A083345 -id:A083345

A373258

a(n) = 1 if n' / gcd(n,n') is a multiple of 4 and A276085(n) is a multiple of 8, otherwise 0, where n' stands for the arithmetic derivative of n, A003415, and A276085 is the primorial base log-function.

+10
3

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

(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) = A369004(n) * A373137(n).

PROG

(PARI)

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)); };

A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };

A373258(n) = (!(A083345(n)%4) && !(A276085(n)%8));

CROSSREFS

Characteristic function of A373259.

Cf. A002110, A003415, A083345, A276085, A369004, A373137.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 02 2024

STATUS

approved

A373264

a(n) = 1 if n' / gcd(n,n') is of the form 4m+2, otherwise 0, where n' stands for the arithmetic derivative of n, A003415.

+10
3

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

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

OFFSET

1

LINKS

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

Index entries for characteristic functions

FORMULA

a(n) = [A083345(n) == 2 (mod 4)], where [ ] is the Iverson bracket.

a(n) = A369001(n) - A369004(n).

PROG

(PARI)

A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };

A373264(n) = (2==(A083345(n)%4));

CROSSREFS

Characteristic function of A373265.

Cf. A003415, A083345, A369001, A369004.

KEYWORD

nonn,easy

AUTHOR

Antti Karttunen, Jun 01 2024

STATUS

approved

A189036

a(n)= lcm(n,n')/gcd(n,n'), where n' is the arithmetic derivative of n.

+10
2

0, 2, 3, 1, 5, 30, 7, 6, 6, 70, 11, 12, 13, 126, 120, 2, 17, 42, 19, 30, 210, 286, 23, 66, 10, 390, 1, 56, 29, 930, 31, 10, 462, 646, 420, 15, 37, 798, 624, 170, 41, 1722, 43, 132, 195, 1150, 47, 21, 14, 90, 1020, 182, 53, 6, 880, 322, 1254, 1798, 59, 345, 61, 2046, 357, 3, 1170, 4026, 67, 306, 1794, 4130, 71, 78, 73, 2886, 165, 380, 1386, 5538, 79, 55, 12, 3526, 83, 651, 1870, 3870, 2784, 770, 89, 1230, 1820, 552, 3162, 4606, 2280, 102, 97, 154, 825, 35

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

OFFSET

1,2

COMMENTS

Least common multiple of n and its arithmetic derivative divided by greatest common divisor of n and its arithmetic derivative.

LINKS

Table of n, a(n) for n=1..100.

FORMULA

a(n) = lcm(n,n')/gcd(n,n') = A086130(n)/A085731(n).

EXAMPLE

n = 8, n'= 12, lcm(8,12)= 24, gcd(8,12)= 4, hence a(8)=24/4 = 6.

CROSSREFS

Cf. A003415, A085731, A086130, A083345, A083346.

KEYWORD

nonn

AUTHOR

Giorgio Balzarotti, Apr 15 2011

STATUS

approved

A369006

a(n) = 1 if there is no prime p such that p^p divides n' / gcd(n,n'), and 0 otherwise, where n' stands for the arithmetic derivative of n, A003415(n).

+10
2

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

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

OFFSET

1

COMMENTS

Question: Is there a formula for the asymptotic mean, which seems to be around 0.813...? Consider A369004 and A369007.

LINKS

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

Index entries for characteristic functions

FORMULA

a(1) = 0, for n > 1, a(n) = A359550(A083345(n)).

A368914(n) <= a(n) <= 1 - A369004(n).

PROG

(PARI)

A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };

A359550(n) = { my(f = factor(n)); prod(k=1, #f~, (f[k, 2]<f[k, 1])); };

A369006(n) = if(1==n, 0, A359550(A083345(n)));

CROSSREFS

Cf. A003415, A083345, A359550, A368914, A368916 [= a(A276086(n))], A369004, A369007.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jan 14 2024

STATUS

approved

A369457

Lexicographically earliest infinite sequence such that a(i) = a(j) => A369456(i) = A369456(j) for all i, j >= 0.

+10
2

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

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

OFFSET

0,2

COMMENTS

Restricted growth sequence transform of A369456.

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..65537

PROG

(PARI)

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; };

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };

A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };

A369456(n) = A083345(A005940(1+n));

v369457 = rgs_transform(vector(1+up_to, n, A369456(n-1)));

A369457(n) = v369457[1+n];

CROSSREFS

Cf. A005940, A083345, A369456.

Cf. also A366802, A369065, A369067.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jan 27 2024

STATUS

approved

A369961

a(n) = 1 if gcd(n, A003415(n)) is equal to gcd(n, A276086(n)), otherwise 0, where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

+10
2

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

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

OFFSET

0

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..100000

Index entries for characteristic functions

Index entries for sequences related to primorial base

FORMULA

a(n) = [A085731(n) == A324198(n)], where [ ] is the Iverson bracket.

PROG

(PARI)

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); };

A369961(n) = (gcd(n, A003415(n))==gcd(n, A276086(n)));

(PARI)

A085731(n) = { my(f=factor(n)); for(i=1, #f~, if (f[i, 2] % f[i, 1], f[i, 2]--); ); factorback(f); };

A324198(n) = { my(m=1, p=2, orgn=n); while(n, m *= (p^min(n%p, valuation(orgn, p))); n = n\p; p = nextprime(1+p)); (m); };

A369961(n) = ((n>0) && (A085731(n)==A324198(n)));

CROSSREFS

Characteristic function of A369962.

