A082299 -id:A082299

A373364

a(n) = gcd(A001414(n), A003415(n)), where A001414 is the sum of prime factors with repetition, and A003415 is the arithmetic derivative.

+10
13

0, 1, 1, 4, 1, 5, 1, 6, 6, 7, 1, 1, 1, 9, 8, 8, 1, 1, 1, 3, 10, 13, 1, 1, 10, 15, 9, 1, 1, 1, 1, 10, 14, 19, 12, 10, 1, 21, 16, 1, 1, 1, 1, 3, 1, 25, 1, 1, 14, 3, 20, 1, 1, 1, 16, 1, 22, 31, 1, 4, 1, 33, 1, 12, 18, 1, 1, 3, 26, 1, 1, 12, 1, 39, 1, 1, 18, 1, 1, 1, 12, 43, 1, 2, 22, 45, 32, 1, 1, 1, 20, 3, 34, 49, 24

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

OFFSET

1,4

COMMENTS

For n >= 1, a(n) is a multiple of A373363(n).

LINKS

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

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]));

A373364(n) = gcd(A001414(n), A003415(n));

CROSSREFS

Cf. A001414, A003415, A373375 (positions of even terms), A373376 (of odd terms).

Cf. also A082299, A373362, A373363.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 02 2024

STATUS

approved

A373362

a(n) = gcd(A001414(n), A276085(n)), where A001414 is the sum of prime factors with repetition, and A276085 is the primorial base log-function.

+10
11

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, 3, 1, 1, 1, 1, 1, 1, 2, 1, 4, 17, 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, 9, 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

As A001414 and A276085 are both fully additive sequences, all sequences that give the positions of multiples of some k > 1 in this sequence are closed under multiplication: For example, A373373, which gives the indices of multiples of 3.

LINKS

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

PROG

(PARI)

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

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)); };

A373362(n) = gcd(A001414(n), A276085(n));

CROSSREFS

Cf. A001414, A002110, A276085.

Cf. A345452 (positions of even terms), A373373 (positions of multiples of 3).

Cf. also A082299, A373361, A373363, A373367.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 02 2024

STATUS

approved

A373363

a(n) = gcd(A001414(n), A083345(n)), where A001414 is the sum of prime factors with repetition, and A083345 is the numerator of the sum of the inverses of prime factors with repetition.

+10
10

0, 1, 1, 1, 1, 5, 1, 3, 2, 7, 1, 1, 1, 9, 8, 2, 1, 1, 1, 3, 10, 13, 1, 1, 2, 15, 1, 1, 1, 1, 1, 5, 14, 19, 12, 5, 1, 21, 16, 1, 1, 1, 1, 3, 1, 25, 1, 1, 2, 3, 20, 1, 1, 1, 16, 1, 22, 31, 1, 1, 1, 33, 1, 3, 18, 1, 1, 3, 26, 1, 1, 1, 1, 39, 1, 1, 18, 1, 1, 1, 4, 43, 1, 1, 22, 45, 32, 1, 1, 1, 20, 3, 34, 49, 24, 1, 1, 1, 1, 7

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

OFFSET

1,6

LINKS

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

PROG

(PARI)

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

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

A373363(n) = gcd(A001414(n), A083345(n));

CROSSREFS

Cf. A001414, A083345.

Cf. A345452 (positions of even terms), A353374 (their characteristic function).

Cf. also A082299, A373362.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 02 2024

STATUS

approved

A099636

a(n) = gcd(sum of distinct prime factors of n, product of distinct prime factors of n).

+10
7

1, 2, 3, 2, 5, 1, 7, 2, 3, 1, 11, 1, 13, 1, 1, 2, 17, 1, 19, 1, 1, 1, 23, 1, 5, 1, 3, 1, 29, 10, 31, 2, 1, 1, 1, 1, 37, 1, 1, 1, 41, 6, 43, 1, 1, 1, 47, 1, 7, 1, 1, 1, 53, 1, 1, 1, 1, 1, 59, 10, 61, 1, 1, 2, 1, 2, 67, 1, 1, 14, 71, 1, 73, 1, 1, 1, 1, 6, 79, 1, 3, 1, 83, 6, 1, 1, 1, 1, 89, 10, 1, 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

FORMULA

From Antti Karttunen, Feb 01 2021: (Start)

a(n) = gcd(A007947(n), A008472(n)).

a(n) = A007947(n) / A340677(n) = A008472(n) / A340678(n).

