A336918 -id:A336918 - cp's OEIS frontend

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

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]

A336919 Numbers k such that A000005(k) does not divide A003973(k); numbers k for which A336839(k) > 1.

Original entry on oeis.org

4, 9, 16, 18, 20, 32, 36, 44, 45, 48, 49, 64, 68, 72, 80, 81, 90, 98, 99, 100, 112, 116, 124, 144, 153, 160, 162, 164, 169, 176, 180, 192, 196, 198, 208, 220, 225, 236, 240, 244, 245, 252, 256, 261, 279, 284, 288, 292, 304, 306, 320, 324, 336, 338, 340, 352, 356, 360, 361, 369, 392, 396, 400, 404, 405, 428, 432, 441

Offset: 1

Antti Karttunen, Aug 12 2020

nonn

Numbers k such that A003961(k) is not in A003601, but in A049642.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Cf. A000005, A003601, A003961, A003973, A049642, A336839.

Cf. A336918 (complement).

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA336919(n) = !!(sigma(A003961(n))%numdiv(n));

A336930 Numbers k such that the 2-adic valuation of A003973(k), the sum of divisors of the prime shifted k is equal to the 2-adic valuation of the number of divisors of k.

Original entry on oeis.org

1, 3, 4, 9, 11, 12, 13, 16, 23, 25, 27, 31, 33, 36, 37, 39, 44, 47, 48, 49, 52, 59, 64, 69, 71, 75, 81, 83, 89, 92, 93, 97, 99, 100, 107, 108, 109, 111, 117, 121, 124, 131, 132, 139, 141, 143, 144, 147, 148, 151, 156, 167, 169, 176, 177, 179, 188, 191, 192, 193, 196, 207, 208, 213, 225, 227, 229, 236, 239, 243, 249, 251

Offset: 1

Antti Karttunen, Aug 17 2020

nonn

Numbers k for which A295664(k) is equal to A336932(k). Note that A295664(A003961(n)) = A295664(n).
Numbers k such that A003961(A007913(k)) [or equally, A007913(A003961(k))] is in A004613, i.e., has only prime divisors of the form 4k+1.
Subsequences include squares (A000290), and also primes p which when prime-shifted [as A003961(p)] become primes of the form 4k+1 (A002144), and all their powers as well as the products between these.

Positions of zeros in A336931.

Cf. A000290, A002144, A003961, A003973, A004613, A007814, A007913, A295664, A336918, A336932, A336937.

A007814(n) = valuation(n, 2);
A336931(n) = { my(f=factor(n)); sum(i=1, #f~, (f[i, 2]%2) * (A007814(1+nextprime(1+f[i, 1]))-1)); };
isA336930(n) = !A336931(n);

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA004613(n) = (1==(n%4) && 1==factorback(factor(n)[, 1]%4)); \\ After code in A004613.
isA336930(n) = isA004613(A003961(core(n)));

from math import prod
from itertools import count, islice
from sympy import factorint, nextprime, divisor_count
def A336930_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:(~(m:=prod(((q:=nextprime(p))**(e+1)-1)//(q-1) for p,e in factorint(n).items()))& m-1).bit_length()==(~(k:=int(divisor_count(n))) & k-1).bit_length(),count(max(startvalue,1)))
A336930_list = list(islice(A336930_gen(),30)) # Chai Wah Wu, Jul 05 2022

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

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

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A336919 Numbers k such that A000005(k) does not divide A003973(k); numbers k for which A336839(k) > 1.

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A336930 Numbers k such that the 2-adic valuation of A003973(k), the sum of divisors of the prime shifted k is equal to the 2-adic valuation of the number of divisors of k.

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