A069359 -id:A069359 - cp's OEIS frontend

A329029 a(n) = A069359(A276086(n)), where A276086 is the primorial base exp-function and A069359(n) = n * Sum_{p|n} 1/p.

0, 1, 1, 5, 3, 15, 1, 7, 8, 31, 24, 93, 5, 35, 40, 155, 120, 465, 25, 175, 200, 775, 600, 2325, 125, 875, 1000, 3875, 3000, 11625, 1, 9, 10, 41, 30, 123, 12, 59, 71, 247, 213, 741, 60, 295, 355, 1235, 1065, 3705, 300, 1475, 1775, 6175, 5325, 18525, 1500, 7375, 8875, 30875, 26625, 92625, 7, 63, 70, 287, 210, 861, 84, 413, 497, 1729

Offset: 0

Antti Karttunen, Nov 07 2019

nonn

A380535 gives the indices n where a(n) is a multiple of A053669(n). This can be seen from the formula a(n) = A003557(A276086(n)) * A069359(A328571(n)). The left hand side of the product is a multiple of A053669(n) if and only if A276088(n) > 1, while the right hand side is never a multiple of A053669(n), as it is equal to A329031(n) = A003415(A007947(A276086(n))). - Antti Karttunen, Feb 11 2025

Antti Karttunen, Table of n, a(n) for n = 0..2310
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..32768
Index entries for sequences related to primorial base

Cf. A003415, A003557, A053669, A069359, A276086, A276088, A328571, A328572, A329031, A380535.

Coincides with A327860 on the positions given by A276156.

A329029(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); if(e, m *= (p^e); s += (1/p)); n = n\p; p = nextprime(1+p)); (s*m); };

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A069359(n) = (n*sumdiv(n, d, isprime(d)/d));
A329029(n) = A069359(A276086(n));

a(n) = A069359(A276086(n)).

a(n) = A328572(n) * A329031(n) = A003557(A276086(n)) * A069359(A328571(n)). - Antti Karttunen, Feb 11 2025

A347956 Dirichlet convolution of A003602 with A069359.

Original entry on oeis.org

0, 1, 1, 3, 1, 8, 1, 7, 5, 11, 1, 22, 1, 14, 13, 15, 1, 35, 1, 31, 16, 20, 1, 50, 8, 23, 20, 40, 1, 81, 1, 31, 22, 29, 19, 95, 1, 32, 25, 71, 1, 106, 1, 58, 62, 38, 1, 106, 11, 77, 31, 67, 1, 134, 25, 92, 34, 47, 1, 217, 1, 50, 78, 63, 28, 156, 1, 85, 40, 151, 1, 215, 1, 59, 95, 94, 28, 181, 1, 151, 74, 65, 1, 286, 34

Offset: 1

Antti Karttunen, Sep 20 2021

nonn

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

Cf. A003602, A069359.

Cf. also A347954, A347955, A347957.

A003602(n) = (1+(n>>valuation(n,2)))/2;
A069359(n) = (n*sumdiv(n, d, isprime(d)/d)); \\ From A069359
A347956(n) = sumdiv(n,d,A003602(d)*A069359(n/d));

a(n) = Sum_{d|n} A003602(d) * A069359(n/d).

A340070 a(n) = gcd(A003415(n), A069359(n)).

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 1, 4, 3, 7, 1, 2, 1, 9, 8, 8, 1, 3, 1, 2, 10, 13, 1, 4, 5, 15, 9, 2, 1, 31, 1, 16, 14, 19, 12, 30, 1, 21, 16, 4, 1, 41, 1, 2, 3, 25, 1, 8, 7, 5, 20, 2, 1, 9, 16, 4, 22, 31, 1, 2, 1, 33, 3, 32, 18, 61, 1, 2, 26, 59, 1, 12, 1, 39, 5, 2, 18, 71, 1, 8, 27, 43, 1, 2, 22, 45, 32, 4, 1, 3, 20, 2, 34, 49, 24

Offset: 1

Antti Karttunen, Dec 30 2020

nonn

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

Cf. A003415, A005117, A069359, A329039.

Cf. also A085731, A300253, A327858.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A069359(n) = (n*sumdiv(n, d, isprime(d)/d)); \\ From A069359
A340070(n) = gcd(A003415(n), A069359(n));

a(n) = gcd(A003415(n), A069359(n)) = gcd(A003415(n), A329039(n)).

For all squarefree k, a(k) = A003415(k) = A069359(k).

A347133 a(n) = Sum_{d|n} A003415(n/d) * A069359(d).

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 6, 1, 2, 0, 16, 0, 2, 2, 24, 0, 19, 0, 20, 2, 2, 0, 72, 1, 2, 9, 24, 0, 40, 0, 80, 2, 2, 2, 111, 0, 2, 2, 96, 0, 48, 0, 32, 25, 2, 0, 256, 1, 29, 2, 36, 0, 117, 2, 120, 2, 2, 0, 244, 0, 2, 29, 240, 2, 64, 0, 44, 2, 56, 0, 446, 0, 2, 31, 48, 2, 72, 0, 352, 54, 2, 0, 308, 2, 2, 2, 168, 0, 304, 2, 56

Offset: 1

Antti Karttunen, Aug 23 2021

nonn

Dirichlet convolution of A003415 (arithmetic derivative) with A069359.

Dirichlet convolution of A001221 (omega, number of distinct prime factors of n) with A347131.

Antti Karttunen, Table of n, a(n) for n = 1..10080
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A001221, A003415, A069359, A347131.

Cf. also A300251, A305809, A347132, A347134, A347135.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A069359(n) = (n*sumdiv(n, d, isprime(d)/d)); \\ From A069359
A347133(n) = sumdiv(n,d,A003415(n/d)*A069359(d));

a(n) = Sum_{d|n} A003415(n/d) * A069359(d).

a(n) = Sum_{d|n} A001221(n/d) * A347131(d).

A347135 a(n) = Sum_{d|n} A001615(n/d) * A069359(d).

Original entry on oeis.org

0, 1, 1, 5, 1, 12, 1, 16, 7, 16, 1, 51, 1, 20, 18, 44, 1, 68, 1, 71, 22, 28, 1, 156, 11, 32, 33, 91, 1, 167, 1, 112, 30, 40, 26, 277, 1, 44, 34, 220, 1, 215, 1, 131, 110, 52, 1, 420, 15, 140, 42, 151, 1, 300, 34, 284, 46, 64, 1, 673, 1, 68, 138, 272, 38, 311, 1, 191, 54, 295, 1, 836, 1, 80, 162, 211, 38, 359, 1, 596

Offset: 1

Antti Karttunen, Aug 23 2021

nonn

Dirichlet convolution of A001615 (Dedekind psi function) with A069359.

Dirichlet convolution of A001221 (omega, number of distinct prime factors of n) with A322577.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Wikipedia, Dirichlet convolution

Cf. A001221, A001615, A069359, A322577.

Cf. also A347132, A347133, A347134.

Table[DivisorSum[n,PrimeNu[n/#]*Sum[DirichletConvolve[j,MoebiusMu[j]^2,j,#/d] EulerPhi[d],{d,Divisors[#]}]&],{n,80}] (* Giorgos Kalogeropoulos, Oct 28 2021 *)

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A069359(n) = (n*sumdiv(n, d, isprime(d)/d)); \\ From A069359
A347135(n) = sumdiv(n,d,A001615(n/d)*A069359(d));

a(n) = Sum_{d|n} A001615(n/d) * A069359(d).

a(n) = Sum_{d|n} A001221(n/d) * A322577(d).

A260310 Pairs with balanced sums of prime divisors (A008472) and inverse prime divisors (A069359), ordered by larger members.