(End)

EXAMPLE

n=84: a(84) = gcd(2*3*7, 2+3+7) = gcd(42, 12) = 6.

MATHEMATICA

PrimeFactors[n_Integer] := Flatten[ Table[ # [[1]], {1}] & /@ FactorInteger[n]]; f[n_] := Block[{pf = PrimeFactors[n]}, GCD[Plus @@ pf, Times @@ pf]]; Table[ f[n], {n, 93}] (* Robert G. Wilson v, Nov 04 2004 *)

PROG

(PARI)

A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947

A008472(n) = vecsum(factor(n)[, 1]); \\ From A008472

A099636(n) = gcd(A007947(n), A008472(n));

CROSSREFS

Cf. A007947, A008472, A082299, A099634, A340677, A340678.

Differs from related A099635 for the first time at n=84, where a(84) = 6, while A099635(84) = 12.

Differs from A014963 for the first time at n=30, where a(30) = 10, while A014963(30) = 1.

KEYWORD

nonn

AUTHOR

Labos Elemer, Oct 28 2004

EXTENSIONS

Name clarified by Antti Karttunen, Feb 01 2021

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

A102217

3-Suzanne numbers; composite multiples of 3 whose sum of prime factors with multiplicity is a multiple of 3.

+10
6

9, 24, 27, 42, 60, 72, 78, 81, 105, 114, 126, 132, 150, 180, 186, 192, 195, 204, 216, 222, 231, 234, 243, 258, 276, 285, 315, 330, 336, 342, 348, 357, 366, 375, 378, 396, 402, 429, 438, 450, 465, 474, 480, 483, 492, 510, 540, 555, 558, 564, 576, 582, 585

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

OFFSET

1,1

COMMENTS

Composite numbers k such that the sum of digits of k (A007953) and the sum of sums of digits of the prime factors of k (taken with multiplicity, A118503) are both divisible by 3. - Amiram Eldar, Apr 23 2021

The new secondary definition is equal to the original because taking the decimal digit sum preserves congruence modulo 3. This is a multiplicative semigroup: if m and n are in the sequence, then so is m*n. - Antti Karttunen, Jun 08 2024

REFERENCES

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384.

James J. Tattersall, Elementary Number Theory in Nine Chapters, 2nd ed., Cambridge University Press, 2005, p. 93.

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

Michael Smith, Cousins of Smith Numbers: Monica and Suzanne Sets, Fibonacci Quarterly, Vol. 34, No. 2 (1996), pp. 102-104.

Eric Weisstein's World of Mathematics, Suzanne Set.

FORMULA

a(n) = 3*A289142(1+n). - Antti Karttunen, Jun 08 2024

EXAMPLE

From Antti Karttunen, Jun 08 2024: (Start)

42 = 2*3*7 is a term as it is a multiple of 3, and also 2+3+7 = 12 is a multiple of 3.

60 = 2*2*3*5 is a term is it is a multiple of 3, and also 2+2+3+5 = 12 is a multiple of 3.

(End)

MATHEMATICA

s[n_] := Plus @@ IntegerDigits[n]; f[p_, e_] := e*s[p]; sp[n_] := Plus @@ f @@@ FactorInteger[n]; suz3Q[n_] := CompositeQ[n] && And @@ Divisible[{s[n], sp[n]}, 3]; Select[Range[600], suz3Q] (* Amiram Eldar, Apr 23 2021 *)

PROG

(PARI) isA102217(n) = if(n<=3 || (n%3), 0, my(f=factor(n)); 0==(sum(i=1, #f~, f[i, 2]*sumdigits(f[i, 1]))%3)); \\ Antti Karttunen, Jun 08 2024

(PARI) isA102217(n) = (n>3 && !(n%3) && A373371(n)); \\ Antti Karttunen, Jun 08 2024

CROSSREFS

Subsequence of A177927.

