where A001414 is the sum of prime factors with repetition, and A003415 is the arithmetic derivative.

+10
9

1, 8, 9, 14, 20, 26, 27, 35, 38, 44, 50, 62, 64, 65, 68, 72, 74, 77, 81, 86, 92, 95, 110, 112, 116, 119, 122, 125, 126, 134, 143, 146, 155, 158, 160, 161, 164, 170, 180, 185, 188, 194, 196, 203, 206, 208, 209, 212, 215, 216, 218, 221, 230, 234, 236, 242, 243, 254, 275, 278, 280, 284, 287, 290, 299, 302, 304, 305

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

A multiplicative semigroup; if m and n are in the sequence then so is m*n.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..20000

PROG

(PARI) isA373478 = A373477;

CROSSREFS

Cf. A001414, A003415, A373477 (characteristic function).

Positions of multiples of 3 in A373364.

Intersection of A289142 and A327863.

Disjoint union of A373475 and A373479.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 07 2024

STATUS

approved

A373369

a(n) = gcd(A001414(n), A059975(n)), where A001414 and A059975 are fully additive with a(p) = p and a(p) = p-1, respectively.

+10
7

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 2, 4, 1, 1, 1, 3, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 5, 2, 1, 2, 2, 1, 1, 2, 1, 1, 3, 1, 3, 1, 1, 1, 1, 2, 3, 2, 1, 1, 1, 2, 1, 2, 1, 1, 4, 1, 1, 1, 6, 2, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 4, 1, 1, 2, 2, 1, 2, 1, 1, 1, 2, 3, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

LINKS

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

PROG

(PARI)

A001414(n) = ((n=factor(n))[, 1]~*n[, 2]);

A059975(n) = {my(f = factor(n)); sum(i = 1, #f~, f[i, 2]*(f[i, 1] - 1)); };

A373369(n) = gcd(A001414(n), A059975(n));

CROSSREFS

Cf. A001414, A059975, A345452 (positions of even terms).

Cf. also A082299, A306709, A373362, A373363, A373364, A373365.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 05 2024

STATUS

approved

A373375

Numbers whose sum of prime factors (with repetition, A001414) and arithmetic derivative (A003415) are both even.

+10
5

1, 4, 8, 9, 15, 16, 21, 25, 32, 33, 35, 36, 39, 49, 51, 55, 57, 60, 64, 65, 69, 72, 77, 81, 84, 85, 87, 91, 93, 95, 100, 111, 115, 119, 120, 121, 123, 128, 129, 132, 133, 135, 140, 141, 143, 144, 145, 155, 156, 159, 161, 168, 169, 177, 183, 185, 187, 189, 196, 200, 201, 203, 204, 205, 209, 213, 215, 217, 219, 220, 221

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Numbers k such that sopfr(k) and k' are both even.

A multiplicative semigroup; if m and n are in the sequence then so is m*n.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000

PROG

(PARI) isA373375 = A373374;

CROSSREFS

Intersection of A036349 and A235992.

Disjoint union of A345452 and 8*A345452.

Positions of even terms in A373364.

Cf. A001414, A003415, A373374 (characteristic function), A373376 (complement).

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 03 2024

STATUS

approved

A373365

a(n) = gcd(A001414(n), A064097(n)), where A001414 is the sum of prime factors with repetition, and A064097 is a quasi-logarithm defined inductively by a(1) = 0 and a(p) = 1 + a(p-1) if p is prime and a(n*m) = a(n) + a(m) if m,n > 1.

+10
4

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 3, 1, 1, 2, 1, 5, 7, 1, 1, 2, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 8, 1, 2, 1, 1, 1, 1, 1, 1, 6, 2, 8, 1, 7, 1, 2, 1, 1, 1, 1, 1, 1, 9, 2, 1, 1, 4, 1, 1, 2, 2, 9, 1, 1, 1, 1, 1, 9, 1, 1, 3, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

As A001414 and A064097 are both fully additive sequences, all sequences that give the positions of multiples of some k > 1 in this sequence are closed under multiplication.

LINKS

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

PROG

(PARI)

A001414(n) = ((n=factor(n))[, 1]~*n[, 2]); \\ From A001414.

A064097(n) = if(1==n, 0, 1+A064097(n-(n/vecmin(factor(n)[, 1]))));

A373365(n) = gcd(A001414(n), A064097(n));

CROSSREFS

Cf. A001414, A064097.

Cf. also A082299, A373362, A373363, A373364, A373366.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 02 2024

STATUS

approved

A373367

a(n) is the greatest common divisor of A001414(n), A003415(n), and A276085(n).

+10
4

0, 1, 1, 2, 1, 1, 1, 3, 2, 7, 1, 1, 1, 1, 8, 4, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 5, 2, 1, 12, 2, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 1, 8, 1, 2, 1, 1, 2, 1, 1, 1, 6, 6, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 4, 1, 1, 2, 2, 1, 8, 1, 1, 1, 20, 1, 2, 1, 12, 1, 1, 1, 1, 14, 1, 1, 1, 1, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

All sequences that give the positions of multiples of some natural number k in this sequence are closed under multiplication because the constituent sequences A001414, A003415, and A276085 also have the same property.

A345452 gives the positions of even terms in this sequence, because it gives them for A373362, and because for A373145 and A373364 the positions of even terms are given by A368998 (union of A345452 and 2*A358776) and A373375 (union of A345452 and 8*A345452), thus both are supersets of A345452.

