A179119 -id:A179119 - cp's OEIS frontend

A349326 a(n) is the number of prime powers (not including 1) that are bi-unitary divisors of n.

0, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 3, 2, 1, 3, 1, 5, 2, 2, 2, 2, 1, 2, 2, 4, 1, 3, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 2, 4, 2, 2, 1, 3, 1, 2, 2, 5, 2, 3, 1, 2, 2, 3, 1, 4, 1, 2, 2, 2, 2, 3, 1, 4, 3, 2, 1, 3, 2, 2, 2, 4, 1, 3, 2, 2, 2, 2, 2, 6, 1, 2, 2, 2, 1, 3, 1, 4, 3

Offset: 1

Amiram Eldar, Nov 15 2021

The total number of prime powers (not including 1) that divide n is A001222(n).

The least number k such that a(k) = m is A122756(m).

			12 has 4 bi-unitary divisors, 1, 3, 4 and 12. Two of these divisors, 3 and 4 = 2^2 are prime powers. Therefore a(12) = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000961, A001222, A122756, A162641, A222266, A246655, A286324, A268335, A350390.
Similar sequences: A001222, A349258, A349281.
Cf. A083342, A179119.

f[p_, e_] := If[OddQ[e], e, e - 1]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecsum(apply(x -> if(x%2, x, x-1), factor(n)[, 2])); \\ Amiram Eldar, Sep 29 2023

Additive with a(p^e) = e if e is odd, and e-1 if e is even.
a(n) <= A001222(n), with equality if and only if n is an exponentially odd number (A268335).
a(n) <= A286324(n) - 1, with equality if and only if n is a prime power (including 1, A000961).
a(n) = A001222(n) - A162641(n). - Amiram Eldar, May 18 2023
From Amiram Eldar, Sep 29 2023: (Start)
a(n) = A001222(A350390(n)) (the number of prime factors of the largest exponentially odd number dividing n, counted with multiplicity).
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B_2 - C), where B_2 = A083342 and C = A179119. (End)

A375144 Numbers whose prime factorization has exactly two exponents that equal 2 and has no higher exponents.

Original entry on oeis.org

36, 100, 180, 196, 225, 252, 300, 396, 441, 450, 468, 484, 588, 612, 676, 684, 700, 828, 882, 980, 1044, 1089, 1100, 1116, 1156, 1225, 1260, 1300, 1332, 1444, 1452, 1476, 1521, 1548, 1575, 1692, 1700, 1900, 1908, 1980, 2028, 2100, 2116, 2124, 2156, 2178, 2196

Offset: 1

Amiram Eldar, Aug 01 2024

Numbers of the form m * p^2 * q^2, where p < q are primes, and m is a squarefree number such that gcd(m, p*q) = 1.
Numbers whose powerful part (A057521) is a square of a squarefree semiprime (A085986).
The asymptotic density of this sequence is ((Sum_{p prime} 1/(p*(p+1)))^2 - Sum_{p prime} 1/(p*(p+1))^2)/(2*zeta(2)) = 0.022124574473271163980012... .

			36 = 2^2 * 3^2 is a term since its prime factorization has exactly two exponents and both are equal to 2.

Cf. A057521, A060687, A085986, A179119.

Subsequence: A179643.

q[n_] := Module[{e = Sort[FactorInteger[n][[;; , 2]], Greater]}, Length[e] > 1 && e[[1;;2]] == {2, 2} && If[Length[e] > 2, e[[3]] == 1, True]]; Select[Range[2200], q]

is(k) = {my(e = vecsort(factor(k)[,2], , 4)~); #e > 1 && e[1..2] == [2,2] && if(#e > 2, e[3] == 1, 1);}

A382562 Decimal expansion of Sum_{p prime} 1/(p^2*(p + 1)).

Original entry on oeis.org

1, 2, 2, 0, 1, 7, 4, 9, 3, 7, 7, 6, 8, 6, 2, 2, 5, 7, 4, 9, 1, 4, 4, 8, 7, 7, 6, 7, 4, 5, 5, 0, 3, 1, 7, 3, 5, 2, 8, 9, 7, 1, 9, 2, 3, 7, 6, 5, 8, 2, 1, 9, 6, 5, 5, 8, 6, 1, 9, 0, 3, 2, 9, 5, 8, 8, 7, 4, 2, 8, 6, 0, 5, 7, 1, 0, 0, 2, 9, 3, 0, 1, 2, 7, 2, 5, 4, 3, 3, 6, 8, 4, 1, 6, 7, 1, 7, 5, 5, 1, 0

Offset: 0

Artur Jasinski, Mar 31 2025

			0.1220174937768622574914487767455031735289719237658219655861903295887428605710029...

