A328390 -id:A328390

A328114

Maximal digit value used when n is written in primorial base (cf. A049345).

+10
54

0, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

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

OFFSET

0,5

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..30030

Index entries for sequences related to primorial base

FORMULA

a(n) = A051903(A276086(n)).

a(A276156(n)) = 1 for all n >= 1.

a(n) <= A276150(n) for all n >= 0.

From Antti Karttunen, Oct 29 2019: (Start)

a(n) = A061395(A328835(n)).

For n >= 1, a(n) < A000040(A235224(n)) and a(n) <= 1 + A328391(n).

For all n >= 1, a(n) = 1+A051903(A328572(n)).

a(A276086(n)) = A328389(n), a(A276087(n)) = A328394(n), a(A328403(n)) = A328398(n).

a(A327860(n)) = A328392(n), a(A003415(n)) = A328390(n), a(A328316(n)) = A328322(n).

(End)

EXAMPLE

For n = 2105, which could be expressed in primorial base for example as "T0021" (where T here stands for the digit value ten), or maybe more elegantly as [10,0,0,2,1] as 2105 = 10*A002110(4) + 2*A002110(1) + 1*A002110(0). The maximum value of these digits is 10, thus a(2105) = 10.

MATHEMATICA

With[{b = MixedRadix[Reverse@ Prime@ Range@ 20]}, Array[Max@ IntegerDigits[#, b] &, 105, 0]] (* Michael De Vlieger, Oct 30 2019 *)

PROG

(PARI) A328114(n) = { my(i=0, m=0, pr=1, nextpr); while((n>0), i=i+1; nextpr = prime(i)*pr; if((n%nextpr), m = max(m, (n%nextpr)/pr); n-=(n%nextpr)); pr=nextpr); (m); };

(PARI) A328114(n) = { my(s=0, p=2); while(n, s = max(s, (n%p)); n = n\p; p = nextprime(1+p)); (s); }; \\ (Faster, no unnecessary construction of primorials) - Antti Karttunen, Oct 29 2019

CROSSREFS

Cf. A002110, A049345, A051903, A061395, A276086, A267263, A276150, A327969, A328316, A328322, A328389, A328390, A328392, A328394, A328398, A328403, A328835.

Cf. A276156 (indices of terms < 2).

KEYWORD

nonn

AUTHOR

Antti Karttunen, Oct 12 2019

STATUS

approved

A328389

Maximal digit value in primorial base expansion of A276086(n): a(n) = A328114(A276086(n)).

+10
13

1, 1, 1, 1, 1, 3, 2, 2, 2, 1, 2, 3, 4, 3, 2, 5, 2, 2, 4, 2, 5, 4, 5, 10, 6, 6, 8, 6, 5, 9, 1, 2, 3, 2, 2, 4, 2, 2, 3, 1, 3, 3, 5, 4, 3, 5, 7, 4, 4, 8, 3, 3, 4, 9, 9, 8, 7, 11, 4, 8, 3, 3, 4, 4, 3, 4, 2, 2, 3, 7, 10, 10, 5, 4, 6, 3, 8, 9, 7, 5, 10, 10, 10, 8, 5, 5, 8, 6, 9, 7, 4, 4, 6, 9, 4, 7, 8, 5, 3, 5, 7, 4, 7, 7, 11, 9

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

OFFSET

0,6

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..32589

Index entries for sequences related to primorial base

FORMULA

a(n) = A328114(A276086(n)).

a(n) = A051903(A276087(n)).

PROG

(PARI)

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };

A328114(n) = { my(s=0, p=2); while(n, s = max(s, (n%p)); n = n\p; p = nextprime(1+p)); (s); };

A328389(n) = A328114(A276086(n));

CROSSREFS

Cf. A002110, A051903, A143293, A276086, A276087, A328114, A328322, A328390, A328391, A328392, A328395.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Oct 15 2019

STATUS

approved

A328391

Maximal exponent in the prime factorization of A327860(n): a(n) = A051903(A327860(n)).

+10
13

0, 0, 1, 1, 1, 0, 1, 3, 1, 1, 1, 1, 2, 1, 1, 4, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 0, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 2, 2, 7, 2, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2

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

OFFSET

1,8

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..30030

Index entries for sequences related to primorial base

FORMULA

a(n) = A051903(A327860(n)) = A051903(A003415(A276086(n))).

a(A002110(n)) = 0 for all n >= 0.

For all n >= 1, a(n) >= A328114(n)-1. [Because arithmetic derivative will decrease the maximal prime exponent (A051903) of its argument by at most one]

PROG

(PARI)

A051903(n) = if((1==n), 0, vecmax(factor(n)[, 2]));

A327860(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= (p^e); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };

A328391(n) = A051903(A327860(n));

CROSSREFS

Cf. A002110, A003415, A051903, A276086, A327860, A327969, A328114, A328388, A328389, A328390, A328392.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Oct 15 2019

STATUS

approved

A328392

Maximal digit value in primorial base expansion of A327860(n): a(n) = A328114(A327860(n)).

