A034387 -id:A034387 - cp's OEIS frontend

A063959 Sum of the primes from 1 to n!.

0, 0, 2, 10, 100, 1593, 41741, 1578242, 80294846, 5356015580, 451223209946, 46900682786541, 5891009442510166, 879657744587755114, 153967535281046615774, 31216213430872403460411, 7256556722488434503836458, 1917031284234466887065107947, 571083301099266868435687532291

Offset: 0

Jason Earls, Sep 03 2001

nonn

Sum of prime factors (without repetition) of (n!)!.

			a(4) = sum of primes <= 24. They are 2, 3, 5, 7, 11, 13, 17, 19 and 23. This sum is 100.

Amiram Eldar, Table of n, a(n) for n = 0..22 (extended using Kim Walisch's primesum; terms 0..20 from Daniel Suteu)
Kim Walisch, primesum program.

Cf. A000142, A034387, A006990, A294194.

NextPrim[n_] := (k = n + 1; While[ ! PrimeQ[k], k++ ]; k); s = 0; p = 1; Do[ Do[p = NextPrim[p]; s = s + p, {i, PrimePi[(n - 1)! ] + 1, PrimePi[(n)! ]}]; Print[s], {n, 1, 12} ]
Do[ Print[ Sum[ Prime[k], {k, 1, PrimePi[n! ]}]], {n, 0, 10} ]
Table[Total[Prime[Range[PrimePi[n!]]]],{n,0,9}] (* The program generates the first 10 terms of the sequence. *) (* Harvey P. Dale, Aug 17 2025 *)

sumprime(n,s,fac,i)=fac=factor(n); for(i=1,matsize(fac)[1],s=s+fac[i,1]); return(s); for(n=0,22,print(sumprime(n!!)))

a(n) = A034387(A000142(n)). - Michel Marcus, Jun 14 2024

Better description and more terms from Robert G. Wilson v, Oct 04 2001
a(13)-a(15) from Donovan Johnson, May 03 2010
a(16)-a(18) from Daniel Suteu, Nov 15 2018

A063960 Sum of non-unitary prime divisors of n!: sum of those prime divisors for which the exponent in the prime factorization exceeds 1.

Original entry on oeis.org

0, 0, 0, 2, 2, 5, 5, 5, 5, 10, 10, 10, 10, 17, 17, 17, 17, 17, 17, 17, 17, 28, 28, 28, 28, 41, 41, 41, 41, 41, 41, 41, 41, 58, 58, 58, 58, 77, 77, 77, 77, 77, 77, 77, 77, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 129, 129, 129, 129, 160, 160, 160, 160

Offset: 1

Labos Elemer, Sep 04 2001

Sum of the prime numbers among the smallest parts of the partitions of n into two parts. For example, a(8)=5; the partitions of 8 into two parts are (7,1), (6,2), (5,3) and (4,4). The prime numbers among the smallest parts are 2 and 3, so 2 + 3 = 5. - Wesley Ivan Hurt, Nov 01 2017

Number of distinct rectangles with integer length and prime width such that L + W = n, W <= L. For a(14)=17; the rectangles are 2 X 12, 3 X 11, 5 X 9, and 7 X 7. The sum of the lengths are then 2+3+5+7 = 17. - Wesley Ivan Hurt, Nov 08 2017

			20! = (2^18)*(3^8)*(5^4)*(7^2)*11*13*17*19, the non-unitary prime divisors are {2, 3, 5, 7}, so a(20) = 2 + 3 + 5 + 7 = 17.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A010051, A034387, A034444, A056169, A056170, A056171, A056172, A063958.

seq(add(j, j=select(isprime, [$1..iquo(n,2)])), n=1..65); # Peter Luschny, Nov 28 2022

Join[{0,0,0},Table[Total[Transpose[Select[FactorInteger[n!], Last[#]>1&]][[1]]],{n,4,70}]] (* Harvey P. Dale, Jun 19 2013 *)

{ for (n=1, 1000, f=factor(n!)~; a=0; for (i=1, length(f), if (f[2, i]>1, a+=f[1, i])); write("b063960.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 04 2009

a(n) = Sum_{i=1..floor(n/2)} i * A010051(i). - Wesley Ivan Hurt, Oct 31 2017
a(n) = A034387(floor(n/2)) for n >= 2. - Georg Fischer, Nov 28 2022
a(n) = A063958(n!). - Amiram Eldar, Jul 24 2024

A074793 Sum of prime powers less than or equal to n.

Original entry on oeis.org

0, 2, 5, 9, 14, 14, 21, 29, 38, 38, 49, 49, 62, 62, 62, 78, 95, 95, 114, 114, 114, 114, 137, 137, 162, 162, 189, 189, 218, 218, 249, 281, 281, 281, 281, 281, 318, 318, 318, 318, 359, 359, 402, 402, 402, 402, 449, 449, 498, 498, 498, 498, 551, 551, 551, 551, 551

Offset: 1

Benoit Cloitre, Sep 07 2002

nonn

			a(10)=38 because 2,3,4,5,7,8,9 are the prime powers less than or equal to 10 and 2+3+4+5+7+8+9 = 38.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A034387, A081738, A000196.