Index to constants which are prime zeta sums {2,0,1}.

Cf. A085548, A179119.

sumeulerrat(1/(p^2*(p+1))) \\ Amiram Eldar, Apr 01 2025

Equals A085548 - A179119.

Equals Sum_{k>=3} (-1)^(k+1) * P(k), where P is the prime zeta function. - Amiram Eldar, Apr 01 2025

A382567 Decimal expansion of Sum_{p prime} 1/(p*(p + 1)^2).

Original entry on oeis.org

0, 8, 5, 7, 5, 2, 6, 2, 2, 1, 0, 7, 6, 0, 9, 9, 3, 4, 0, 6, 3, 1, 4, 6, 2, 1, 6, 7, 3, 9, 3, 9, 3, 7, 9, 3, 0, 0, 6, 8, 8, 5, 7, 6, 7, 2, 9, 5, 3, 0, 8, 6, 5, 7, 1, 9, 6, 5, 3, 2, 4, 5, 0, 2, 7, 8, 6, 0, 4, 0, 2, 1, 7, 2, 8, 6, 5, 3, 6, 9, 7, 9, 9, 6, 2, 2, 2, 1, 8, 2, 2, 4, 9, 3, 3, 9, 5, 5, 8, 6, 7, 7, 9, 5

Offset: 0

Artur Jasinski, Mar 31 2025

			0.0857526221076099340631462167393937930068857672953086571965324502786040217286...

Index to constants which are prime zeta sums {1,0,2}.

Cf. A179119, A382554.

sumeulerrat(1/(p*(p+1)^2)) \\ Amiram Eldar, Apr 01 2025

Equals A179119 - A382554.

Equals Sum_{k>=3} (-1)^(k+1) * (k-2) * P(k), where P is the prime zeta function. - Amiram Eldar, Apr 01 2025

A382584 Decimal expansion of Sum_{p prime} 1/((p - 1)^2p(p + 1)).

Original entry on oeis.org

1, 9, 0, 2, 2, 2, 4, 7, 7, 1, 5, 3, 0, 2, 2, 1, 0, 8, 3, 1, 4, 1, 2, 4, 6, 1, 7, 3, 9, 0, 9, 4, 9, 2, 4, 3, 0, 3, 6, 8, 0, 8, 8, 3, 2, 8, 9, 3, 7, 8, 6, 8, 0, 7, 1, 5, 8, 8, 9, 7, 2, 6, 7, 6, 1, 8, 6, 9, 1, 6, 2, 6, 9, 0, 2, 0, 7, 9, 5, 6, 5, 4, 2, 0, 0, 3, 0, 5, 5, 8, 9, 6, 9, 1, 1, 2, 2, 1, 9, 2, 9, 7, 3

Offset: 0

Artur Jasinski, Mar 31 2025

			0.1902224771530221083141246173909492430368088328937868071588972676186916269020795654200305...

Index to constants which are prime zeta sums {1,2,1}.

Cf. A086242, A136141, A179119.

sumeulerrat(1/((p-1)^2*p*(p+1))) \\ Amiram Eldar, Apr 02 2025

Equals -3*A136141/4 + A086242/2 + A179119/4.

Equals Sum_{k>=2} (k-1) * (P(2*k) + P(2*k+1)), where P is the prime zeta function. - Amiram Eldar, Apr 02 2025

A284748 Decimal expansion of the sum of reciprocals of composite powers.