+10
8

0, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 4, 2, 2, 4, 6, 1, 3, 2, 2, 4, 5, 6, 7, 3, 7, 10, 7, 6, 9, 1, 1, 2, 2, 3, 5, 2, 4, 2, 1, 3, 4, 3, 5, 2, 6, 9, 4, 3, 10, 3, 6, 8, 2, 8, 5, 7, 7, 5, 10, 2, 2, 3, 4, 2, 6, 4, 2, 6, 9, 3, 4, 4, 5, 6, 9, 6, 7, 5, 7, 9, 5, 11, 10, 4, 3, 5, 11, 16, 7, 4, 3, 5, 2, 5, 6, 5, 7, 4, 6, 8, 10, 5, 9, 12, 11

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

OFFSET

0,4

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..30030

Index entries for sequences related to primorial base

FORMULA

a(n) = A328114(A327860(n)) = A328390(A276086(n)) = A328114(A003415(A276086(n))).

PROG

(PARI)

A327860(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= (p^e); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };

A328114(n) = { my(s=0, p=2); while(n, s = max(s, (n%p)); n = n\p; p = nextprime(1+p)); (s); };

A328392(n) = A328114(A327860(n));

CROSSREFS

Cf. A002110, A003415, A276086, A327859, A327860, A328114, A328389, A328390, A328391, A328396.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Oct 15 2019

STATUS

approved

A341518

Numbers k such that the primorial base representation of their arithmetic derivative does not contain digits larger than 1.

+10
6

0, 1, 2, 3, 5, 7, 9, 10, 11, 13, 14, 15, 16, 17, 19, 23, 28, 29, 30, 31, 37, 41, 43, 45, 47, 53, 58, 59, 61, 62, 67, 71, 73, 74, 79, 83, 87, 89, 97, 101, 103, 107, 108, 109, 112, 113, 127, 131, 136, 137, 139, 149, 151, 155, 157, 161, 163, 167, 173, 179, 181, 189, 191, 193, 197, 198, 199, 203, 209, 210, 211, 212, 217

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

OFFSET

1,3

COMMENTS

Numbers k for which A328390(k) <= 1, numbers k such that A003415(k) is in A276156.

Numbers k such that A327859(k) = A276086(A003415(k)) is squarefree.

LINKS

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

Index entries for sequences related to primorial base

FORMULA

For all n > 2, A328390(a(n)) = A328114(A003415(a(n))) = 1.

PROG

(PARI)

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

ismaxprimobasedigit_at_most(n, k) = { my(s=0, p=2); while(n, if((n%p)>k, return(0)); n = n\p; p = nextprime(1+p)); (1); };

isA341518(n) = ismaxprimobasedigit_at_most(A003415(n), 1); \\ Antti Karttunen, Feb 03 2024

CROSSREFS

Cf. A003415, A005117, A049345, A276086, A276156, A327859, A328114, A328390.

Positions of nonzero terms in A341517.

Subsequences: A000040, A327978, A328232, A369647 (terms k where A051903(k) obtains novel values).

Cf. also A327969.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Feb 28 2021

STATUS

approved

A341517

a(n) = mu(A327859(n)), where mu is the Möbius function, A008683.

+10
5

1, 1, -1, -1, 0, -1, 0, -1, 0, -1, 1, -1, 0, -1, -1, 1, 1, -1, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, -1, 0, 1, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1, -1, 0, -1, -1, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, -1, 1, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, -1, 0, 0

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

OFFSET

0

LINKS

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

Index entries for sequences related to primorial base

FORMULA

a(n) = A008683(A327859(n)) = A008683(A276086(A003415(n))).

For all n > 1, abs(a(n)) = [A328390(n)==1], where [ ] is the Iverson bracket.

a(p) = -1 for all primes p.

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); };

A327859(n) = A276086(A003415(n));

A341517(n) = moebius(A327859(n));

CROSSREFS

Cf. A003415, A008683, A276086, A327859, A328390.

Absolute values give the characteristic function sequence for A341518.

KEYWORD

sign

AUTHOR

Antti Karttunen, Feb 28 2021

STATUS

approved

A351098

Numbers k such that the maximal digit value in primorial base expansion of the arithmetic derivative of k is less than the maximal exponent in the prime factorization of k.

