A336851 -id:A336851 - cp's OEIS frontend

A003973 Inverse Möbius transform of A003961; a(n) = sigma(A003961(n)), where A003961 shifts the prime factorization of n one step towards the larger primes.

1, 4, 6, 13, 8, 24, 12, 40, 31, 32, 14, 78, 18, 48, 48, 121, 20, 124, 24, 104, 72, 56, 30, 240, 57, 72, 156, 156, 32, 192, 38, 364, 84, 80, 96, 403, 42, 96, 108, 320, 44, 288, 48, 182, 248, 120, 54, 726, 133, 228, 120, 234, 60, 624, 112, 480, 144, 128, 62, 624, 68

Offset: 1

Marc LeBrun

Sum of the divisors of the prime shifted n, or equally, sum of the prime shifted divisors of n. - Antti Karttunen, Aug 17 2020

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from indices in prime factorization.
Index entries for sequences related to sigma(n).

Cf. A000203, A000290 (positions of odd terms), A003961, A007814, A048673, A108228, A151800, A295664, A336840.
Permutation of A008438.
Used in the definitions of the following sequences: A326042, A336838, A336841, A336844, A336846, A336847, A336848, A336849, A336850, A336851, A336852, A336856, A336931, A336932.
Cf. also A003972.

b[1] = 1; b[p_?PrimeQ] := b[p] = Prime[ PrimePi[p] + 1]; b[n_] := b[n] = Times @@ (b[First[#]]^Last[#] &) /@ FactorInteger[n]; a[n_] := Sum[ b[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 70}]  (* Jean-François Alcover, Jul 18 2013 *)

aPrime(p,e)=my(q=nextprime(p+1));(q^(e+1)-1)/(q-1)
a(n)=my(f=factor(n));prod(i=1,#f~,aPrime(f[i,1],f[i,2])) \\ Charles R Greathouse IV, Jul 18 2013

A003973(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); sigma(factorback(f)); }; \\ Antti Karttunen, Aug 06 2020

from math import prod
from sympy import factorint, nextprime
def A003973(n): return prod(((q:=nextprime(p))**(e+1)-1)//(q-1) for p,e in factorint(n).items()) # Chai Wah Wu, Jul 05 2022

Multiplicative with a(p^e) = (q^(e+1)-1)/(q-1) where q = nextPrime(p). - David W. Wilson, Sep 01 2001
From Antti Karttunen, Aug 06-12 2020: (Start)
a(n) = Sum_{d|n} A003961(d) = Sum_{d|A003961(n)} d.
a(n) = A000203(A003961(n)) = A000593(A003961(n)).
a(n) = 2*A336840(n) - A000005(n) = 2*Sum_{d|n} (A048673(d) - (1/2)).
a(n) = A008438(A108228(n)) = A008438(A048673(n)-1).
a(n) = A336838(n) * A336856(n).
a(n) is odd if and only if n is a square.
(End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} p^3/((p+1)*(p^2-nextprime(p))) = 3.39513795..., where nextprime is A151800. - Amiram Eldar, Dec 08 2022, May 30 2025

More terms from David W. Wilson, Aug 29 2001

Secondary name added by Antti Karttunen, Aug 06 2020

A286385 a(n) = A003961(n) - A000203(n).

Original entry on oeis.org

0, 0, 1, 2, 1, 3, 3, 12, 12, 3, 1, 17, 3, 9, 11, 50, 1, 36, 3, 21, 23, 3, 5, 75, 18, 9, 85, 43, 1, 33, 5, 180, 17, 3, 29, 134, 3, 9, 29, 99, 1, 69, 3, 33, 97, 15, 5, 281, 64, 54, 23, 55, 5, 255, 19, 177, 35, 3, 1, 147, 5, 15, 171, 602, 35, 51, 3, 45, 49, 87, 1, 480, 5, 9, 121, 67, 47, 87, 3, 381, 504, 3, 5, 271, 25, 9, 35, 171, 7, 291, 75, 93, 57, 15, 41, 963

Offset: 1

Antti Karttunen, May 09 2017

nonn

Are all terms nonnegative? This question is equivalent to the question posed in A285705.
From Antti Karttunen, Aug 05 2020: (Start)
The answer to the above question is yes. Because both A000203 and A003961 are multiplicative sequences, it suffices to prove that for any prime p, and e >= 1, q^e >= sigma(p^e) = ((p^(1+e))-1) / (p-1), where q = A151800(p), i.e., the next larger prime after p. If p is a lesser twin prime, then q = p+2 (and this difference can't be less than 2, apart from case p=2), and it is easy to see that (n+2)^e > ((n^(e+1)) - 1) / (n-1), for all n >= 2, e >= 1.
See comments in A326042.
(End)
This is the inverse Möbius transform of A337549, from which it is even easier to see that all terms are nonnegative. - Antti Karttunen, Sep 22 2020

Antti Karttunen, Table of n, a(n) for n = 1..16383

Cf. A000203, A001359, A003961, A001065, A031924, A033879, A048673, A151800, A285705, A326042, A336702, A336851, A336852, A337549 (Möbius transform).

Cf. A326057 [= gcd(a(n), A252748(n))].

