A276651 -id:A276651

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A276655

Numbers j such that Sum_{p|j} 0.p is an integer where p ranges over the prime divisors of j.

+10
9

1, 21, 30, 60, 63, 90, 120, 147, 150, 180, 189, 240, 270, 300, 360, 441, 450, 480, 540, 567, 600, 720, 750, 810, 900, 960, 979, 1029, 1080, 1200, 1323, 1350, 1411, 1440, 1463, 1500, 1547, 1620, 1701, 1742, 1800, 1920, 1947, 2059, 2090, 2160, 2210, 2250, 2318

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

OFFSET

1,2

COMMENTS

Here 0.p means the decimal fraction obtained by writing p after the decimal point, e.g., 0.11 = 11/100.

The first few values of Sum_{p|n} 0.p for n >= 1 are 0, 1/5, 3/10, 1/5, 1/2, 1/2, 7/10, 1/5, 3/10, 7/10, ...

Numbers j such that Sum_{p|j} 0.p (where p ranges over the prime divisors of j) = numbers j such that A276651(j) / A276652(j) is an integer.

See A276513 - the smallest number k such that Sum_{p|k} 0.p = n where p ranges over the prime divisors of k.

Sum_{p|a(n)} 0.p = 1 for first 133 terms of this sequence; Sum_{p|a(134)} 0.p = Sum_{p|16102} 0.p = 2. For number 16102 with set of prime divisors {2, 83, 97} holds: 0.2 + 0.83 + 0.97 = 2.

It is clear from the definition that if j is in the sequence so are all numbers m with rad(m) = rad(j). For example, since 21 is in the sequence, so are 63, 147, 189, 441, 567, 1029, 1323, 1701, etc. - Charles R Greathouse IV, Sep 10 2016

LINKS

Chai Wah Wu, Table of n, a(n) for n = 1..10000

FORMULA

A276652(a(n)) = 1.

EXAMPLE

The prime divisors of 60 are 2, 3, and 5, and 0.2 + 0.3 + 0.5 = 1, so 60 is a term.

MATHEMATICA

{1}~Join~Select[Range[2400], IntegerQ@ Total[# 10^(-Floor@ Log10@ # - 1) &@ FactorInteger[#][[All, 1]]] &] (* Michael De Vlieger, Sep 12 2016 *)

PROG

(Magma) [n: n in [1..1000000] | Denominator(&+[d/(10^(#Intseq(d))): d in PrimeDivisors(n)]) eq 1]

(PARI) is(n)=my(f=factor(n)[, 1]); denominator(sum(i=1, #f, f[i]/10^#Str(f[i])))==1 \\ Charles R Greathouse IV, Sep 10 2016

CROSSREFS

Cf. A276513, A276651, A276652, A276653, A276654.

KEYWORD

nonn,base

AUTHOR

Jaroslav Krizek, Sep 10 2016

EXTENSIONS

a(1) inserted by Charles R Greathouse IV, Sep 10 2016

STATUS

approved

A276513

a(n) = the smallest number k>1 such that Sum_{p|k} 0.p = n where p runs through the prime divisors of k.

+10
6

21, 16102, 281785, 275867515, 9178864590, 8533159052845, 9404411107962990

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

OFFSET

1,1

COMMENTS

Here 0.p means the decimal fraction obtained by writing p after the decimal point, e.g. 0.11 = 11/100.

a(n) = the smallest number k>1 such that A276651(k) / A276652(k) = n.

The first few values of Sum_{p|n} 0.p are: 1/5, 3/10, 1/5, 1/2, 1/2, 7/10, 1/5, 3/10, 7/10, ...

Conjecture: a(4) = 730610790; Sum_{p|730610790} 0.p = 0.2 + 0.3 + 0.5 + 0.7 + 0.13 + 0.31 + 0.89 + 0.97 = 4.

Subsequence of A005117. - Chai Wah Wu, Sep 15 2016

a(8) <= 8646420251472669505, a(9) <= 1879755659507289195345, a(10) <= 3625424828481802325595910. - Giovanni Resta, Aug 19 2019

a(11) <= 771700218558425481527617170, a(12) <= 3840490537418012461017296489710. - Chai Wah Wu, Feb 07 2022

LINKS

Table of n, a(n) for n=1..7.

EXAMPLE

Number 16102 is the smallest number k with Sum_{p|k} 0.p = 2; set of prime divisors of 16102: {2, 83, 97}; Sum_{p|16102} 0.p = 0.2 + 0.83 + 0.97 = 2.

MATHEMATICA

Table[k = 1;

While[f = FactorInteger[k][[All, 1]];

Total[f*10^-IntegerLength[f]] != n, k++];

k, {n, 1, 4}] (* Robert Price, Sep 20 2019 *)

PROG

(Magma) A276513:=func<n|exists(r){k:k in[2..10^6] | (&+[d / (10^(#Intseq(d))): d in PrimeDivisors(k)]) eq n}select r else 0>; [A276513(n): n in[1..3]]

CROSSREFS

Cf. A005117, A276651, A276652, A276653, A276654, A276655.