+10
5

8, 9, 16, 24, 28, 32, 40, 45, 48, 81, 96, 108, 112, 120, 125, 128, 136, 160, 184, 189, 192, 198, 208, 212, 225, 236, 244, 250, 256, 270, 288, 296, 352, 361, 459, 507, 625, 640, 768, 800, 832, 864, 896, 928, 960, 972, 1008, 1024, 1056, 1088, 1104, 1120, 1152, 1168, 1184, 1232, 1272, 1280, 1320, 1344, 1350, 1408, 1440

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

OFFSET

1,1

COMMENTS

Numbers k such that A328390(k) < A051903(k).

Numbers k for which A051903(A327859(n)) < A051903(k).

These seem to be rarer than A351075. All terms are nonsquarefree.

LINKS

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

Index entries for sequences related to primorial base

PROG

(PARI)

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

A051903(n) = if((1==n), 0, vecmax(factor(n)[, 2]));

A328114(n) = { my(s=0, p=2); while(n, s = max(s, (n%p)); n = n\p; p = nextprime(1+p)); (s); };

isA351098(n) = (A328114(A003415(n)) < A051903(n));

(PARI) \\ Or see A369637.

CROSSREFS

Positions of negative terms in A351097. Subsequence of A013929 and of A351099.

Cf. A003415, A051903, A276086, A327859, A328390, A369637 (characteristic function).

Cf. also A351075.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Feb 03 2022

STATUS

approved

A351097

Difference between {the maximal digit value in primorial base expansion of the arithmetic derivative of n} and {the maximal exponent in the prime factorization of n}.

+10
4

0, 0, 0, 0, 0, 1, 0, -1, -1, 0, 0, 0, 0, 0, 0, -3, 0, 1, 0, 2, 1, 1, 0, -1, 0, 1, 1, -1, 0, 0, 0, -2, 1, 2, 1, 0, 0, 2, 1, -1, 0, 1, 0, 1, -1, 3, 0, -1, 0, 0, 2, 2, 0, 0, 1, 0, 2, 0, 0, 1, 0, 0, 1, 0, 2, 1, 0, 0, 3, 3, 0, 2, 0, 0, 2, 1, 2, 1, 0, 1, -1, 1, 0, 2, 2, 1, 0, 1, 0, 2, 2, 1, 1, 2, 3, -3, 0, 0, 0, 2, 0, 2, 0, 2, 1

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

OFFSET

1,16

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..30030

FORMULA

a(n) = A328390(n) - A051903(n) = A051903(A327859(n)) - A051903(n).

PROG

(PARI)

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

A051903(n) = if((1==n), 0, vecmax(factor(n)[, 2]));

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };

A327859(n) = A276086(A003415(n));

A351097(n) = (A051903(A327859(n)) - A051903(n));

CROSSREFS

Cf. A003415, A051903, A276086, A327859, A328390, A351098 (positions of negative terms), A351099.

Cf. also A350074, A351074.

KEYWORD

sign

AUTHOR

Antti Karttunen, Feb 03 2022

STATUS

approved

A351099

Composite numbers k such that the maximal digit value in primorial base expansion of the arithmetic derivative of k is not larger than the maximal exponent in the prime factorization of k.

+10
2

4, 8, 9, 10, 12, 14, 15, 16, 24, 25, 28, 30, 32, 36, 40, 45, 48, 49, 50, 54, 56, 58, 62, 64, 68, 74, 81, 87, 96, 98, 99, 108, 112, 120, 125, 128, 136, 155, 156, 160, 161, 162, 184, 189, 192, 196, 198, 203, 204, 208, 209, 210, 212, 217, 220, 221, 224, 225, 236, 244, 246, 247, 250, 252, 256, 268, 270, 272, 280, 282, 288

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

OFFSET

1,1

COMMENTS

Composite k such that A328390(k) <= A051903(k).

Composite k for which A051903(A327859(n)) <= A051903(k).

LINKS

Table of n, a(n) for n=1..71.

PROG

(PARI)

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

A051903(n) = if((1==n), 0, vecmax(factor(n)[, 2]));

A328114(n) = { my(s=0, p=2); while(n, s = max(s, (n%p)); n = n\p; p = nextprime(1+p)); (s); };

isA351099(n) = (n>1&&!isprime(n)&&(A328114(A003415(n)) <= A051903(n)));

CROSSREFS

Cf. A003415, A051903, A276086, A327859, A328390, A351097, A351098 (subsequence).

KEYWORD

nonn

AUTHOR

Antti Karttunen, Feb 03 2022

STATUS

approved

A369637

a(n) = 1 if the maximal digit in the primorial base representation of n' is less than the maximal exponent in the prime factorization of n, where n' stands for the arithmetic derivative of n, A003415.

+10
2

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

Index entries for sequences related to primorial base

FORMULA

a(n) = [A328390(n) < A051903(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]));