Original entry on oeis.org

2, 2, 6, 8, 4, 3, 3, 3, 0, 9, 5, 0, 2, 0, 4, 8, 7, 2, 1, 3, 5, 6, 3, 2, 5, 4, 0, 1, 4, 4, 0, 5, 7, 6, 0, 4, 3, 8, 1, 2, 5, 8, 6, 6, 3, 9, 1, 6, 8, 1, 3, 9, 5, 1, 6, 8, 8, 9, 9, 3, 9, 3, 2, 6, 4, 3, 2, 9, 0, 9, 7, 1, 5, 1, 0, 7, 6, 6, 6, 0, 2, 1, 6, 6, 2, 0, 1, 2, 4, 1, 1, 7, 6, 6, 7, 9, 1, 8, 1, 6, 7, 1, 0, 6, 2, 1

Offset: 0

Terry D. Grant, Apr 01 2017

			Equals 1/(4*3)+1/(6*5)+1/(8*7)+1/(9*8)+1/(10*9)+...
= 0.226843330950204872135632540144057604...

Cf. A066247, A077761, A179119, A185380.
Decimal expansion of the sum of reciprocal powers: A136141 (primes), A154945 (primes at even powers), A152447 (semiprimes), A154932 (squarefree semiprimes).
Decimal expansion of the 'nonprime zeta function': A275647 (at 2), A278419 (at 3).

RealDigits[ NSum[Zeta[n]-1-PrimeZetaP[n], {n, 2, Infinity}], 10, 105] [[1]]

1 - sumeulerrat(1/(p*(p-1))) \\ Amiram Eldar, Mar 18 2021

Equals Sum_{n>=1} 1/A002808(n)^(n+1) = (A275647 - 1) + (A278419 - 1) + ...
Equals Sum_{n>=1} 1/A002808(n)*(A002808(n)-1).
Equals Sum_{n>=2} (Zeta(n) - PrimeZeta(n) - 1) = Sum_{n>=2} CompositeZeta(n).
Equals 1 - A136141.

More digits from Vaclav Kotesovec, Jan 13 2021

A382563 Decimal expansion of Sum_{p prime} 1/(p^3*(p + 1)).

Original entry on oeis.org

0, 5, 2, 7, 4, 5, 1, 4, 5, 5, 2, 2, 5, 8, 1, 2, 7, 8, 9, 3, 1, 6, 6, 4, 5, 3, 7, 9, 2, 0, 2, 0, 3, 5, 2, 7, 4, 4, 6, 4, 4, 0, 1, 9, 8, 1, 6, 0, 3, 2, 7, 3, 2, 4, 3, 0, 2, 4, 8, 1, 6, 8, 7, 1, 1, 2, 8, 8, 8, 7, 2, 8, 9, 5, 7, 1, 2, 6, 6, 5, 7, 5, 0, 8, 3, 7, 9, 8, 6, 3, 6, 9, 5, 3, 0, 0, 8, 0, 5, 4, 4, 5, 1

Offset: 0

Artur Jasinski, Mar 31 2025

			0.0527451455225812789316645379202035274464401981603273243024816...

Index to constants which are prime zeta sums {3,0,1}.

Cf. A085541, A085548, A179119.

sumeulerrat(1/(p^3*(p+1))) \\ Amiram Eldar, Apr 01 2025

Equals -A085548 + A085541 + A179119.

Equals Sum_{k>=4} (-1)^k * P(k), where P is the prime zeta function. - Amiram Eldar, Apr 01 2025

A382565 Decimal expansion of Sum_{p prime} 1/((p - 1)^2*(p + 1)).

Original entry on oeis.org

4, 1, 1, 6, 8, 5, 8, 4, 8, 5, 4, 5, 8, 1, 8, 0, 5, 1, 7, 3, 8, 7, 6, 1, 0, 5, 3, 2, 7, 5, 5, 4, 8, 0, 6, 0, 5, 2, 4, 0, 4, 9, 7, 9, 1, 1, 9, 8, 3, 4, 4, 6, 0, 3, 2, 3, 9, 2, 8, 6, 0, 0, 0, 9, 1, 5, 8, 3, 5, 0, 5, 7, 7, 5, 0, 2, 4, 2, 2, 9, 2, 1, 7, 6, 0, 7, 8, 3, 3, 4, 6, 2, 4, 1, 5, 5, 0, 7, 6, 9, 0, 1, 2, 7

Offset: 0

Artur Jasinski, Mar 31 2025

			0.411685848545818051738761053275548060524049791198344603239286000...

Index to constants which are prime zeta sums {0,2,1}.

