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.

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A295878 Multiplicative with a(p^(2e)) = 1, a(p^(2e-1)) = prime(e).

Original entry on oeis.org

1, 2, 2, 1, 2, 4, 2, 3, 1, 4, 2, 2, 2, 4, 4, 1, 2, 2, 2, 2, 4, 4, 2, 6, 1, 4, 3, 2, 2, 8, 2, 5, 4, 4, 4, 1, 2, 4, 4, 6, 2, 8, 2, 2, 2, 4, 2, 2, 1, 2, 4, 2, 2, 6, 4, 6, 4, 4, 2, 4, 2, 4, 2, 1, 4, 8, 2, 2, 4, 8, 2, 3, 2, 4, 2, 2, 4, 8, 2, 2, 1, 4, 2, 4, 4, 4, 4, 6, 2, 4, 4, 2, 4, 4, 4, 10, 2, 2, 2, 1, 2, 8, 2, 6, 8, 4, 2, 3, 2, 8, 4, 2, 2, 8, 4, 2, 2, 4, 4, 12
Offset: 1

Views

Author

Antti Karttunen, Nov 29 2017

Keywords

Comments

This sequence can be used as a filter. It matches at least to the following sequence, as for all i, j:
a(i) = a(j) => A162642(i) = A162642(j), as A162642(n) = A001222(a(n)).
a(i) = a(j) => A056169(i) = A056169(j), as A056169(n) = A007814(a(n)).
a(i) = a(j) => A295883(i) = A295883(j), as A295883(n) = A007949(a(n)).
a(i) = a(j) => A295662(i) = A295662(j).
a(i) = a(j) => A295664(i) = A295664(j).

Links

Crossrefs

Cf. A294873, A295879.

Programs

  • Mathematica
    Array[Apply[Times, FactorInteger[#] /. {p_, e_} /; p > 0 :> Which[p == 1, 1, EvenQ@ e, 1, True, Prime[(e + 1)/2]]] &, 120] (* Michael De Vlieger, Nov 29 2017 *)

Formula

a(1) = 1; for n>1, if n = Product prime(i)^e(i), then a(n) = Product prime((e(i)+1)/2)^A000035(e(i)).

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

Views

Author

Antti Karttunen, Aug 17 2020

Keywords

Comments

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.

Crossrefs

Positions of zeros in A336931.
Cf. A000290, A002144, A003961, A003973, A004613, A007814, A007913, A295664, A336918, 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)); };
isA336930(n) = !A336931(n);
  • PARI
    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)));
  • Python
    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

A386258 Exponent of the highest power of 2 dividing the product of exponents of the prime factorization of n.

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Jul 17 2025

Keywords

Comments

First differs from A386259 at n = 36.
First differs from A370078 at n = 64.
The first occurrence of k = 0, 1, 2, ... is at n = A085629(2^k) = 1, 4, 16, 144, 1296, 20736, 518400, ... .
The asymptotic density of the occurrences of 1 in this sequence is the asymptotic density of numbers whose prime factorization has only odd exponents except for one exponent that is of the form 4*k+2 (k >= 0) which equals A065463 * Sum_{p prime} p^2/(p^4+p^3+p-1) = 0.22670657681840536721... .

Links

Crossrefs

Cf. A005361, A007814, A065463, A085629, A268335, A295664, A370078, A386259.

Programs

  • Mathematica
    a[n_] := IntegerExponent[Times @@ FactorInteger[n][[;; , 2]], 2]; Array[a, 100]
  • PARI
    a(n) = vecsum(apply(x -> valuation(x, 2), factor(n)[, 2]));

Formula

a(n) = A007814(A005361(n)).
Additive with a(p^e) = A007814(e).
a(n) = 0 if and only if n is an exponentially odd number (A268335).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} f(1/p) = 0.37572872586497617473..., where f(x) = Sum_{k>=1} x^(2^k)/(1-x^(2^k)).
Previous Showing 11-13 of 13 results.

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