cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 13 results. Next

A295663 a(n) = A295664(n) - A056169(n); 2-adic valuation of tau(n) minus the number of unitary prime divisors of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2
Offset: 1

Views

Author

Antti Karttunen, Nov 28 2017

Keywords

Links

Crossrefs

Cf. A000005, A007814, A056169, A295662, A295664.
Cf. A295661 (positions of nonzero terms).

Programs

  • Mathematica
    Table[IntegerExponent[DivisorSigma[0, n], 2] - DivisorSum[n, 1 &, And[PrimeQ@ #, CoprimeQ[#, n/#]] &], {n, 105}] (* Michael De Vlieger, Nov 28 2017 *)
  • PARI
    a(n) = vecsum(apply(x -> if(x == 1, 0, valuation(x+1, 2)), factor(n)[, 2])); \\ Amiram Eldar, Sep 28 2023

Formula

Additive with a(p) = 0, a(p^e) = A007814(1+e) if e > 1.
a(1) = 0; and for n > 1, if A067029(n) = 1, a(n) = a(A028234(n)), otherwise A007814(1+A067029(n)) + a(A028234(n)).
a(n) = A295664(n) - A056169(n).
a(n) = 0 iff A295662(n) = 0, and when A295662(n) > 0, a(n) >= A295662(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} f(1/p) = 0.22852676306472099280..., where f(x) = -1 + (1-x)*(-x + Sum_{k>=0} x^(2^k-1)/(1-x^(2^k))). - Amiram Eldar, Sep 28 2023

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.

Original entry on oeis.org

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

Views

Author

Marc LeBrun

Keywords

Comments

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

Links

Crossrefs

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.

Programs

  • Mathematica
    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 *)
  • PARI
    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
  • PARI
    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
  • Python
    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

Formula

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

Extensions

More terms from David W. Wilson, Aug 29 2001
Secondary name added by Antti Karttunen, Aug 06 2020

A056169 Number of unitary prime divisors of n.

Original entry on oeis.org

0, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 1, 0, 2, 0, 1, 1, 3, 1, 0, 2, 2, 2, 0, 1, 2, 2, 1, 1, 3, 1, 1, 1, 2, 1, 1, 0, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 0, 2, 3, 1, 1, 2, 3, 1, 0, 1, 2, 1, 1, 2, 3, 1, 1, 0, 2, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 2, 1, 1, 1, 1, 0, 1, 3, 1, 1, 3
Offset: 1

Views

Author

Labos Elemer, Jul 27 2000

Keywords

Comments

The zeros of this sequences are the powerful numbers (A001694). There are no arbitrarily long subsequences with a given upper bound; for example, every sequence of 4 values includes one divisible by 2 but not 4, so there are no more than 3 consecutive zeros. Similarly, there can be no more than 23 consecutive values with none divisible by both 2 and 3 but neither 4 nor 9 (so a(n) >= 2), etc. In general, this gives an upper bound that is a (relatively) small multiple of the k-th primorial number (prime(k)#). One suspects that the actual upper bounds for such subsequences are quite a bit lower; e.g., Erdős conjectured that there are no three consecutive powerful numbers. - Franklin T. Adams-Watters, Aug 08 2006
In particular, for every A048670(k)*A002110(k) consecutive terms, at least one is greater than or equal to k. - Charlie Neder, Jan 03 2019
Following Catalan's conjecture (which became Mihăilescu's theorem in 2002), the first case of two consecutive zeros in this sequence is for a(8) and a(9), because 8 = 2^3 and 9 = 3^2, and there are no other consecutive zeros for consecutive powers. However, there are other pairs of consecutive zeros at powerful numbers (A001694, A060355). The next example is a(288) = a(289) = 0, because 288 = 2^5 * 3^2 and 289 = 17^2, then also a(675) and a(676). - Bernard Schott, Jan 06 2019
a(2k-1) is the number of primes p such that p || x + y and p^2 || x^(2k-1) + y^(2k-1) for some positive integers x and y. For any positive integers x, y and k > 1, there is no prime p such that p || x + y and p^2 || x^(2k) + y^(2k). - Jinyuan Wang, Apr 08 2020

Examples

			9 = 3^2 so a(9) = 0; 10 = 2 * 5 so a(10) = 2; 11 = 11^1 so a(11) = 1.