Cf. A086242, A136141, A179119.

sumeulerrat(1/((p-1)^2*(p+1))) \\ Amiram Eldar, Apr 02 2025

Equals -A136141/4 + A086242/2 - A179119/4.

Equals Sum_{k>=2} (k-1) * (P(2*k-1) + P(2*k)), where P is the prime zeta function. - Amiram Eldar, Apr 02 2025

A382566 Decimal expansion of Sum_{p prime} 1/((p - 1)^3*(p + 1)).

Original entry on oeis.org

3, 6, 7, 9, 2, 1, 6, 2, 4, 6, 0, 6, 3, 8, 0, 9, 9, 7, 5, 9, 7, 2, 6, 1, 3, 7, 6, 5, 0, 3, 2, 9, 1, 1, 7, 0, 5, 6, 0, 3, 5, 7, 4, 7, 2, 1, 6, 3, 3, 8, 3, 8, 2, 6, 0, 1, 8, 2, 6, 9, 0, 6, 1, 5, 1, 7, 3, 4, 9, 2, 6, 6, 3, 5, 2, 4, 0, 1, 5, 6, 9, 6, 4, 8, 2, 8, 6, 9, 7, 6, 3, 7, 4, 2, 6, 7, 9, 3, 4, 9, 2, 5, 2

Offset: 0

Artur Jasinski, Mar 31 2025

			0.36792162460638099759726137650329117056035747216338382601826906151734926635240156964828...

Index to constants which are prime zeta sums {0,3,1}.

Cf. A086242, A136141, A179119, A380840.

sumeulerrat(1/((p-1)^3*(p+1))) \\ Amiram Eldar, Apr 02 2025

Equals A136141/8 - A086242/4 + A380840/2 + A179119/8.

Equals Sum_{k>=2} (k-1)^2 * P(2*k) + (k-1)*k * P(2*k+1), where P is the prime zeta function. - Amiram Eldar, Apr 02 2025

A382568 Decimal expansion of Sum_{p prime} 1/(p^2*(p + 1)^2).

Original entry on oeis.org

0, 3, 6, 2, 6, 4, 8, 7, 1, 6, 6, 9, 2, 5, 2, 3, 2, 3, 4, 2, 8, 3, 0, 2, 5, 6, 0, 0, 0, 6, 1, 0, 9, 3, 8, 0, 5, 2, 2, 0, 8, 6, 1, 5, 6, 4, 7, 0, 5, 1, 3, 3, 0, 8, 3, 8, 9, 6, 5, 7, 8, 7, 9, 3, 1, 0, 1, 3, 8, 8, 3, 8, 8, 4, 2, 3, 4, 9, 2, 3, 2, 1, 3, 1, 0, 3, 2, 1, 5, 4, 5, 9, 2, 3, 3, 2, 1, 9, 6, 4, 2, 3, 6, 7, 4

Offset: 0

Artur Jasinski, Mar 31 2025

			0.03626487166925232342830256000610938052208615647051330838965787931013...

Index to constants which are prime zeta sums {2,0,2}.

Cf. A085548, A179119, A382554.

sumeulerrat(1/(p^2*(p+1)^2)) \\ Amiram Eldar, Apr 01 2025

Equals A085548 - 2*A179119 + A382554.

Equals Sum_{k>=4} (-1)^k * (k-3) * P(k), where P is the prime zeta function. - Amiram Eldar, Apr 01 2025

cp's OEIS Frontend

A349326 a(n) is the number of prime powers (not including 1) that are bi-unitary divisors of n.

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A375144 Numbers whose prime factorization has exactly two exponents that equal 2 and has no higher exponents.

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A382562 Decimal expansion of Sum_{p prime} 1/(p^2*(p + 1)).

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A382567 Decimal expansion of Sum_{p prime} 1/(p*(p + 1)^2).

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A382584 Decimal expansion of Sum_{p prime} 1/((p - 1)^2*p*(p + 1)).

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A284748 Decimal expansion of the sum of reciprocals of composite powers.

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A382563 Decimal expansion of Sum_{p prime} 1/(p^3*(p + 1)).

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A382565 Decimal expansion of Sum_{p prime} 1/((p - 1)^2*(p + 1)).

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A382584 Decimal expansion of Sum_{p prime} 1/((p - 1)^2p(p + 1)).