A061397 -id:A061397 - cp's OEIS frontend

A130739 Sum of primes < 2^n.

0, 5, 17, 41, 160, 501, 1720, 6081, 22548, 80189, 289176, 1070091, 3908641, 14584641, 54056763, 202288087, 761593692, 2867816043, 10862883985, 41162256126, 156592635694, 596946687124, 2280311678414, 8729068693022

Offset: 1

Graeme McRae, Jul 06 2007

nonn

			a(3) is 17 because the sum of primes less than 2^3 is 2+3+5+7=17.

Kim Walisch, Table of n, a(n) for n = 1..80 (terms n=1..42 from Cino Hilliard)
Kim Walisch, primesum program.

Cf. A046731, A099824.

Table[Sum[Prime[i], {i, PrimePi[2^n-1]}], {n, 1, 10}]

a(n) = {s = 0; forprime(p=2, 2^n-1, s +=p); return (s);} \\ Michel Marcus, Jul 17 2013

a(n) = Sum_{i=2..2^n-1} A061397(i).

A219224 G.f.: exp( Sum_{n>=1} A005063(n)*x^n/n ), where A005063(n) = sum of squares of primes dividing n.

Original entry on oeis.org

1, 0, 2, 3, 3, 11, 10, 26, 32, 51, 90, 117, 198, 283, 417, 610, 890, 1284, 1848, 2615, 3716, 5217, 7289, 10222, 14158, 19514, 26882, 36805, 50131, 68428, 92466, 125128, 168093, 225775, 302171, 402876, 536730, 711601, 942009, 1243513, 1638395, 2152828, 2823004

Offset: 0

Paul D. Hanna, Nov 15 2012

nonn

Euler transform of A061397. - Peter Luschny, Nov 21 2022

			G.f.: A(x) = 1 + 2*x^2 + 3*x^3 + 3*x^4 + 11*x^5 + 10*x^6 + 26*x^7 + 32*x^8 +...
where
log(A(x)) = 4*x^2/2 + 9*x^3/3 + 4*x^4/4 + 25*x^5/5 + 13*x^6/6 + 49*x^7/7 + 4*x^8/8 + 9*x^9/9 + 29*x^10/10 + 121*x^11/11 + 13*x^12/12 + 169*x^13/13 + 53*x^14/14 + 34*x^15/15 +...+ A005063(n)*x^n/n +...

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Cf. A000607, A005063, A220427, A061397.

# The function EulerTransform is defined in A358369.
a := EulerTransform(n -> ifelse(isprime(n), n, 0)):
seq(a(n), n = 0..42); # Peter Luschny, Nov 21 2022

a[n_] := SeriesCoefficient[ Exp[ Sum[ DivisorSum[k, Boole[PrimeQ[#]] * #^2&] * x^k/k, {k, 1, n+1}]], {x, 0, n}]; Table[a[n], {n, 0, 42}] (* Jean-François Alcover, Jul 11 2017, from PARI *)

{a(n)=polcoeff(exp(sum(k=1,n+1,sumdiv(k,d,isprime(d)*d^2)*x^k/k)+x*O(x^n)),n)}
for(n=0,50,print1(a(n),", "))

A319131 a(n) = Sum_{d|n} Sum_{p|d, p prime} p.

Original entry on oeis.org

0, 2, 3, 4, 5, 10, 7, 6, 6, 14, 11, 17, 13, 18, 16, 8, 17, 18, 19, 23, 20, 26, 23, 24, 10, 30, 9, 29, 29, 40, 31, 10, 28, 38, 24, 30, 37, 42, 32, 32, 41, 48, 43, 41, 27, 50, 47, 31, 14, 26, 40, 47, 53, 26, 32, 40, 44, 62, 59, 64, 61, 66, 33, 12, 36, 64, 67, 59, 52, 56

Offset: 1

Ilya Gutkovskiy, Sep 11 2018

Inverse Möbius transform of A008472.

			a(12) = 13 as 12 has 6 divisors and 2 * 6 * (2/3) + 3 * 6 * (1/2) = 17. - _David A. Corneth_, Oct 08 2019

David A. Corneth, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms

Cf. A000005, A007429, A008472, A061397, A062799, A319132.