Links

Crossrefs

Cf. A001221, A001694, A002110, A034444, A056170, A055231, A076445, A162642, A275812, A295659, A295662, A295664, A001694.
See also A060355.
Cf. A077761, A085548.

Programs

  • Haskell
    a056169 = length . filter (== 1) . a124010_row
-- Reinhard Zumkeller, Sep 10 2013
  • Maple
    a:= n-> nops(select(i-> i[2]=1, ifactors(n)[2])):
seq(a(n), n=1..120);  # Alois P. Heinz, Mar 27 2017
  • Mathematica
    Join[{0},Table[Count[Transpose[FactorInteger[n]][[2]],1],{n,2,110}]] (* Harvey P. Dale, Mar 15 2012 *)
Table[DivisorSum[n, 1 &, And[PrimeQ@ #, CoprimeQ[#, n/#]] &], {n, 105}] (* Michael De Vlieger, Nov 28 2017 *)
  • PARI
    a(n)=my(f=factor(n)[,2]); sum(i=1,#f,f[i]==1) \\ Charles R Greathouse IV, Apr 29 2015
  • Python
    from sympy import factorint
def a(n):
    f=factorint(n)
    return 0 if n==1 else sum(1 for i in f if f[i]==1)
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 19 2017
  • Scheme
    ;; With memoization-macro definec.
(definec (A056169 n) (if (= 1 n) 0 (+ (if (= 1 (A067029 n)) 1 0) (A056169 (A028234 n))))) ;; Antti Karttunen, Nov 28 2017

Formula

A prime factor of n is unitary iff its exponent is 1 in prime factorization of n. In general, gcd(p, n/p) = 1 or = p.
Additive with a(p^e) = 1 if e = 1, 0 otherwise.
a(n) = #{k: A124010(n,k) = 1, k = 1..A001221}. - Reinhard Zumkeller, Sep 10 2013
From Antti Karttunen, Nov 28 2017: (Start)
a(1) = 0; for n > 1, a(n) = A063524(A067029(n)) + a(A028234(n)).
a(n) = A001221(A055231(n)) = A001222(A055231(n)).
a(n) = A001221(n) - A056170(n) = A001221(n) - A001221(A000188(n)).
a(n) = A001222(n) - A275812(n).
a(n) = A162642(n) - A295662(n).
a(n) <= A162642(n) <= a(n) + A295659(n).
a(n) <= A295664(n).
(End)
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B - C), where B is Mertens's constant (A077761) and C = Sum_{p prime} (1/p^2) = 0.452247... (A085548). - Amiram Eldar, Sep 28 2023

A162642 Number of odd exponents in the canonical prime factorization of n.

Original entry on oeis.org

0, 1, 1, 0, 1, 2, 1, 1, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 2, 0, 2, 1, 1, 1, 3, 1, 1, 2, 2, 2, 0, 1, 2, 2, 2, 1, 3, 1, 1, 1, 2, 1, 1, 0, 1, 2, 1, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 0, 2, 3, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 0, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 1, 1, 1, 0, 1, 3, 1, 2, 3
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 08 2009

Keywords

Comments

a(n) is also known as the squarefree rank of n. - Jason Kimberley, Jul 08 2017
The number of primes that are infinitary divisors of n. - Amiram Eldar, Oct 01 2023

Links

Crossrefs

Cf. A000290 (positions of zeros), A001221, A001620, A002035, A007913, A056169, A162641, A295316, A295659, A295662, A295664.