LINKS

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

FORMULA

a(n) = gcd(A373145(n), A373362(n)) = gcd(A373145(n), A373364(n)) = gcd(A373362(n), A373364(n)).

PROG

(PARI)

A001414(n) = ((n=factor(n))[, 1]~*n[, 2]); \\ From A001414.

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))); };

A373367(n) = gcd([A001414(n), A003415(n), A276085(n)]);

CROSSREFS

Cf. A001414, A003415, A276085, A345452 (gives the positions of even terms).

Greatest common divisor of any two of these three: A373145, A373362, A373364.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 03 2024

STATUS

approved

A373376

Numbers k such that gcd(A001414(k), A003415(k)) is odd, where A001414 is the sum of prime factors with repetition, and A003415 is the arithmetic derivative.

+10
4

2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 17, 18, 19, 20, 22, 23, 24, 26, 27, 28, 29, 30, 31, 34, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 52, 53, 54, 56, 58, 59, 61, 62, 63, 66, 67, 68, 70, 71, 73, 74, 75, 76, 78, 79, 80, 82, 83, 86, 88, 89, 90, 92, 94, 96, 97, 98, 99, 101, 102, 103, 104, 105, 106, 107, 108, 109

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Numbers k such that sopfr(k) and k' are not both even.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000

PROG

(PARI) isA373376(n) = !A373374(n);

CROSSREFS

Positions of odd terms in A373364, positions of 0's in A373374.

Complement of A373375.

Cf. A001414, A003415.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 03 2024

STATUS

approved

A373378

a(n) = gcd(A003415(n), A059975(n)), where A003415 is the arithmetic derivative and A059975 is fully additive with a(p) = p-1.

+10
4

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 4, 1, 1, 2, 4, 1, 1, 1, 6, 2, 1, 1, 1, 2, 1, 3, 8, 1, 1, 1, 5, 2, 1, 2, 6, 1, 1, 2, 1, 1, 1, 1, 12, 1, 1, 1, 2, 2, 9, 2, 14, 1, 1, 2, 1, 2, 1, 1, 4, 1, 1, 1, 6, 2, 1, 1, 18, 2, 1, 1, 1, 1, 1, 5, 20, 2, 1, 1, 8, 4, 1, 1, 2, 2, 1, 2, 1, 1, 3, 2, 24, 2, 1, 2, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16384

FORMULA

For n >= 1, a(n) is a multiple of A373377(n).

PROG

(PARI)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

A059975(n) = {my(f = factor(n)); sum(i = 1, #f~, f[i, 2]*(f[i, 1] - 1)); };

A373378(n) = gcd(A003415(n), A059975(n));

CROSSREFS

Cf. A003415, A059975.

Cf. A368998 (positions of even terms), A368999 (of odd terms).

Cf. also A373364, A373368, A373369, A373377.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 05 2024

STATUS

approved

A373374

a(n) = 1 if both A001414(n) and A003415(n) are even, otherwise 0, where A001414 is the sum of prime factors with repetition, and A003415 is the arithmetic derivative.

+10
3

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1

LINKS

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

Index entries for characteristic functions

FORMULA

a(n) = A059841(A373364(n)).

a(n) = A356163(n) * A358680(n).

a(n) = A353374(n) + A253513(n)*A353374(n/8). [With shortcut + and *]

PROG

(PARI)

A353374(n) = (!(bigomega(n)%2) && !(valuation(n, 2)%2));

A373374(n) = (A353374(n) || (!(n%8) && A353374(n/8)));

(PARI)

A001414(n) = ((n=factor(n))[, 1]~*n[, 2]); \\ From A001414.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

A373374(n) = !(gcd(A001414(n), A003415(n))%2);

CROSSREFS

Characteristic function of A373375, whose complement A373376 gives the positions of 0's.

Positions of even terms in A373364.

Cf. A001414, A003415, A059841, A253513, A353374, A356163, A358680.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 03 2024

STATUS

approved

A373379

Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j), A085731(i) = A085731(j) and A107463(i) = A107463(j), for all i, j >= 1.

+10
3

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, 47, 56, 2, 57, 58, 59, 2, 60, 41, 61, 62, 63, 2, 64, 37, 65, 66, 67, 68, 69, 2, 70, 71, 72

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Restricted growth sequence transform of the triple [A003415(n), A085731(n), A107463(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) => A373363(i) = A373363(j),

a(i) = a(j) => A373364(i) = A373364(j).

Starts to differ from A300235 at n=153. - R. J. Mathar, Jun 06 2024

LINKS

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

PROG

(PARI)

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);

A001414(n) = ((n=factor(n))[, 1]~*n[, 2]);

A107463(n) = if(n<=1, n, if(isprime(n), 1, A001414(n)));

Aux373379(n) = [A003415(n), A085731(n), A107463(n)];

v373379 = rgs_transform(vector(up_to, n, Aux373379(n)));

A373379(n) = v373379[n];

CROSSREFS

Cf. A001414, A003415, A085731, A107463, A305800, A369051, A373363, A373364.

Differs from A305895, A327931, and A353560 for the first time at n=1610, where a(1610) = 1112, while A305895(1610) = A327931(1610) = A353560(1610) = 1210.

Cf. also A373150, A373152, A373380.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 03 2024

STATUS

approved