Array[Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1] - DivisorSigma[1, #] &, 96] (* Michael De Vlieger, Oct 05 2020 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A286385(n) = (A003961(n) - sigma(n));
for(n=1, 16384, write("b286385.txt", n, " ", A286385(n)));

from sympy import factorint, nextprime, divisor_sigma as D
from operator import mul
def a048673(n):
    f = factorint(n)
    return 1 if n==1 else (1 + reduce(mul, [nextprime(i)**f[i] for i in f]))/2
def a(n): return 2*a048673(n) - D(n) - 1 # Indranil Ghosh, May 12 2017

(define (A286385 n) (- (A003961 n) (A000203 n)))

a(n) = A285705(A048673(n)) - 1 = 2*A048673(n) - A000203(n) - 1.
a(n) = A336852(n) - A336851(n). - Antti Karttunen, Aug 05 2020
a(n) = Sum_{d|n} A337549(d). - Antti Karttunen, Sep 22 2020
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((p^2-p)/(p^2-q(p))) - Pi^2/12 = 1.24152934..., where q(p) = nextprime(p) (A151800). - Amiram Eldar, Dec 21 2023

A336850 a(n) = gcd(A003961(n), sigma(A003961(n))), where A003961 is the prime shift towards larger primes.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 15, 1, 3, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 13, 1, 3, 1, 3, 1, 3, 5, 9, 1, 3, 7, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 5, 3, 1, 5, 1, 3, 1, 3, 1, 3, 1, 1, 1, 1, 1, 9, 1, 3, 1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 1, 15, 1, 9, 1

Offset: 1

Antti Karttunen, Aug 05 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A003961, A003973, A009194, A336849, A336851.

A336850(n) = gcd(A003961(n),sigma(A003961(n)));

a(n) = gcd(A003961(n), A003973(n)) = gcd(A003961(n), A336851(n)).
a(n) = A009194(A003961(n)).
a(n) = A003961(n) / A336849(n).

A336852 a(n) = sigma(A003961(n)) - sigma(n).

Original entry on oeis.org

0, 1, 2, 6, 2, 12, 4, 25, 18, 14, 2, 50, 4, 24, 24, 90, 2, 85, 4, 62, 40, 20, 6, 180, 26, 30, 116, 100, 2, 120, 6, 301, 36, 26, 48, 312, 4, 36, 52, 230, 2, 192, 4, 98, 170, 48, 6, 602, 76, 135, 48, 136, 6, 504, 40, 360, 64, 38, 2, 456, 6, 56, 268, 966, 60, 192, 4, 134, 84, 240, 2, 1045, 6, 54, 218, 172, 72, 264, 4, 782

Offset: 1

Antti Karttunen, Aug 05 2020

nonn

Inverse Möbius transform of A336853(n) = (A003961(n) - n).

Antti Karttunen, Table of n, a(n) for n = 1..16383
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000203, A003961, A003973, A286385, A336851, A336853.

Cf. A001105 (positions of odd terms), A001359 (positions of 2's).

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336852(n) = (sigma(A003961(n)) - sigma(n));

PARI
```
A336852(n) = sumdiv(n,d,A003961(d)-d);
```

a(n) = Sum_{d|n} (A003961(d)-d).
a(n) = A003973(n) - A000203(n) = A000203(A003961(n)) - A000203(n).
a(n) = A336851(n) + A286385(n).

A344587 Deficiency of prime-shifted n: a(n) = 2*A003961(n) - sigma(A003961(n)).

Original entry on oeis.org

1, 2, 4, 5, 6, 6, 10, 14, 19, 10, 12, 12, 16, 18, 22, 41, 18, 26, 22, 22, 38, 22, 28, 30, 41, 30, 94, 42, 30, 18, 36, 122, 46, 34, 58, 47, 40, 42, 62, 58, 42, 42, 46, 52, 102, 54, 52, 84, 109, 66, 70, 72, 58, 126, 70, 114, 86, 58, 60, 6, 66, 70, 178, 365, 94, 54, 70, 82, 110, 78, 72, 110, 78, 78, 148, 102, 118, 78

Offset: 1

Antti Karttunen, May 28 2021

First negative value occurs as a(120) = -30.

Questions: Which subsets of natural numbers generate the "cut sigmoid" graph(s) that cross the X-axis in the (lowermost) scatter plot?

Cf. A000203, A003961, A003973, A033879, A153881, A336851, A337386 (positions of terms <= 0), A346246 (Dirichlet inverse), A349387, A378216, A378231 [= a(n^2)].

Inverse Möbius transform of A337544.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A344587(n) = { my(u=A003961(n)); (u+u - sigma(u)); };

a(n) = A033879(A003961(n)) = 2*A003961(n) - A003973(n).
a(n) = Sum_{d|n} A337544(d).
From Antti Karttunen, Nov 23 2024: (Start)
a(n) = Sum_{d|n} A003961(d)*A153881(n/d) = A003961(n) - A336851(n).
a(n) = Sum_{d|n} A033879(d)*A349387(n/d).
a(n) = Sum_{d|n} A003972(d)*A378216(n/d).
(End)

cp's OEIS Frontend

A003973 Inverse Möbius transform of A003961; a(n) = sigma(A003961(n)), where A003961 shifts the prime factorization of n one step towards the larger primes.

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A286385 a(n) = A003961(n) - A000203(n).

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A336850 a(n) = gcd(A003961(n), sigma(A003961(n))), where A003961 is the prime shift towards larger primes.

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A336852 a(n) = sigma(A003961(n)) - sigma(n).

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A344587 Deficiency of prime-shifted n: a(n) = 2*A003961(n) - sigma(A003961(n)).

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