Original entry on oeis.org

3, 8, 7, 16, 11, 18, 7, 27, 17, 45, 29, 50, 41, 54, 53, 60, 31, 64, 71, 84, 29, 99, 107, 132, 61, 147, 41, 153, 131, 162, 53, 207, 157, 220, 113, 225, 179, 228, 239, 240, 131, 242, 79, 243, 73, 245, 127, 255, 127, 256, 229, 264, 223, 280, 113, 297, 199, 315, 73, 325, 317, 336, 181, 338, 43, 343, 269, 348

Offset: 1

Juri-Stepan Gerasimov, Jul 22 2015

nonn

Consider pairs (x,y) of numbers where sum(p|x) p + sum(q|y) q = x*sum(p|x) 1/p + y*sum(q|y) 1/q where p, q are primes and sum(p|x) p > sum(q|y) q.
Or, pairs of numbers x and y where A008472(x) + A008472(y) = A069359(x) + A069359(y) where A008472(x) > A008472(y).
A001222(a(2n -1)) = 1 and A001222(a(2n)) >= 3.
For the vast majority of the time, a(2n-1) is prime. There seems to be about 1 pair per decade.
Conjecture: a(2n) < a(2n+2) for all n>0, but there are many times (1/10.84) that a(2n) + 1 = a(2n+2).
Conjecture: if a(2n-1) is prime then a(2n) is composite and vice versa. And when a(2n-1) is composite, it is congruent to 0 (mod 6). - Robert G. Wilson v, Jul 22 2015
The first conjecture appears to be satisfied because if both x and y were prime then the sum of the A008472 were the sum of the two primes and the sum of the A069359 were two. - R. J. Mathar, Aug 03 2015

			3 and 8 is first pair of this sequence because A008472(3) + A008472(8) = 3 + 2 = 5 is equal to A069359(3) + A069359(8) = 1 + 4 = 5.

Robert G. Wilson v, Table of n, a(n) for n = 1..19812

Cf. A001222, A008472, A069359.

f[n_] := f[n] = Block[{fi = FactorInteger[n][[All, 1]]}, {Plus @@ fi, n*Plus @@ (1/fi)}] /; n > 0; k =3; lst = {}; While[ k < 400, j = 2; While[ j < k, If[ f[k][[1]] + f[j][[1]] == f[k][[2]] + f[j][[2]] && f[k][[1]] != f[k][[2]], AppendTo[lst, {j,k}]]; j++]; k++]; lst // Flatten (* Robert G. Wilson v, Jul 22 2015 *)

Corrected and edited by Robert G. Wilson v, Jul 22 2015

A344756 a(n) = A003415(n) / gcd(A003415(n), A069359(n)).

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 8, 1, 1, 1, 4, 1, 7, 1, 12, 1, 1, 1, 11, 2, 1, 3, 16, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 17, 1, 1, 1, 24, 13, 1, 1, 14, 2, 9, 1, 28, 1, 9, 1, 23, 1, 1, 1, 46, 1, 1, 17, 6, 1, 1, 1, 36, 1, 1, 1, 13, 1, 1, 11, 40, 1, 1, 1, 22, 4, 1, 1, 62, 1, 1, 1, 35, 1, 41, 1, 48, 1, 1, 1, 17, 1, 11, 25

Offset: 2

Antti Karttunen, May 28 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 2..12012
Antti Karttunen, Data supplement: n, a(n) computed for n = 2..65537

Cf. A003415, A005117 (for n > 1 gives the positions of ones), A069359, A340070, A344757.

Cf. also A344696.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A069359(n) = (n*sumdiv(n, d, isprime(d)/d)); \\ From A069359
A344756(n) = { my(u=A003415(n)); (u/gcd(u,A069359(n))); };

a(n) = A003415(n) / A340070(n) = A003415(n) / gcd(A003415(n), A069359(n)).

A347134 a(n) = Sum_{d|n} phi(n/d) * A069359(d), where phi is Euler totient function.

Original entry on oeis.org

0, 1, 1, 3, 1, 8, 1, 8, 5, 12, 1, 23, 1, 16, 14, 20, 1, 36, 1, 35, 18, 24, 1, 60, 9, 28, 21, 47, 1, 87, 1, 48, 26, 36, 22, 103, 1, 40, 30, 92, 1, 119, 1, 71, 66, 48, 1, 148, 13, 92, 38, 83, 1, 144, 30, 124, 42, 60, 1, 247, 1, 64, 86, 112, 34, 183, 1, 107, 50, 183, 1, 268, 1, 76, 110, 119, 34, 215, 1, 228, 81, 84

Offset: 1

Antti Karttunen, Aug 23 2021

nonn

Dirichlet convolution of A000010 (Euler totient function phi) with A069359.