[0] cat  [&+[&+[PrimeDivisors(d)[i]:i in [1..#PrimeDivisors(d)]]:d in Set(Divisors(n)) diff {1}]:n in [2..70]]; // Marius A. Burtea, Oct 08 2019

[0] cat [&+[p*#Divisors(n div p):p in PrimeDivisors(n)]:n in [2..70]]; // Marius A. Burtea, Oct 08 2019 (According to the formula given by Ridouane Oudra)

with(numtheory): seq(add(p*tau(n/p), p in factorset(n)), n=1..80); # Ridouane Oudra, Oct 08 2019

Table[Sum[Total[Select[Divisors[d], PrimeQ]], {d, Divisors[n]}], {n, 70}]
nmax = 70; Rest[CoefficientList[Series[Sum[DivisorSum[k, # &, PrimeQ[#] &] x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 70; Rest[CoefficientList[Series[-Log[Product[(1 - x^k)^(DivisorSum[k, # &, PrimeQ[#] &]/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]

a(n) = sumdiv(n, d, my(f=factor(d)); vecsum(f[,1])); \\ Michel Marcus, Oct 08 2019

a(n) = my(f = factor(n), nd = numdiv(f)); sum(i = 1, #f~, f[i, 1] * nd / (f[i, 2] + 1) * f[i, 2]) \\ David A. Corneth, Oct 08 2019

G.f.: Sum_{k>=1} A008472(k)*x^k/(1 - x^k).
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(A008472(k)/k)) = Sum_{n>=1} a(n)*x^n/n.
a(p^k) = p*k, where p is a prime.
a(n) = Sum_{p|n} p*tau(n/p), where p is a prime and tau(n) = A000005(n). - Ridouane Oudra, Oct 08 2019
a(n) = Sum_{p|n} p*tau(n)*(e_p-1)/(e_p) where e_p is the exponent of p in the factorization of n. - David A. Corneth, Oct 08 2019
a(n) = Sum_{d|n} sopf(d). - Wesley Ivan Hurt, May 23 2021

A369744 a(n) = Sum_{p|n, p prime} p * omega(n/p).

Original entry on oeis.org

0, 0, 0, 2, 0, 5, 0, 2, 3, 7, 0, 7, 0, 9, 8, 2, 0, 8, 0, 9, 10, 13, 0, 7, 5, 15, 3, 11, 0, 20, 0, 2, 14, 19, 12, 10, 0, 21, 16, 9, 0, 24, 0, 15, 11, 25, 0, 7, 7, 12, 20, 17, 0, 8, 16, 11, 22, 31, 0, 22, 0, 33, 13, 2, 18, 32, 0, 21, 26, 28, 0, 10, 0, 39, 13, 23, 18, 36

Offset: 1

Wesley Ivan Hurt, Jan 30 2024

Dirichlet convolution of A061397(n) and A001221(n). - Wesley Ivan Hurt, Apr 24 2025

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A001221 (omega), A010051, A061397, A323599, A345354, A345355.

Cf. also A369911.

Table[DivisorSum[n, #*PrimeNu[n/#] &, PrimeQ[#] &], {n, 100}]

A369744(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, f[i,1]*omega(n/f[i, 1]))); \\ Antti Karttunen, Jan 23 2025

a(p^k) = 1 for p prime and k = 1, else p if k >= 2. - Wesley Ivan Hurt, Jun 26 2024

a(n) = Sum_{d|n} d * omega(n/d) * c(d), where c = A010051. - Wesley Ivan Hurt, Apr 15 2025

A073046 Write 2n = p+q (p,q prime), pq minimal; then a(n) = p*q.

Original entry on oeis.org

4, 9, 15, 21, 35, 33, 39, 65, 51, 57, 95, 69, 115, 161, 87, 93, 155, 217, 111, 185, 123, 129, 215, 141, 235, 329, 159, 265, 371, 177, 183, 305, 427, 201, 335, 213, 219, 365, 511, 237, 395, 249, 415, 581, 267, 445, 623, 1501, 291, 485, 303, 309, 515, 321, 327

Offset: 2

Werner D. Sand, Aug 31 2002

nonn

Least semiprime whose sum of prime factors equals 2*n.