Cf. A003415, A083345, A085731, A276086, A324198, A351251.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Feb 07 2024

STATUS

approved

A369977

Numbers that are found in A369002, but not in its subsequence A369976.

+10
2

81, 144, 225, 256, 400, 441, 625, 729, 784, 972, 1089, 1215, 1225, 1521, 1620, 1701, 1728, 1936, 2160, 2268, 2401, 2601, 2673, 2700, 2704, 2835, 2880, 3024, 3025, 3072, 3159, 3249, 3375, 3564, 3600, 3840, 4032, 4096, 4131, 4212, 4225, 4455, 4500, 4617, 4624, 4725, 4752, 4761, 4800, 5040, 5120, 5265, 5292, 5376

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

OFFSET

1,1

COMMENTS

a(10) = 972 = 2^2 * 3^5 is the first term that is not a square.

LINKS

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

PROG

(PARI)

A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };

A369001(n) = !(A083345(n)%2);

memoA369974 = Map();

A369974(n) = if(1==n, 1, my(v); if(mapisdefined(memoA369974, n, &v), v, v = -sumdiv(n, d, if(d<n, A369001(n/d)*A369974(d), 0)); mapput(memoA369974, n, v); (v)));

A369975(n) = (A369974(n)%2);

isA369977(n) = (A369001(n) && !A369975(n));

CROSSREFS

Setwise difference A369002 \ A369976.

Cf. A083345, A369001, A369974, A369975.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Feb 09 2024

STATUS

approved

A369980

Run lengths in A369001.

+10
2

1, 7, 1, 2, 1, 2, 2, 3, 2, 3, 1, 2, 1, 4, 1, 1, 1, 3, 1, 4, 1, 4, 1, 1, 2, 2, 1, 1, 1, 7, 1, 2, 2, 6, 2, 3, 1, 3, 1, 1, 1, 3, 3, 1, 1, 12, 1, 2, 1, 3, 2, 2, 1, 1, 1, 1, 2, 4, 1, 3, 1, 1, 1, 5, 1, 1, 3, 2, 1, 6, 1, 3, 1, 1, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 2, 1, 1, 1, 1, 3, 2, 1, 8, 1, 1, 1, 1, 1, 3, 1, 2, 2, 1, 1

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

OFFSET

1,2

LINKS

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

PROG

(PARI)

up_to = 105;

A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };

A369001(n) = !(A083345(n)%2);

A369980list(up_to) = { my(v=vector(up_to), oc=A369001(1), nc, n=0, on=1, k=0); while(k<up_to, n++; nc = A369001(n); if(nc!=oc, oc=nc; k++; v[k] = (n-on); on=n)); (v); }

v369980 = A369980list(up_to);

A369980(n) = v369980[n];

CROSSREFS

Cf. A083345, A369001.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Feb 09 2024

STATUS

approved

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.

+10
2

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

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

OFFSET

1,2

COMMENTS

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).

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]));

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];

CROSSREFS

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.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 09 2024

STATUS

approved

A374040

Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A003415(n), A085731(n), A007814(n), A007949(n)], for all i, j >= 1.

+10
2

1, 2, 3, 4, 5, 6, 5, 7, 8, 9, 5, 10, 5, 11, 12, 13, 5, 14, 5, 15, 16, 17, 5, 18, 19, 20, 21, 22, 5, 23, 5, 24, 25, 26, 27, 28, 5, 29, 30, 31, 5, 32, 5, 33, 34, 35, 5, 36, 37, 38, 39, 40, 5, 41, 42, 43, 44, 45, 5, 46, 5, 47, 48, 49, 50, 51, 5, 52, 53, 54, 5, 55, 5, 56, 57, 58, 50, 59, 5, 60, 61, 62, 5, 63, 64, 65, 66, 67, 5, 68, 69, 70, 71, 72, 73, 74, 5, 75

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

OFFSET

1,2

COMMENTS

Restricted growth sequence transform of the quadruple [A003415(n), A085731(n), A007814(n), A007949(n)].

For all i, j >= 1:

A305900(i) = A305900(j) => a(i) = a(j),

a(i) = a(j) => A322026(i) = A322026(j),

a(i) = a(j) => A369051(i) = A369051(j) => A083345(i) = A083345(j),

a(i) = a(j) => b(i) = b(j), where b can be any of the sequences listed at the crossrefs-section, under "some of the other matched sequences".

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]));

Aux374040(n) = { my(d=A003415(n)); [d, gcd(n, d), valuation(n, 2), valuation(n, 3)]; };

v374040 = rgs_transform(vector(up_to, n, Aux374040(n)));

A374040(n) = v374040[n];

CROSSREFS

Cf. A003415, A007814, A007949, A085731.

Some of the other matched sequences (see comments): A083345, A359430, A369001, A369004, A369643, A369658, A373143, A373474, A373483.

Cf. also A322026, A353521, A369051, A373268, A372573, A374131 for similar and related constructions.

Differs from A305900 first at n=77, where a(77) = 50, while A305900(77) = 59.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jul 01 2024

STATUS

approved