A089026 -id:A089026 - cp's OEIS frontend

A349215 a(n) = Sum_{d|n} d^c(d), where c is the prime characteristic (A010051).

1, 3, 4, 4, 6, 7, 8, 5, 5, 9, 12, 9, 14, 11, 10, 6, 18, 9, 20, 11, 12, 15, 24, 11, 7, 17, 6, 13, 30, 15, 32, 7, 16, 21, 14, 12, 38, 23, 18, 13, 42, 17, 44, 17, 12, 27, 48, 13, 9, 11, 22, 19, 54, 11, 18, 15, 24, 33, 60, 19, 62, 35, 14, 8, 20, 21, 68, 23, 28, 19, 72, 15, 74, 41, 12

Offset: 1

Wesley Ivan Hurt, Nov 10 2021

nonn

For each divisor d of n, add d if d is prime, otherwise add 1. For example, a(6) = 1 + 2 + 3 + 1 = 7.

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

Inverse Moebius transform of A089026.

Cf. A010051, A000005 (tau), A008472 (sopf), A001221 (omega).

a[n_] := DivisorSum[n, #^Boole[PrimeQ[#]] &]; Array[a, 75] (* Amiram Eldar, Nov 11 2021 *)

a(n) = sumdiv(n, d, if (isprime(d), d, 1)); \\ Michel Marcus, Nov 11 2021

from math import prod
from sympy import factorint
def A349215(n):
    fs = factorint(n)
    return sum(a-1 for a in fs.keys())+prod(1+d for d in fs.values()) # Chai Wah Wu, Nov 11 2021

a(n) = A000005(n)+A008472(n)-A001221(n). - Chai Wah Wu, Nov 11 2021

a(p) = p+1 for primes p. - Wesley Ivan Hurt, Nov 28 2021

A369119 a(n) = (n + 1)^[is_prime(n + 1)] * n!.

Original entry on oeis.org

1, 2, 6, 6, 120, 120, 5040, 5040, 40320, 362880, 39916800, 39916800, 6227020800, 6227020800, 87178291200, 1307674368000, 355687428096000, 355687428096000, 121645100408832000, 121645100408832000, 2432902008176640000, 51090942171709440000

Offset: 0

Peter Luschny, Jan 14 2024

nonn

Cf. A089026, A000142, A347425, A000367/A002445 (Bernoulli(2n)).

A369119[n_] := n! If[PrimeQ[n+1], n+1, 1];
Array[A369119, 25, 0] (* Paolo Xausa, Jan 15 2024 *)

def A369119(n): return (n+1)^is_prime(n+1)*factorial(n)

a(2*n)*Bernoulli(2*n) = A347425(n).

A372112 Rad-transform of the squarefree numbers A005117 (see Comments).

Original entry on oeis.org

1, 2, 3, 5, 1, 7, 1, 11, 13, 1, 1, 17, 19, 1, 1, 23, 1, 29, 1, 31, 1, 1, 1, 37, 1, 1, 41, 1, 43, 1, 47, 1, 53, 1, 1, 1, 59, 61, 1, 1, 1, 67, 1, 1, 71, 73, 1, 1, 1, 79, 1, 83, 1, 1, 1, 89, 1, 1, 1, 1, 97, 101, 1, 103, 1, 1, 107, 109, 1, 1, 113, 1, 1, 1, 1, 1, 1, 127, 1, 1

Offset: 1

David James Sycamore, Apr 19 2024

nonn

For a sequence A with terms a(1), a(2), a(3).... , let R(0) = 1, and for k >= 1 let R(k) = rad(a(1)*a(2)*...*a(k)). Define the Rad-transform of A to be R(n)/R(n-1); n >= 1, where rad is A007947. This sequence is the Rad transform of the squarefree numbers, A = A005117; see Example.
The sequence consists of only 1's and primes.
Sequence is obtained directly from A005117 by leaving all primes and a(1) = 1 untouched, and replacing all composite squarefree numbers with 1. Alternatively: in A000027, delete all squarefull numbers (A013929), replace all squarefree composites with 1, leave primes untouched and concatenate.

			For A005117, R(k) (k >= 0) is 1 U A128040. That is, 1,1,2,6,30,30,210,... from which: a(1) = 1/1 = 1, a(2) = 2/1 = 2, a(3) = 6/2 = 3, 4(4) = 30/6 = 5, a(5) = 30/30 = 1, and so on. Note that the first term of R(k) is 1, the empty product (product of the first 0 terms of A005117).