Assuming Goldbach's conjecture, a(n) exists for all n >= 2. - David James Sycamore, Jan 08 2019

			n=13: 2n=26; 26 = 23 + 3 = 19 + 7 = 13 + 13; 23*3 = minimal => p*q = 23*3 = 69.

Reinhard Zumkeller, Table of n, a(n) for n = 2..10000

Cf. A000040, A061397.

Cf. A102084, A193315, A288814.

a073046 n = head $ dropWhile (== 0) $
                   zipWith (*) prims $ map (a061397 . (2*n -)) prims
   where prims = takeWhile (<= n) a000040_list
-- Reinhard Zumkeller, Aug 28 2011

Array[Block[{p = 2, q}, While[! PrimeQ@ Set[q, 2 # - p], p = NextPrime[p]]; p q] &, 55, 2] (* Michael De Vlieger, Aug 02 2020 *)

For all n except 3, a(n) = A288814(2*n). - David James Sycamore, Jan 08 2019

Corrected by Ray Chandler, Jun 11 2005

A120007 Mobius transform of sum of prime factors of n with multiplicity (A001414).

Original entry on oeis.org

0, 2, 3, 2, 5, 0, 7, 2, 3, 0, 11, 0, 13, 0, 0, 2, 17, 0, 19, 0, 0, 0, 23, 0, 5, 0, 3, 0, 29, 0, 31, 2, 0, 0, 0, 0, 37, 0, 0, 0, 41, 0, 43, 0, 0, 0, 47, 0, 7, 0, 0, 0, 53, 0, 0, 0, 0, 0, 59, 0, 61, 0, 0, 2, 0, 0, 67, 0, 0, 0, 71, 0, 73, 0, 0, 0, 0, 0, 79, 0, 3, 0, 83, 0, 0, 0, 0, 0, 89, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Franklin T. Adams-Watters, Jun 02 2006

nonn

Same as A014963, except this function is zero when n is not a prime power, whereas A014963 is one.

Moreover, this sequence, A014963, A297108 and A297109 partition the natural numbers to identical equivalence classes: For all i, j >= 1, a(i) = a(j) <=> A014963(i) = A014963(j) <=> A297108(i) = A297108(j) <=> A297109(i) = A297109(j). - Antti Karttunen, Feb 01 2021

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Factor.
Eric Weisstein's World of Mathematics, Prime Zeta Function.

Cf. A000040, A001414, A007947, A014963, A010051, A010055, A061397, A070939, A140508 (Möbius transform of this sequence), A297108, A297109.

a120007 1 = 0
a120007 n | until ((> 0) . (`mod` spf)) (`div` spf) n == 1 = spf
          | otherwise = 0
          where spf = a020639 n
-- Reinhard Zumkeller, Sep 19 2011

Table[If[Length@ # == 1, #[[1, 1]], 0] &@ FactorInteger@ n, {n, 96}] /. 1 -> 0 (* Michael De Vlieger, Jun 19 2016 *)
Table[If[PrimePowerQ[n],FactorInteger[n][[1,1]],0],{n,100}] (* Harvey P. Dale, Jan 25 2020 *)

A120007(n) = { my(v); if(isprimepower(n, &v), v, 0); }; \\ Antti Karttunen, Jan 31 2021

If n is a prime power p^k, k>0, a(n) = p; otherwise a(n) = 0.
Dirichlet g.f. sum_{p prime} p/(p^s-1) = sum_{k>0} primezeta(ks-1).
a(n) = A010055(n) * A007947(n). - Reinhard Zumkeller, Mar 26 2010
a(n) = A061397(A007947(n)). - Reinhard Zumkeller, Sep 19 2011, corrected by Antti Karttunen, Jan 31 2021
a(n) = Sum_{k=2..n} k*A010051(k)*(floor(k^n/n)-floor((k^n -1)/n)). - Anthony Browne, Jun 17 2016
If A297109(n) = 0, then a(n) = 0, otherwise a(n) = A000040(A297109(n)). - Antti Karttunen, Feb 01 2021

A182936 Greatest common divisor of the proper divisors of n, 0 if there are none.