Dirichlet convolution of A001221 (omega) with A029935 (the convolution square of Euler phi).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000010, A001221, A069359, A029935.

Cf. also A347131, A347133, A347135.

Table[DivisorSum[n,EulerPhi[n/#]*#*DivisorSum[#,1/#&,PrimeQ]&],{n,82}] (* Giorgos Kalogeropoulos, Oct 28 2021 *)

A069359(n) = (n*sumdiv(n, d, isprime(d)/d)); \\ From A069359
A347134(n) = sumdiv(n,d,eulerphi(n/d)*A069359(d));

a(n) = Sum_{d|n} A000010(n/d) * A069359(d)
a(n) = Sum_{d|n} A001221(n/d) * A029935(d).
a(n) = Sum_{k=1..n} A069359(gcd(n,k)). - Antti Karttunen, Oct 17 2021

A344028 a(n) = A069359(A005940(1+n)).

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 3, 4, 1, 7, 8, 10, 5, 15, 9, 8, 1, 9, 10, 14, 12, 31, 24, 20, 7, 35, 40, 30, 25, 45, 27, 16, 1, 13, 14, 18, 16, 41, 30, 28, 18, 59, 71, 62, 60, 93, 72, 40, 11, 63, 70, 70, 84, 155, 120, 60, 49, 175, 200, 90, 125, 135, 81, 32, 1, 15, 16, 26, 18, 61, 42, 36, 20, 87, 103, 82, 80, 123, 90, 56, 24, 113, 131

Offset: 0

Antti Karttunen, May 11 2021

Coincides with A344026 on Fibbinary numbers, A003714.

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for sequences related to binary expansion of n

Cf. A000079 (positions of ones), A003714, A005940, A069359, A344026, A344182.

A069359(n) = (n*sumdiv(n, d, isprime(d)/d)); \\ From A069359
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A344028(n) = A069359(A005940(1+n));

a(n) = A069359(A005940(1+n)).

A344757 a(n) = A069359(n) / gcd(A003415(n), A069359(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 5, 1, 7, 1, 1, 1, 5, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 13, 8, 1, 1, 5, 1, 7, 1, 15, 1, 5, 1, 9, 1, 1, 1, 31, 1, 1, 10, 1, 1, 1, 1, 19, 1, 1, 1, 5, 1, 1, 8, 21, 1, 1, 1, 7, 1, 1, 1, 41, 1, 1, 1, 13, 1, 31, 1, 25, 1, 1, 1, 5, 1, 9, 14, 1, 1, 1, 1, 15

Offset: 2

Antti Karttunen, May 28 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 2..12012
Antti Karttunen, Data supplement: n, a(n) computed for n = 2..65537

Cf. A003415, A069359, A340070, A344756.

Cf. also A344697.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A069359(n) = (n*sumdiv(n, d, isprime(d)/d)); \\ From A069359
A344757(n) = { my(u=A069359(n)); (u/gcd(u,A003415(n))); };

a(n) = A069359(n) / A340070(n) = A069359(n) / gcd(A003415(n), A069359(n)).

cp's OEIS Frontend

A329029 a(n) = A069359(A276086(n)), where A276086 is the primorial base exp-function and A069359(n) = n * Sum_{p|n} 1/p.

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A347956 Dirichlet convolution of A003602 with A069359.

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A340070 a(n) = gcd(A003415(n), A069359(n)).

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A347133 a(n) = Sum_{d|n} A003415(n/d) * A069359(d).

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A347135 a(n) = Sum_{d|n} A001615(n/d) * A069359(d).

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A260310 Pairs with balanced sums of prime divisors (A008472) and inverse prime divisors (A069359), ordered by larger members.

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A344756 a(n) = A003415(n) / gcd(A003415(n), A069359(n)).

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A347134 a(n) = Sum_{d|n} phi(n/d) * A069359(d), where phi is Euler totient function.

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