KEYWORD

nonn,base,more

AUTHOR

Jaroslav Krizek, Sep 14 2016

EXTENSIONS

a(4) from Chai Wah Wu, Sep 16 2016

a(5)-a(7) from Giovanni Resta, Aug 19 2019

STATUS

approved

A276652

a(n) = denominator of Sum_{p|n} 0.d where p runs through the prime divisors of n.

+10
6

5, 10, 5, 2, 2, 10, 5, 10, 10, 100, 2, 100, 10, 5, 5, 100, 2, 100, 10, 1, 100, 100, 2, 2, 100, 10, 10, 100, 1, 100, 5, 100, 100, 5, 2, 100, 100, 100, 10, 100, 5, 100, 100, 5, 100, 100, 2, 10, 10, 100, 100, 100, 2, 100, 10, 100, 100, 100, 1, 100, 100, 1, 5, 100

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

OFFSET

2,1

COMMENTS

The first few values of Sum_{p|n} 0.d are: 1/5, 3/10, 1/5, 1/2, 1/2, 7/10, 1/5, 3/10, 7/10, ...

See A276655 - numbers n such that Sum_{p|n} 0.d is an integer.

LINKS

Jaroslav Krizek, Table of n, a(n) for n = 2..1000

FORMULA

a(n) = A276651(n) / (Sum_{p|n} 0.d) where p = prime divisors of n.

EXAMPLE

For n=12; Sum_{p|12} 0.d = 0.2 + 0.3 = 0.5 = 5/10 = 1/2; a(12) = 2.

MATHEMATICA

Denominator[Table[f = FactorInteger[i][[All, 1]];

Total[f*10^-IntegerLength[f]], {i, 2, 65}]] (* Robert Price, Sep 20 2019 *)

PROG

(Magma) [Denominator(&+[d/(10^(#Intseq(d))): d in PrimeDivisors(n)]): n in [2..1000]]

CROSSREFS

Cf. A276651, A276653, A276654, A276655, A276513.

KEYWORD

nonn,base,frac

AUTHOR

Jaroslav Krizek, Sep 10 2016

STATUS

approved

A276653

a(n) = floor(Sum_{p|n} 0.p) where p runs through the prime divisors of n>1.

+10
6

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

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

OFFSET

COMMENTS

Here 0.p means the decimal fraction obtained by writing p after the decimal point, e.g., 0.11 = 11/100.

See A276654(n) = the smallest number k>1 such that floor(Sum_{p|k} 0.p) = n where p runs through the prime divisors of k.

LINKS

Jaroslav Krizek, Table of n, a(n) for n = 2..1000

EXAMPLE

For n = 35: floor(Sum_{p|35} 0.p) = floor(0.5 + 0.7) = floor(1.2) = 1.

MATHEMATICA

Table[Floor@ Total[# 10^(-Floor@ Log10@ # - 1) &@ FactorInteger[n][[All, 1]]], {n, 2, 120}] (* Michael De Vlieger, Sep 12 2016 *)

PROG

(Magma) [Floor(&+[d/(10^(#Intseq(d))): d in PrimeDivisors(n)]): n in [2..1000]]

CROSSREFS

Cf. A276513, A276651, A276652, A276654, A276655.

KEYWORD

nonn,base

AUTHOR

Jaroslav Krizek, Sep 11 2016

STATUS

approved

A276654

a(n) = the smallest number k>1 such that floor(Sum_{p|k} 0.p) = n where p runs through the prime divisors of k.

+10
6

2, 21, 2905, 281785, 47740490, 9178864590, 8533159052845, 1817562878255985, 1801204812351681135, 787408225243814333670

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

OFFSET

0,1

COMMENTS

Here 0.p means the decimal fraction obtained by writing p after the decimal point, e.g. 0.11 = 11/100.

The first few values of Sum_{p|n} 0.p are: 1/5, 3/10, 1/5, 1/2, 1/2, 7/10, 1/5, 3/10, 7/10, ...

Subsequence of A005117. - Chai Wah Wu, Sep 15 2016

LINKS

Table of n, a(n) for n=0..9.

EXAMPLE

Number 2905 is the smallest number k with floor(Sum_{p|k} 0.p) = 2; set of prime divisors of 2905: {5, 7, 83}; floor(Sum_{p|2905} 0.p) = 0.5 + 0.7 + 0.83 = floor(2.03) = 2.

MATHEMATICA

Table[k = 2; While[f = FactorInteger[k][[All, 1]];

Floor[Total[f*10^-IntegerLength[f]]] != n, k++];

k, {n, 0, 3}] (* Robert Price, Sep 20 2019 *)

PROG

(Magma) A276654:=func<n|exists(r){k:k in[2..1000000] | Floor(&+[d / (10^(#Intseq(d))): d in PrimeDivisors(k)]) eq n}select r else 0>; [A276654(n): n in[0..3]]

CROSSREFS

Cf. A005117, A276513, A276651, A276652, A276653, A276655.

KEYWORD

nonn,base,more

AUTHOR

Jaroslav Krizek, Sep 11 2016

EXTENSIONS

a(4) from Michel Marcus, Sep 11 2016

a(5)-a(9) from Giovanni Resta, Aug 31 2019

STATUS

approved

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