Original entry on oeis.org

0, 0, 0, 2, 0, 1, 0, 2, 3, 1, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 1, 0, 1, 5, 1, 3, 1, 0, 1, 0, 2, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 7, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 3, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1

Offset: 1

Peter Luschny, Mar 22 2011

nonn

Here a proper divisor d of n is a divisor of n such that 1 < d < n.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A014963, A020500, A048671, A034387, A061397, A072107, A380118.

A182936 := n -> igcd(op(numtheory[divisors](n) minus {1,n}));
seq(A182936(i), i=1..79); # Peter Luschny, Mar 22 2011

Join[{0}, Table[GCD@@Most[Rest[Divisors[n]]],{n,2,110}]] (* Harvey P. Dale, May 04 2018 *)
(* From Peter Luschny, Jan 31 2025: (Start) *)
Join[{0}, Table[Exp[MangoldtLambda[n]] - If[PrimeQ[n], n, 0], {n,2,110}]]
(* or *)
Table[Cyclotomic[n, 1] - If[PrimeQ[n], n, 0], {n,1,110}] (* End *)

A182936(n) = { my(divs=divisors(n)); if(#divs<3,0,gcd(vector(numdiv(n)-2,k,divs[k+1]))); }; \\ Antti Karttunen, Sep 23 2017

a(n) = 0 if n is not composite, p if n is a proper power of prime p, and 1 otherwise. - Franklin T. Adams-Watters, Mar 22 2011
Conjecture: Sum_{k=1..n} a(k) = A072107(n) - A034387(n) - 1. - Vaclav Kotesovec, Jan 29 2025
From Peter Luschny, Jan 31 2025: (Start)
a(n) = A014963(n) - A061397(n) for n > 1. In other words, this sequence is the exponential von Mangoldt function restricted to proper divisors of n. See A380118. This implies the above conjecture.
a(n) = A020500(n) - A061397(n). (End)

More terms from Antti Karttunen, Sep 23 2017

A191558 a(n) = 0 if n prime, otherwise n.

Original entry on oeis.org

1, 0, 0, 4, 0, 6, 0, 8, 9, 10, 0, 12, 0, 14, 15, 16, 0, 18, 0, 20, 21, 22, 0, 24, 25, 26, 27, 28, 0, 30, 0, 32, 33, 34, 35, 36, 0, 38, 39, 40, 0, 42, 0, 44, 45, 46, 0, 48, 49, 50, 51, 52, 0, 54, 55, 56, 57, 58, 0, 60, 0, 62, 63, 64, 65, 66, 0, 68, 69, 70, 0, 72, 0, 74, 75, 76, 77, 78, 0, 80, 81, 82, 0, 84, 85, 86, 87, 88, 0, 90, 91, 92, 93, 94

Offset: 1

Vincenzo Librandi, Jun 07 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Cf. A002808.

[IsPrime(n) select 0 else n: n in [1..100]]; // Vincenzo Librandi, May 17 2014

A191558 := proc(n) if isprime(n) then 0; else n; end if; end proc:
seq(A191558(n),n=1..80) ; # R. J. Mathar, Jun 11 2011

Table[If[PrimeQ[n], 0, n], {n, 150}] (* Vincenzo Librandi, May 17 2014 *)
(*recurrence*)
Clear[t];
nn = 94;
t[1, 1] = 1;
t[n_, k_] :=
  t[n, k] =
   If[n == k, n*(1 - Product[t[n, k - i], {i, 1, k - 1}]),
    If[n > k, t[n - k, k], 1]];
Table[t[n, n], {n, 1, nn}](* Mats Granvik, Jul 05 2014 *)
Table[n (1 - PrimePi[n] + PrimePi[n - 1]), {n, 50}] (* Wesley Ivan Hurt, Jul 06 2014 *)

a(n) = n * A005171(n) = n - A061397(n). - R. J. Mathar, Jun 11 2011

A328260 a(n) = n * omega(n).