Programs

Formula

a(n) = A001221(n) - A162641(n).
a(n) = A001221(A007913(n)). - Jason Kimberley, Jan 06 2016
a(A000290(n)) = 0, n > 0. - Michel Marcus, Jan 08 2016
G.f.: Sum_{i>=1} Sum_{j>=1} (-1)^j x^(prime(i)^j)/(x^(prime(i)^j) - 1). - Robert Israel, Jan 15 2016
From Antti Karttunen, Nov 28 2017: (Start)
Additive with a(p^e) = A000035(e).
a(n) = A056169(n) + A295662(n).
A056169(n) <= a(n) <= A056169(n) + A295659(n).
a(n) <= A295664(n).
(End)
Sum_{k=1..n} a(k) = n * log(log(n)) + c * n + O(n/log(n)), where c = gamma + Sum_{p prime} (log(1-1/p) + 1/(p+1)) = A077761 - A179119 = -0.0687327134... and gamma is Euler's constant (A001620). - Amiram Eldar, Dec 25 2021

A295662 Number of odd exponents larger than one in the canonical prime factorization of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
Offset: 1

Views

Author

Antti Karttunen, Nov 28 2017

Keywords

Examples

			For n = 24 = 2^3 * 3^1 there are two odd exponents, but only the other is larger than 1, thus a(24) = 1.
For n = 216 = 2^3 * 3^3 there are two odd exponents larger than 1, thus a(216) = 2.

Links

Crossrefs

Cf. A056169, A162642, A295659, A295663, A295664.
Cf. A295661 (positions of nonzero terms).

Programs

Formula

Additive with a(p) = 0, a(p^e) = A000035(e) if e > 1.
a(1) = 0; and for n > 1, if A067029(n) = 1, a(n) = a(A028234(n)), otherwise A000035(A067029(n)) + a(A028234(n)).
a(n) = A162642(n) - A056169(n).
a(n) <= A295659(n).
a(n) = 0 iff A295663(n) = 0, and when A295663(n) > 0, a(n) <= A295663(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} 1/(p^2*(p+1)) = 0.122017493776862257491... . - Amiram Eldar, Sep 28 2023

A336937 The 2-adic valuation of sigma(n), the sum of divisors of n.

Original entry on oeis.org

0, 0, 2, 0, 1, 2, 3, 0, 0, 1, 2, 2, 1, 3, 3, 0, 1, 0, 2, 1, 5, 2, 3, 2, 0, 1, 3, 3, 1, 3, 5, 0, 4, 1, 4, 0, 1, 2, 3, 1, 1, 5, 2, 2, 1, 3, 4, 2, 0, 0, 3, 1, 1, 3, 3, 3, 4, 1, 2, 3, 1, 5, 3, 0, 2, 4, 2, 1, 5, 4, 3, 0, 1, 1, 2, 2, 5, 3, 4, 1, 0, 1, 2, 5, 2, 2, 3, 2, 1, 1, 4, 3, 7, 4, 3, 2, 1, 0, 2, 0, 1, 3, 3, 1, 6
Offset: 1

Views

Author

Antti Karttunen, Aug 12 2020

Keywords

Comments

Equally, the 2-adic valuation of A000593(n), the sum of odd divisors of n.
Proof for the given additive formula: It's easy to see that for all powers of 2 and all even powers of odd primes the result is zero. Thus assuming p is an odd prime, factorize sigma(p^(2e-1)) = (1 + p + p^2 + ... + p^(2e-1)) as (1+p)*(1 + u + u^2 + u^3 + ... + u^(e-1)), where u=p^2. Note that u [and its powers] are always of the form 4k+1, thus the 2-adic valuation of that sum is A007814(e) [see my Aug 15 2020 comment there] which when added to the 2-adic valuation of 1+p then gives the 2-adic valuation for whole sigma(p^(2e-1)).

Links

Crossrefs

One less than A286357.
Cf. A000035, A000203, A000265, A000593, A007814, A028982 (positions of zeros), A161942, A295664, A336932.

Programs

  • Mathematica
    a[n_] := IntegerExponent[DivisorSigma[1, n], 2]; Array[a, 100] (* Amiram Eldar, Jul 04 2022 *)
  • PARI
    A336937(n) = valuation(sigma(n),2);
  • PARI
    A007814(n) = valuation(n,2);
A336937(n) = { my(f=factor(n)); sum(i=1,#f~,(f[i,1]%2) * (f[i,2]%2) * (A007814(1+f[i,1])+A007814(1+f[i,2])-1)); };
  • Python
    from sympy import divisor_sigma
def A336937(n): return (~(m:=int(divisor_sigma(n))) & m-1).bit_length() # Chai Wah Wu, Jul 01 2022

Formula

a(n) = A007814(A000203(n)) = A007814(A000593(n)).
Additive with a(2^e) = 0, a(p^2e) = 0, a(p^(2e-1)) = A007814(1+p) + A007814(e).
For n with factorization p_1^e_1 * ... * p_k^e_k, a(n) = Sum_{i=1..k} A000035(p_i) * A000035(e_i) * (A007814(1+p_i)+A007814(1+e_i)-1).
a(n) = A286357(n)-1.
a(n) = a(2*n) = a(A000265(n)).
For any odd n, a(n) >= A295664(n).
a(A003961(n)) = A336932(n).
A161942(n) = A000203(n) / (2^a(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

Views

Author

Antti Karttunen, Aug 07 2020

Keywords

Comments

Also denominator of A336841(n) / A000005(n).
All terms are odd because A336932(n) = A007814(A003973(n)) >= A295664(n) for all n.

Links

Crossrefs

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).

Programs

  • PARI
    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));

Formula

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.

A212181 Largest odd divisor of tau(n): a(n) = A000265(A000005(n)).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 5, 3, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 7, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 3, 3, 1, 1, 1, 5, 5, 1, 1, 3, 1, 1, 1, 1
Offset: 1

Views

Author

Matthew Vandermast, Jun 04 2012

Keywords

Comments

Completely determined by the exponents >=2 in the prime factorization of n (cf. A212172).
Not the same as the number of odd divisors of n (A001227(n)); see example.
Multiplicative because A000005 is multiplicative and A000265 is completely multiplicative. - Andrew Howroyd, Aug 01 2018
a(n) = 1 iff the number of divisors of n is a power of 2 (A036537). - Bernard Schott, Nov 04 2022

Examples

			48 has a total of 10 divisors (1, 2, 3, 4, 6, 8, 12, 16, 24 and 48). Since the largest odd divisor of 10 is 5, a(48) = 5.

Links

Crossrefs

Cf. A000005, A000079, A000265, A036537, A108951, A212172, A295664, A331286 (applied to primorial inflation of n).

Programs

  • Mathematica
    Table[Block[{nd=DivisorSigma[0, n]}, nd/2^IntegerExponent[nd, 2]], {n, 100}] (* Indranil Ghosh, Jul 19 2017, after PARI code *)
  • PARI
    a(n) = my(nd = numdiv(n)); nd/2^valuation(nd, 2); \\ Michel Marcus, Jul 19 2017
  • Python
    from sympy import divisor_count, divisors
def a(n): return [i for i in divisors(divisor_count(n)) if i%2][-1]
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jul 19 2017

Formula

a(n) = A000265(A000005(n)).
From Antti Karttunen, Jan 14 2020: (Start)
a(n) = A000005(n) / A000079(A295664(n)).
a(A108951(n)) = A331286(n).
(End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p odd prime} ((1 - 1/p)*(1 + Sum_{k>=1} a(k+1)/p^k)) = 2.076325817863586... . - Amiram Eldar, Oct 15 2022

A336932 The 2-adic valuation of A003973(n), the sum of divisors of prime shifted n.

Original entry on oeis.org

0, 2, 1, 0, 3, 3, 2, 3, 0, 5, 1, 1, 1, 4, 4, 0, 2, 2, 3, 3, 3, 3, 1, 4, 0, 3, 2, 2, 5, 6, 1, 2, 2, 4, 5, 0, 1, 5, 2, 6, 2, 5, 4, 1, 3, 3, 1, 1, 0, 2, 3, 1, 2, 4, 4, 5, 4, 7, 1, 4, 2, 3, 2, 0, 4, 4, 3, 2, 2, 7, 1, 3, 4, 3, 1, 3, 3, 4, 2, 3, 0, 4, 1, 3, 5, 6, 6, 4, 1, 5, 3, 1, 2, 3, 6, 3, 1, 2, 1, 0, 3, 5, 2, 4, 6
Offset: 1

Views

Author

Antti Karttunen, Aug 16 2020

Keywords

Links

Crossrefs

Cf. A000203, A000290 (positions of zeros), A003961, A003973, A007814, A295664, A336937.

Programs

  • PARI
    A336932(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); valuation(sigma(factorback(f)),2); };
  • PARI
    A007814(n) = valuation(n, 2);
A336932(n) = { my(f=factor(n)); sum(i=1, #f~, (f[i, 2]%2) * (A007814(1+nextprime(1+f[i, 1]))+A007814(1+f[i, 2])-1)); };
  • Python
    from math import prod
from sympy import factorint, nextprime
def A336932(n): return (~(m:=prod(((q:=nextprime(p))**(e+1)-1)//(q-1) for p,e in factorint(n).items()))& m-1).bit_length() # Chai Wah Wu, Jul 05 2022

Formula

Additive with a(p^2e) = 0, a(p^(2e-1)) = A007814(1+A003961(p)) + A007814(e).
a(n) = A007814(A003973(n)).
a(n) = A336937(A003961(n)).
For all n >= 1, a(n) >= A295664(n).

A336931 Difference between the 2-adic valuation of A003973(n) [= the sum of divisors of the prime shifted n] and the 2-adic valuation of the number of divisors of n.

Original entry on oeis.org

0, 1, 0, 0, 2, 1, 1, 1, 0, 3, 0, 0, 0, 2, 2, 0, 1, 1, 2, 2, 1, 1, 0, 1, 0, 1, 0, 1, 4, 3, 0, 1, 0, 2, 3, 0, 0, 3, 0, 3, 1, 2, 3, 0, 2, 1, 0, 0, 0, 1, 1, 0, 1, 1, 2, 2, 2, 5, 0, 2, 1, 1, 1, 0, 2, 1, 2, 1, 0, 4, 0, 1, 3, 1, 0, 2, 1, 1, 1, 2, 0, 2, 0, 1, 3, 4, 4, 1, 0, 3, 1, 0, 0, 1, 4, 1, 0, 1, 0, 0, 2, 2, 1, 1, 3
Offset: 1

Views

Author

Antti Karttunen, Aug 17 2020

Keywords

Comments

Note that A295664(n) = A295664(A003961(n)).

Links

Crossrefs

Cf. A003961, A003973, A007814, A007913, A295664, A336930 (positions of zeros), A336932, A336937.

Programs

  • PARI
    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)); };
  • PARI
    A003973(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); sigma(factorback(f)); };
A007814(n) = valuation(n, 2);
A336931(n) = (A007814(A003973(n)) - A007814(numdiv(n)));
  • Python
    from math import prod
from sympy import factorint, nextprime, divisor_count
def A336931(n): return (~(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() # Chai Wah Wu, Jul 05 2022

Formula

Additive with a(p^e) = 0 when e is even, a(p^e) = A007814(1+A003961(p))-1 when e is odd.
a(n) = A336932(n) - A295664(n).
a(n) = a(A007913(n)).
Showing 1-10 of 13 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.