Intersection of A008585 and A289142 without the initial 3.

Positions of multiples of 3 in A082299, after A082299(3).

Cf. A001414, A007953, A018252, A102216, A118503, A373371.

KEYWORD

nonn

AUTHOR

Eric W. Weisstein, Dec 30 2004

EXTENSIONS

Alternative definition added and keyword:base removed by Antti Karttunen, Jun 08 2024

STATUS

approved

A373368

a(n) = gcd(n, A059975(n)), where A059975 is fully additive with a(p) = p-1.

+10
6

1, 1, 1, 2, 1, 3, 1, 1, 1, 5, 1, 4, 1, 7, 3, 4, 1, 1, 1, 2, 1, 11, 1, 1, 1, 13, 3, 4, 1, 1, 1, 1, 3, 17, 5, 6, 1, 19, 1, 1, 1, 3, 1, 4, 1, 23, 1, 6, 1, 1, 3, 2, 1, 1, 1, 1, 1, 29, 1, 4, 1, 31, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 1, 37, 5, 4, 1, 3, 1, 8, 1, 41, 1, 2, 5, 43, 3, 1, 1, 9, 1, 4, 1, 47, 1, 1, 1, 1, 1, 10

(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)

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

A373368(n) = gcd(n, A059975(n));

CROSSREFS

Cf. A059975, A108269 (positions of even terms), A359794 (of odd terms), A359832 (parity of terms).

Cf. also A082299, A373361, A373369.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 05 2024

STATUS

approved

A082343

Numerator of sopfr(n)/n, where sopfr=A001414 is the sum of prime factors (with repetition).

+10
4

0, 1, 1, 1, 1, 5, 1, 3, 2, 7, 1, 7, 1, 9, 8, 1, 1, 4, 1, 9, 10, 13, 1, 3, 2, 15, 1, 11, 1, 1, 1, 5, 14, 19, 12, 5, 1, 21, 16, 11, 1, 2, 1, 15, 11, 25, 1, 11, 2, 6, 20, 17, 1, 11, 16, 13, 22, 31, 1, 1, 1, 33, 13, 3, 18, 8, 1, 21, 26, 1, 1, 1, 1, 39, 13, 23, 18, 3, 1, 13, 4, 43, 1, 1, 22, 45, 32, 17

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

OFFSET

1,6

COMMENTS

Denominator is A082344(n) = n/A082299(n).

LINKS

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

FORMULA

a(n) = A001414(n)/A082299(n).

EXAMPLE

n=200: (2+2+2+5+5)/(2*2*2*5*5) = 16/(2*2*2*5*5) = (2*2*2*2)/(2*2*2*5*5) = 2/25, therefore a(200)=2, A082344(200)=25.

MATHEMATICA

sopfr[n_] := If[n == 1, 0, Total[Times @@@ FactorInteger[n]]];

a[n_] := Numerator[sopfr[n]/n];

Array[a, 100] (* Jean-François Alcover, Dec 03 2021 *)

PROG

(PARI)

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

A082299(n) = gcd(n, A001414(n));

A082343(n) = A001414(n)/A082299(n); \\ Antti Karttunen, Mar 04 2018

CROSSREFS

Cf. A001414, A082299, A082344.

KEYWORD

nonn,frac

AUTHOR

Reinhard Zumkeller, Apr 09 2003

STATUS

approved

A082344

Denominator of sopfr(n)/n, where sopfr=A001414 is the sum of prime factors (with repetition).

+10
4

1, 1, 1, 1, 1, 6, 1, 4, 3, 10, 1, 12, 1, 14, 15, 2, 1, 9, 1, 20, 21, 22, 1, 8, 5, 26, 3, 28, 1, 3, 1, 16, 33, 34, 35, 18, 1, 38, 39, 40, 1, 7, 1, 44, 45, 46, 1, 48, 7, 25, 51, 52, 1, 54, 55, 56, 57, 58, 1, 5, 1, 62, 63, 16, 65, 33, 1, 68, 69, 5, 1, 6, 1, 74, 75, 76, 77, 13, 1, 80, 27, 82, 1, 6

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

OFFSET

1,6

COMMENTS

Numerator is A082343(n) = A001414(n)/A082299(n).

LINKS

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

FORMULA

a(n) = n/A082299(n).

EXAMPLE

n=200: (2+2+2+5+5)/(2*2*2*5*5) = 16/(2*2*2*5*5) = (2*2*2*2)/(2*2*2*5*5) = 2/25, therefore a(200)=25, A082343(200)=2.

MATHEMATICA

sopd[n_]:=Module[{f=Flatten[Table[#[[1]], #[[2]]]&/@FactorInteger[n]]}, Denominator[ Total[f]/n]]; Array[sopd, 90] (* Harvey P. Dale, Jul 24 2018 *)

sopfr[n_] := If[n == 1, 0, Total[Times @@@ FactorInteger[n]]];

a[n_] := Denominator[sopfr[n]/n];

Array[a, 100] (* Jean-François Alcover, Dec 03 2021 *)

PROG

(PARI)

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

A082299(n) = gcd(n, A001414(n));

A082344(n) = (n/A082299(n)); \\ Antti Karttunen, Mar 04 2018

CROSSREFS

Cf. A001414, A082299, A082343.

KEYWORD

nonn,frac

AUTHOR

Reinhard Zumkeller, Apr 09 2003

STATUS

approved

A337490

a(0)=1; for n > 0, a(n) = the greatest common divisor (GCD) of n and the sum of all previous terms if the GCD is not already in the sequence; otherwise a(n) = a(n-1) + n.

+10
4

1, 2, 4, 7, 11, 5, 6, 13, 21, 30, 10, 21, 33, 46, 14, 29, 45, 62, 18, 37, 57, 78, 22, 45, 69, 94, 26, 53, 81, 110, 140, 171, 203, 236, 270, 305, 341, 378, 416, 39, 79, 120, 162, 205, 249, 294, 340, 387, 3, 52, 102, 17, 69, 122, 176, 231, 287, 344, 402, 461, 521, 582, 644, 707, 771, 836, 902, 969

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

OFFSET

0,2

COMMENTS

The sequence displays the unusual behavior of decreasing 53 times in the first 1975 terms, due to the existence of a GCD which has not previously appeared in the sequence, but then not decreasing again for n up to at least 100 million. In this period there are 37 repeated terms, the first being 21 at n=11 and the last 161202 at n=2054. In the same range many values do not appear, for example 16,23,28,32,36. It is unknown when the sequence decreases again, or if all values eventually appear. The 100 millionth term is 4999999948050717.

See the companion sequence A333980 for the sum of the terms from a(0) to a(n).

LINKS

Scott R. Shannon, Table of n, a(n) for n = 0..10000

Scott R. Shannon, Graph of the terms for n=0..2500. This includes the last known decrease in the sequence, n(1974) = 42.

Scott R. Shannon, Graph of the terms for n=0..10000000.

EXAMPLE

a(2) = 4 as the sum of all previous terms is a(0)+a(1) = 3, and the GCD of 3 and 2 is 1, which has already appeared in the sequence. Therefore a(2) = a(1) + n = 2 + 2 = 4.

a(4) = 11 as the sum of all previous terms is a(0)+...+a(3) = 14, and the GCD of 14 and 4 is 2. However 2 has already appeared so a(4) = a(3) + n = 7 + 4 = 11.

a(5) = 5 as the sum of all previous terms is a(0)+...+a(4) = 25, and the GCD of 25 and 5 is 5, and as 5 has not previous appeared a(5) = 5.

PROG

(PARI) lista(nn) = {my(va = vector(nn), s=0); va[1] = 1; s += va[1]; for (n=2, nn, my(g = gcd(n-1, s)); if (#select(x->(x==g), va), va[n] = va[n-1]+n-1, va[n] = g); s += va[n]; ); va; } \\ Michel Marcus, Sep 05 2020

CROSSREFS

Cf. A333980, A333826 (same rules but starting a(1)=1), A165430, A064814, A082299, A005132, A336957.

KEYWORD

nonn

AUTHOR

Scott R. Shannon, Aug 29 2020

STATUS

approved