Original entry on oeis.org

0, 2, 3, 4, 5, 12, 7, 8, 9, 20, 11, 24, 13, 28, 30, 16, 17, 36, 19, 40, 42, 44, 23, 48, 25, 52, 27, 56, 29, 90, 31, 32, 66, 68, 70, 72, 37, 76, 78, 80, 41, 126, 43, 88, 90, 92, 47, 96, 49, 100, 102, 104, 53, 108, 110, 112, 114, 116, 59, 180, 61, 124, 126, 64, 130, 198, 67, 136, 138, 210

Offset: 1

Ilya Gutkovskiy, Oct 09 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Mikhail R. Gabdullin and Vitalii V. Iudelevich, Numbers of the form kf(k), arXiv:2201.09287 [math.NT] (2022).
Eric Weisstein's World of Mathematics, Distinct Prime Factors

Cf. A001221, A001222, A007947, A038040, A061397, A066959, A069359, A246655 (fixed points), A293548, A298473.

[0] cat [n*(#PrimeDivisors(n)):n in [2..70]]; // Marius A. Burtea, Oct 10 2019

Table[n PrimeNu[n], {n, 1, 70}]
nmax = 70; CoefficientList[Series[Sum[Prime[k] x^Prime[k]/(1 - x^Prime[k])^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n)=n*omega(n) \\ Charles R Greathouse IV, Mar 16 2022

G.f.: Sum_{k>=1} prime(k) * x^prime(k) / (1 - x^prime(k))^2.
a(n) = bigomega(rad(n)^n).
a(n) = Sum_{d|n} A061397(n/d) * d.
Define f(x) = #{n <= x: a(n) <= x}. Gabdullin & Iudelevich show that f(x) ~ x/log log x. - Charles R Greathouse IV, Mar 16 2022

A369911 a(n) = Sum_{p|n, p prime} p * sopf(n/p).

Original entry on oeis.org

0, 0, 0, 4, 0, 12, 0, 4, 9, 20, 0, 16, 0, 28, 30, 4, 0, 21, 0, 24, 42, 44, 0, 16, 25, 52, 9, 32, 0, 62, 0, 4, 66, 68, 70, 25, 0, 76, 78, 24, 0, 82, 0, 48, 39, 92, 0, 16, 49, 45, 102, 56, 0, 21, 110, 32, 114, 116, 0, 66, 0, 124, 51, 4, 130, 122, 0, 72, 138, 118, 0, 25

Offset: 1

Wesley Ivan Hurt, Feb 05 2024

Dirichlet convolution of A061397(n) and A008472(n). - Wesley Ivan Hurt, Jul 10 2025

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A008472 (sopf), A061397, A369744.

a[n_] := Sum[p, {p, Select[Divisors[n], PrimeQ]}]; Table[DivisorSum[n, #*a[n/#] &, PrimeQ[#] &], {n, 100}]

A008472(n) = vecsum(factor(n)[, 1]);
A369911(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, f[i,1]*A008472(n/f[i, 1]))); \\ Antti Karttunen, Jan 23 2025

From Wesley Ivan Hurt, Jul 10 2025: (Start)
a(n) = Sum_{d|n} A061397(d) * A008472(n/d).
a(p^k) = p^2 * (1-floor(1/k)) for p prime and k>=1. (End)

cp's OEIS Frontend

A130739 Sum of primes < 2^n.

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A219224 G.f.: exp( Sum_{n>=1} A005063(n)*x^n/n ), where A005063(n) = sum of squares of primes dividing n.

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A319131 a(n) = Sum_{d|n} Sum_{p|d, p prime} p.

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A369744 a(n) = Sum_{p|n, p prime} p * omega(n/p).

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A073046 Write 2*n = p+q (p,q prime), p*q minimal; then a(n) = p*q.

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A120007 Mobius transform of sum of prime factors of n with multiplicity (A001414).

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A182936 Greatest common divisor of the proper divisors of n, 0 if there are none.

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A073046 Write 2n = p+q (p,q prime), pq minimal; then a(n) = p*q.