A336856 -id:A336856 - 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

A336841 Prime-shifted analog of A094471: a(n) = A336845(n) - A003973(n).

Original entry on oeis.org

0, 2, 4, 14, 6, 36, 10, 68, 44, 52, 12, 192, 16, 84, 92, 284, 18, 326, 22, 274, 148, 100, 28, 840, 90, 132, 344, 438, 30, 648, 36, 1094, 176, 148, 212, 1622, 40, 180, 232, 1192, 42, 1032, 46, 520, 802, 228, 52, 3324, 230, 654, 260, 684, 58, 2376, 252, 1896, 316, 244, 60, 3156, 66, 292, 1278, 4010, 332, 1224, 70, 766

Offset: 1

Antti Karttunen, Aug 06 2020

nonn

All terms are even because A003973 and A336845 match parity-wise. Also in the sum formulas, only even terms are summed (only one of which is zero).

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. A000005, A000203, A003961, A048673, A094471, A336837, A336839, A336840, A336845, A336856.
Cf. A336846 [= gcd(a(n), A003973(n))].
Twice the terms of A336854.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336841(n) = ((numdiv(n)*A003961(n)) - sigma(A003961(n)));

A336841(n) = sumdiv(n,d,(A003961(n)-A003961(d)));

A336841(n) = sumdiv(A003961(n),d,(A003961(n)-d));

a(n) = A336845(n) - A003973(n) = (A000005(n)*A003961(n)) - A000203(A003961(n)).
a(n) = A094471(A003961(n)).
a(n) = Sum_{d|n} (A003961(n)-A003961(d)) = Sum_{d|A003961(n)} (A003961(n)-d).
a(n) = 2*A336854(n) = 2*Sum_{d|n} (A048673(n)-A048673(d)).
a(n) = ((A003961(n)+1)*A000005(n)) - 2*A336840(n).
a(n) = 2 * ((A000005(n)*A048673(n)) - A336840(n)).
a(n) = A000005(n) * (A336837(n)/A336839(n)) = A336837(n) * A336856(n).

A336839 Denominator of the arithmetic mean of the divisors of A003961(n).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 5, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Aug 07 2020

Also denominator of A336841(n) / A000005(n).

All terms are odd because A336932(n) = A007814(A003973(n)) >= A295664(n) for all n.

Cf. A000005, A000225, A003961, A003973, A007814, A052940, A057021, A295664, A336840, A336841, A336856, A336931, A336932.
Cf. A336918 (positions of 1's), A336919 (of terms > 1).
Cf. A336837 and A336838 (numerators).

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336839(n) = denominator(sigma(A003961(n))/numdiv(n));

a(n) = denominator(A003973(n)/A000005(n)).
a(n) = d(n)/A336856(n) = d(n)/gcd(d(n),A003973(n)) = d(n)/gcd(d(n),A336841(n)), where d(n) is the number of divisors of n, A000005(n).
a(n) = A057021(A003961(n)).
For all primes p, and e >= 0, a(A000225(e)) = a(p^((2^e) - 1)) = 1. [See A336856]
It seems that for all odd primes p, and with the exponents e=5, 11, 17 or 23 (at least these), a(p^e) = 1.
It seems that a(27^((2^n)-1)) = A052940(n-1) for all n >= 1.

A336838 Numerator of the arithmetic mean of the divisors of A003961(n).

Original entry on oeis.org

1, 2, 3, 13, 4, 6, 6, 10, 31, 8, 7, 13, 9, 12, 12, 121, 10, 62, 12, 52, 18, 14, 15, 30, 19, 18, 39, 26, 16, 24, 19, 182, 21, 20, 24, 403, 21, 24, 27, 40, 22, 36, 24, 91, 124, 30, 27, 363, 133, 38, 30, 39, 30, 78, 28, 60, 36, 32, 31, 52, 34, 38, 62, 1093, 36, 42, 36, 130, 45, 48, 37, 310, 40, 42, 57, 52, 42, 54, 42

Offset: 1

Antti Karttunen, Aug 07 2020

Ratio r(n) = a(n)/A336839(n) is multiplicative. For example r(3) = 3/1, r(4) = 13/3, thus r(12) = r(3)*r(4) = 13/1.

Conjecture: For all primes p with an odd exponent e, a(p^e) is a multiple of A048673(p). Note that q+1 is a divisor of (q+1)^e - sigma(q^e) = (q+1)^e - (1 + q + q^2 + ... + q^e) when e is odd, thus also A048673(p) = (q+1)/2 is, where q = A003961(p), thus the conjecture holds, unless the denominator (A336839) has enough prime factors of A048673(p).

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. A000005, A003961, A003973, A048673, A057020, A336840, A336918.

Cf. A336839 (denominators).

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336838(n) = numerator(sigma(A003961(n))/numdiv(n));

a(n) = A057020(A003961(n)).
a(n) = numerator(A003973(n)/A000005(n)).
a(n) = A003973(n) / A336856(n) = A003973(n) / gcd(A000005(n), A003973(n)).
a(p) = A048673(p) for all primes p.
a(p^3) = 2*A048673(p)^3 - 2*A048673(p)^2 + A048673(p). [The denominator A336839(p^3) = 1 for all p]

A336837 Numerator of ratio A336841(n)/A000005(n).

Original entry on oeis.org

0, 1, 2, 14, 3, 9, 5, 17, 44, 13, 6, 32, 8, 21, 23, 284, 9, 163, 11, 137, 37, 25, 14, 105, 30, 33, 86, 73, 15, 81, 18, 547, 44, 37, 53, 1622, 20, 45, 58, 149, 21, 129, 23, 260, 401, 57, 26, 1662, 230, 109, 65, 114, 29, 297, 63, 237, 79, 61, 30, 263, 33, 73, 213, 4010, 83, 153, 35, 383, 100, 183, 36, 1715, 39, 81

Offset: 1

Antti Karttunen, Aug 07 2020

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

Cf. A000005, A003961, A003973, A336838, A336841, A336856.

Cf. A336839 (denominators).

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336841(n) = ((numdiv(n)*A003961(n)) - sigma(A003961(n)));
A336837(n) = numerator(A336841(n)/numdiv(n));

a(n) = A336841(n) / A336856(n) = A336841(n) / gcd(A000005(n), A336841(n)).

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Python

Formula

Extensions

A336841 Prime-shifted analog of A094471: a(n) = A336845(n) - A003973(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

PARI

Formula

A336839 Denominator of the arithmetic mean of the divisors of A003961(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A336838 Numerator of the arithmetic mean of the divisors of A003961(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A336837 Numerator of ratio A336841(n)/A000005(n).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula