A136141 -id:A136141 - cp's OEIS frontend

A244623 Odd prime powers that are not primes.

1, 9, 25, 27, 49, 81, 121, 125, 169, 243, 289, 343, 361, 529, 625, 729, 841, 961, 1331, 1369, 1681, 1849, 2187, 2197, 2209, 2401, 2809, 3125, 3481, 3721, 4489, 4913, 5041, 5329, 6241, 6561, 6859, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12167, 12769, 14641, 15625, 16129, 16807, 17161, 18769, 19321, 19683

Offset: 1

Jani Melik, Jul 02 2014

nonn

Intersection of A061345 and A014076.

A014076 set minus A061346.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Intersection of A005408 and A025475.

Cf. A061345 (odd prime powers), A061346 (odd neither prime nor prime power), A062739 (odd powerful), A075109 (perfect powers), A136141.

Join[{1},Select[Range[1,20001,2],PrimePowerQ[#]&&(!PrimeQ[#])&]] (* Harvey P. Dale, Dec 11 2018 *)

isok(p) = ((p%2) && !isprime(p) && isprimepower(p)) || (p==1); \\ Michel Marcus, Jul 06 2021

def isA244623(n) :
   return(n % 2 == 1 and is_prime_power(n) == 1 and is_prime(n) == 0)
[n for n in (1..20000) if isA244623(n)]

a(n) = A079290(n) at least in the range n=3..94, and perhaps beyond. - R. J. Mathar, Aug 20 2014

Sum_{n>=1} 1/a(n) = 1/2 + Sum_{p prime} 1/(p*(p-1)) = 1/2 + A136141. - Amiram Eldar, Dec 21 2020

A190146 Decimal expansion of Sum_{k>=2} (1/Sum_{j=2..k} j'), where n' is the arithmetic derivative of n.

Original entry on oeis.org

2, 3, 3, 0, 0, 9

Offset: 1

Paolo P. Lava, May 05 2011

Slow convergence.
a(7) is likely either 3 or 4. Is there a simple proof that this sum converges? - Nathaniel Johnston, May 24 2011
From Husnain Raza, Aug 29 2023: (Start)
The series indeed converges: we have that the series is C = Sum_{k>=2} (1/Sum_{j=2..k} A003415(j)).
Let s_k = Sum_{j=2..k} A003415(j) be the inner sum.
It is known that s_k = (1/2)*T_0*k^2 + O(k^(1+n)) for all real n > 0 where T_0 = A136141.
Therefore, 1/s_k = (2/T_0)*k^(-2) + O(k^(-3+n)) = (2/T_0)*k^(-2) + O(k^(-3)).
Summing both sides from k=2 to infinity, we have that:
C = Sum_{k >= 2} 1/s_k = Sum_{k >= 2} ((2/T_0)*k^(-2) + O(k^(-3))), which converges. (End)

			1/2' + 1/(2'+3') + 1/(2'+3'+4') + 1/(2'+3'+4'+5') + ... = 1 + 1/2 + 1/6 + 1/7 + ... = 2.33009...

Cf. A003415, A190144, A190145, A190147.

with(numtheory);
P:=proc(i)
local a,b,f,n,p,pfs;
a:=0; b:=0;
for n from 2 to i do
  pfs:=ifactors(n)[2];
  f:=n*add(op(2,p)/op(1,p),p=pfs);
  b:=b+f; a:=a+1/b;
od;
print(evalf(a,300));
end:
P(1000);

digits = 5; d[0] = d[1] = 0; d[n_] := d[n] = n*Total[Apply[#2/#1 &, FactorInteger[n], {1}]]; p[m_] := p[m] = Sum[1/Sum[d[j], {j, 2, k}], {k, 2, m}] // RealDigits[#, 10, digits]& // First; p[digits]; p[m = 2*digits]; While[Print["p(", m, ") = ", p[m]]; p[m] != p[m/2], m = 2*m]; p[m] (* Jean-François Alcover, Feb 21 2014 *)

a(6) corrected and a(7) removed by Nathaniel Johnston, May 24 2011

A152447 Decimal expansion of the sum_q 1/(q*(q-1)) over the semiprimes q = A001358.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Dec 04 2008

The semiprime analog of A136141. To obtain the (smaller) sum over the squarefree semiprimes A006881, subtract the prime zeta functions of 4 ( A085964 ), 6, 8 etc. from this constant here. The first term in the representation as the geometric series in powers 1/q^s is in A117543 .

R. J. Mathar, Series of reciprocal powers of k-almost primes, constant B_{2,1} in table 8.

Equals 0.17105189297999663662220256437237421399124661203550059749107997... = 1/(4*3)+1/(6*5)+1/(9*8)+1/(10*9)+...

A369632 Decimal expansion of Sum_{primes p} 1/(p*(p^2 - 1)).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 28 2024

			0.22146337139279594342463643588459881748724095830455...

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

Cf. A127917, A136141, A179119.

RealDigits[NSum[PrimeZetaP[2*k + 1], {k, 1, Infinity}, WorkingPrecision -> 100]][[1]] (* Amiram Eldar, Jan 28 2024 *)

sumeulerrat(1/(p*(p^2-1))) \\ Amiram Eldar, Jan 28 2024

Equals Sum_{i>=1} 1/A127917(i) = (A136141 - A179119)/2.

Equals Sum_{k>=1} P(2*k+1), where P(s) is the prime zeta function. - Amiram Eldar, Jan 28 2024

More terms from Amiram Eldar, Jan 28 2024

A185380 Decimal expansion of sum 1/(p*(p+2)) over the primes p.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 21 2013

If we omit the first term 1/(2*4)=0.125 from the sum, 0.138672... remains, which is an upper limit of A209329 in the sense that we "fake" prime gaps of 2 here [which are actually larger on average].

			0.263672061761153178749842188233776 .. = 1/(2*4) +1/(3*5) + 1/(5*7) + 1/(7*9) + 1/(11*13)+ ...

Cf. A136141 (1/(p(p-1))), A179119 (1/(p(p+1))).

read("transforms") ;
Digits := 300 ;
# insert coding of ZetaM(s,M) and Hurw(a) from A179119 here...
A185380 := proc()
        Hurw(2) ;
end proc:
A185380() ;

sumeulerrat(1/(p*(p+2))) \\ Amiram Eldar, Mar 19 2021

Equals -1/8 + Sum_{k>=2} (-1)^k * 2^(k-2) * P(k), where P is the prime zeta function. - Vaclav Kotesovec, Jan 13 2021

More digits from Vaclav Kotesovec, Jan 13 2021

A343224 a(n) = sigma(n) - A003415(n), where A003415 is the arithmetic derivative, and sigma is the sum of divisors of n.

Original entry on oeis.org

1, 2, 3, 3, 5, 7, 7, 3, 7, 11, 11, 12, 13, 15, 16, -1, 17, 18, 19, 18, 22, 23, 23, 16, 21, 27, 13, 24, 29, 41, 31, -17, 34, 35, 36, 31, 37, 39, 40, 22, 41, 55, 43, 36, 39, 47, 47, 12, 43, 48, 52, 42, 53, 39, 56, 28, 58, 59, 59, 76, 61, 63, 53, -65, 66, 83, 67, 54, 70, 85, 71, 39, 73, 75, 69, 60, 78, 97, 79, 10, 13

Offset: 1

Antti Karttunen, Apr 15 2021

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n).

Cf. A000203, A003415, A013661, A136141.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A343224(n) = sigma(n) - A003415(n);

a(n) = A000203(n) - A003415(n).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Pi^2/6 - Sum_{p prime} 1/(p*(p-1)) = A013661 - A136141 = 0.871777... . - Amiram Eldar, Dec 29 2024

A363217 Odd powerful numbers that are not powers of primes.

Original entry on oeis.org

225, 441, 675, 1089, 1125, 1225, 1323, 1521, 2025, 2601, 3025, 3087, 3249, 3267, 3375, 3969, 4225, 4563, 4761, 5625, 5929, 6075, 6125, 7225, 7569, 7803, 8281, 8575, 8649, 9025, 9261, 9747, 9801, 10125, 11025, 11907, 11979, 12321, 13225, 13689, 14161, 14283, 15125, 15129, 16641, 16875, 17689, 18225, 19773

Offset: 1

Michael De Vlieger, May 21 2023

nonn

This sequence is { A286708 INTERSECT A005408 } = { A001694 INTERSECT A360769 }.

Subset of A001694, A062739, A126706, and A360769.

			a(1) = 225 = 3^2 * 5^2, the smallest odd number with multiple distinct prime factors, each of which have multiplicity exceeding 1.
a(2) = 441 = 3^2 * 7^2,
a(3) = 675 = 3^3 * 5^2, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000961, A001694, A062739, A126706, A286708, A363216.

Cf. A002117, A013661, A013664, A082695, A136141.

With[{nn = 20000}, Select[Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], And[OddQ[#], ! PrimePowerQ[#]] &] ]

isok(k) = (k>1) && (k%2) && ispowerful(k) && !isprimepower(k); \\ Michel Marcus, May 28 2023

This sequence is { k = a^2*b^3 : a >= 1, b >= 1, omega(k) > 1, k mod 2 = 1 }.

Sum_{n>=1} 1/a(n) = 2*zeta(2)*zeta(3)/(3*zeta(6)) - 1/2 - Sum_{p prime} 1/(p*(p-1)) = (2/3) * A082695 - 1/2 - A136141 = 0.0225742... . - Amiram Eldar, May 28 2023

A069812 a(n) = Sum_{k=1..n} (bigomega(k)-omega(k)).

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 3, 4, 4, 4, 5, 5, 5, 5, 8, 8, 9, 9, 10, 10, 10, 10, 12, 13, 13, 15, 16, 16, 16, 16, 20, 20, 20, 20, 22, 22, 22, 22, 24, 24, 24, 24, 25, 26, 26, 26, 29, 30, 31, 31, 32, 32, 34, 34, 36, 36, 36, 36, 37, 37, 37, 38, 43, 43, 43, 43, 44, 44, 44, 44, 47, 47, 47, 48

Offset: 1

Benoit Cloitre, Apr 30 2002

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A001221, A001222, A046660, A136141.

Accumulate[Table[PrimeOmega[n]-PrimeNu[n],{n,80}]] (* Harvey P. Dale, Oct 10 2015 *)

lista(nmax) = {my(s = 0, e); for(n = 1, nmax, e = factor(n)[, 2]; s += vecsum(e) - #e; print1(s, ", "));} \\ Amiram Eldar, Sep 17 2024

a(n) ~ A136141 * n, where A136141 = 0.773156669... - Vaclav Kotesovec, Sep 01 2019

G.f.: (1/(1 - x)) * Sum_{p prime, k>=2} x^(p^k)/(1 - x^(p^k)). - Ilya Gutkovskiy, Sep 03 2019

A226957 Number of prime divisors p of 2n such that p-1 divides n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2

Offset: 1

José María Grau Ribas, Jun 24 2013

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A136141.

fa = FactorInteger; a[n_] := Sum[If[IntegerQ[n/(fa[2 n][[i,1]]-1)],1,0],{i, Length[fa[2 n]]}]; Table[a[n], {n, 1, 100}]

a(n)=my(f=factor(2*n)[,1]);sum(i=1,#f,n%(f[i]-1)==0) \\ Charles R Greathouse IV, Jun 26 2013

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/2 + Sum_{p prime} 1/(p*(p-1)) = 1/2 + A136141 = 1.273156669... . - Amiram Eldar, May 17 2024

Erroneous a(87) added by T. D. Noe, corrected by Charles R Greathouse IV, Jun 26 2013

A229523 Partial sum of the arithmetic derivative A003415 (A190121) up to 10^n.

Original entry on oeis.org

0, 38, 3906, 386517, 38671110, 3865941752, 386580463478, 38657862140521, 3865783461518530, 386578337105347684, 38657833484501788407, 3865783345588492717623, 386578334529872234861944, 38657833452536035472588254, 3865783345249467526546175599

Offset: 0

M. F. Hasler, Sep 25 2013

Hiroaki Yamanouchi, Table of n, a(n) for n = 0..17
E. J. Barbeau, Remark on an arithmetic derivative, Canad. Math. Bull., Vol. 4, No. 2 (May 1961), pp. 117-122.

Cf. A003415, A136141, A190121.

s=0;for(k=0,8,for(n=10^(k-1)+1,10^k,s+=A003415(n));print1(s","));s

a(n) = A190121(10^n).
It seems that a(n)/10^(2n-1) -> 3.865783... as n -> oo.
Note: A190121 ~ 0.374... * n^2 [Barbeau]. - Giorgio Balzarotti, Oct 15 2013
a(n) ~ 0.386578334524897563932183729927 * 100^n. - Hiroaki Yamanouchi, Jul 09 2014
The constant is (1/2) * Sum_{p prime} 1/(p*(p-1)) = A136141 / 2 = 0.3865783345... . This constant was given by Barbeau (1961) but with the wrong value 0.374. - Amiram Eldar, Oct 06 2023

a(8)-a(10) from Donovan Johnson, Sep 25 2013
a(11)-a(12) from Giovanni Resta, Mar 13 2014
a(13)-a(14) from Hiroaki Yamanouchi, Jul 09 2014

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A244623 Odd prime powers that are not primes.

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A190146 Decimal expansion of Sum_{k>=2} (1/Sum_{j=2..k} j'), where n' is the arithmetic derivative of n.

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A152447 Decimal expansion of the sum_q 1/(q*(q-1)) over the semiprimes q = A001358.

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A369632 Decimal expansion of Sum_{primes p} 1/(p*(p^2 - 1)).

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A185380 Decimal expansion of sum 1/(p*(p+2)) over the primes p.

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A343224 a(n) = sigma(n) - A003415(n), where A003415 is the arithmetic derivative, and sigma is the sum of divisors of n.

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A363217 Odd powerful numbers that are not powers of primes.

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A069812 a(n) = Sum_{k=1..n} (bigomega(k)-omega(k)).

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