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000040, A005117, A007947, A089026, A128040.

k = r = s = 1; f[x_] := f[x] = Times @@ FactorInteger[x][[All, 1]]; Reap[Do[While[! SquareFreeQ[k], k++]; s = f[s*f[k]]; Sow[s/r]; r = s; k++, {n, 120}] ][[-1, 1]] (* Michael De Vlieger, Apr 19 2024 *)

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
lista(nn) = my(v = select(issquarefree, [1..nn])); my(w = vector(#v, k, rad(prod(i=1, k, v[i])))); concat(1, vector(#w-1, k, w[k+1]/w[k])); \\ Michel Marcus, Apr 19 2024

from math import isqrt
from sympy import mobius, isprime
def A372112(n):
    def f(x): return int(n-sum(mobius(k)*(x//k**2) for k in range(2, isqrt(x)+1)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m if isprime(m) else 1 # Chai Wah Wu, Dec 23 2024

More terms from Michel Marcus, Apr 19 2024

A133255 Triangle with a minimum occurrence of prime powers for which the least common multiple of the rows will give the terms in A003418.

Original entry on oeis.org

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

Offset: 1

Mats Granvik, Oct 14 2007

Checked up to 29th row. Similar to A133232 and A133233. In this table the prime powers with the same base are in the same column. A prime power occurs in the table: (base of prime power-1)*(the prime power).

			lcm{1}= 1
lcm{1,1} = 1
lcm{1,1,2} = 2
lcm{1,1,2,3} = 6
lcm{1,1,4,3,1} = 12
lcm{1,1,4,3,1,5} = 60
lcm{1,1,4,3,1,5,1} = 60
lcm{1,1,4,3,1,5,1,7} = 420
lcm{1,1,8,3,1,5,1,7,1} = 840
lcm{1,1,8,9,1,5,1,7,1,1} = 2520
1 = 1
1*1 = 1
1*1*2 = 2
1*1*2*3 = 6
1*1*4*3*1 = 12
1*1*4*3*1*5 = 60
1*1*4*3*1*5*1 = 60
1*1*4*3*1*5*1*7 = 420
1*1*8*3*1*5*1*7*1 = 840
1*1*8*9*1*5*1*7*1*1 = 2520

Cf. A089026, A003418, A000961.

=if(column()=1;1; if(and(row()-1>=(A089026(n-1))^0;row()-1<(A089026(n-1))^1);(A089026(n-1))^0; if(and(row()-1>=(A089026(n-1))^1;row()-1<(A089026(n-1))^2);(A089026(n-1))^1; if(and(row()-1>=(A089026(n-1))^2;row()-1<(A089026(n-1))^3);(A089026(n-1))^2; if(and(row()-1>=(A089026(n-1))^3;row()-1<(A089026(n-1))^4);(A089026(n-1))^3; if(and(row()-1>=(A089026(n-1))^4;row()-1<(A089026(n-1))^5);(A089026(n-1))^4; 1)))))) And so on.

T(n,k) = if k=1 then 1 elseif n-1>=(A089026(n-1))^0 and n-1<(A089026(n-1))^1 then (A089026(n-1))^0 elseif n-1>=(A089026(n-1))^1 and n-1<(A089026(n-1))^2 then (A089026(n-1))^1 elseif n-1>=(A089026(n-1))^2 and n-1<(A089026(n-1))^3 then (A089026(n-1))^2 elseif n-1>=(A089026(n-1))^3 and n-1<(A089026(n-1))^4 then (A089026(n-1))^3 elseif n-1>=(A089026(n-1))^4 and n-1<(A089026(n-1))^5 then (A089026(n-1))^4 else 1 (1<=k<=n) And so on, this formula needs to be expanded if one wants to make a bigger table. A089026(n-1) means that the index to that sequence is shifted in this formula so that the first term in A089026 is used in the second column of the table.

A358452 The inverse Euler transform of p(n) = n if n is prime, otherwise 1.

Original entry on oeis.org

1, 1, 1, 1, -3, 3, -3, 5, -8, 5, -11, 36, -45, 41, -72, 142, -223, 311, -493, 851, -1243, 1823, -3204, 5336, -7906, 12083, -20134, 33133, -51685, 81568, -133556, 215363, -340155, 547916, -895895, 1442323, -2300704, 3718260, -6056908, 9787064, -15755664, 25541623

Offset: 0

Peter Luschny, Nov 21 2022

sign

Conjecture: signum(a(n)) + (-1)^n = 0 for n >= 3.

N. J. A. Sloane, Transforms.
Wiki, Euler_transform

Cf. A089026, A219224.

# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(n -> ifelse(isprime(n), n, 1)):
seq(a(n), n = 0..41);
# Using EULERi the sequence is returned without a(0) and has offset 1.
f := n -> ifelse(isprime(n), n, 1): EULERi([seq(f(n), n = 1..41)]);

cp's OEIS Frontend

A349215 a(n) = Sum_{d|n} d^c(d), where c is the prime characteristic (A010051).

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A369119 a(n) = (n + 1)^[is_prime(n + 1)] * n!.

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A372112 Rad-transform of the squarefree numbers A005117 (see Comments).

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A133255 Triangle with a minimum occurrence of prime powers for which the least common multiple of the rows will give the terms in A003418.

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A358452 The inverse Euler transform of p(n) = n if n is prime, otherwise 1.

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