Accumulate[Table[If[PrimePowerQ[n],n,0],{n,60}]] (* Harvey P. Dale, Oct 04 2019 *)

PARI
```
a(n)=sum(k=1,n,k*if(omega(k)-1,0,1))
```

Is a(n) asymptotic to c*n^2/log(n) with c=0.55...?
From Daniel Suteu, Aug 20 2023: (Start)
a(n) = Sum_{k=1..floor(log_2(n))} Sum_{p prime <= n^(1/k)} p^k.
a(n) = A034387(n) + A081738(A000196(n)) + Sum_{p prime <= n^(1/3)} ((p^(floor(log_p(n))+1) - 1)/(p-1) - p^2 - p - 1). (End)

A143537 Triangle read by rows: T(n,k) = number of primes in the interval [k..n], n >= 1, 1 <= k <= n.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Aug 23 2008

Old name: triangle read by rows, A000012 * A143536, 1<=k<=n.

			Triangle T(n,k) begins:
n\k 1  2  3  4  5  6  7  8 ...
1:  0;
2:  1, 1;
3:  2, 2, 1;
4:  2, 2, 1, 0;
5:  3, 3, 2, 1, 1;
6:  3, 3, 2, 1, 1, 0;
7:  4, 4, 3, 2, 2, 1, 1;
8:  4, 4, 3, 2, 2, 1, 1, 0;
...

Row sums are A034387.
Column k=1 gives A000720.
Main diagonal gives A010051.
T(2n,n) gives A035250.
Cf. A143536.

T(n,k) = pi(n) - pi(k-1), where pi = A000720. - Ilya Gutkovskiy, Mar 19 2020

New name and corrected by Ilya Gutkovskiy, Mar 19 2020

A294113 Sum of the smaller parts of the partitions of 2n into two parts with larger part prime.

Original entry on oeis.org

0, 3, 4, 4, 8, 6, 11, 8, 13, 20, 28, 24, 32, 25, 32, 41, 51, 42, 51, 40, 49, 60, 72, 60, 72, 84, 97, 111, 125, 109, 124, 107, 121, 136, 152, 169, 188, 169, 187, 206, 226, 204, 224, 199, 218, 238, 258, 229, 248, 268, 289, 312, 336, 306, 331, 357, 384, 412

Offset: 1

Wesley Ivan Hurt, Oct 22 2017

			For n=7, 2n = 14 can be partitioned into two parts with the larger part prime as 13 + 1, 11 + 3, and 7 + 7. So a(7) = 1 + 3 + 7 = 11. - _Michael B. Porter_, Mar 14 2018

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to partitions

Cf. A000720, A010051, A034387, A294114.

N:= 1000: # to get a(1)..a(n)
P:= select(isprime, [2,seq(i,i=3..2*N,2)]):
S:= ListTools:-PartialSums(P):
f:= proc(n) local k1,k2;
     k1:= numtheory:-pi(2*n);
     k2:= numtheory:-pi(n-1);
     2*n*(k1-k2) - S[k1] + S[k2]
end proc:
f(1):= 0:
seq(f(n),n=1..N); # Robert Israel, Mar 13 2018

Table[Sum[i (PrimePi[2 n - i] - PrimePi[2 n - i - 1]), {i, n}], {n, 80}]

a(n) = sum(k=1, n, k*isprime(2*n-k)); \\ Michel Marcus, Oct 24 2017

a(n) = my(res = 0); forprime(p = n, 2*n, res+=(2*n - p)); res \\ David A. Corneth, Oct 24 2017

a(n) = Sum_{i=1..n} i * A010051(2n-i).

a(n) = 2*n*(A000720(2*n)-A000720(n-1)) - A034387(2*n) + A034387(n-1) for n >= 2. - Robert Israel, Mar 13 2018

A341700 Sum of the primes p satisfying n < p <= 2n.

Original entry on oeis.org

2, 3, 5, 12, 7, 18, 24, 24, 41, 60, 49, 72, 59, 59, 88, 119, 102, 102, 120, 120, 161, 204, 181, 228, 228, 228, 281, 281, 252, 311, 341, 341, 341, 408, 408, 479, 515, 515, 515, 594, 553, 636, 593, 593, 682, 682, 635, 635, 732, 732, 833, 936, 883, 990, 1099, 1099

Offset: 1

Chai Wah Wu, Feb 17 2021

nonn

For n >= 2, a(n) is the sum of the prime numbers appearing in the 2nd row of an n X n square array whose elements are the numbers from 1..n^2, listed in increasing order by rows. - Wesley Ivan Hurt, May 17 2021

			a(7) = 24 = 11+13 (sum of primes larger than 7 and less than or equal to 14).

Cf. A010051, A034387, A073837, A108954, A128076.

Array[Total@ Select[Range[# + 1, 2 #], PrimeQ] &, 56] (* Michael De Vlieger, Feb 17 2021 *)

from sympy import nextprime
def A341700(n):
    s, m = 0, nextprime(n)
    while m <= 2*n:
        s += m
        m = nextprime(m)
    return s

a(n) = A034387(2*n) - A034387(n).
a(n) = A073837(n) if n is not a prime. Otherwise, a(n) = A073837(n)-n.
For n >= 2, a(n) = Sum_{k=(n^2-n+2)/2..(n^2+n-2)/2} A010051(A128076(k)) * A128076(k). - Wesley Ivan Hurt, Jan 08 2022

A380118 a(n) = Sum_{k=1..n} (A014963(k) - A061397(k)).

Original entry on oeis.org

1, 1, 1, 3, 3, 4, 4, 6, 9, 10, 10, 11, 11, 12, 13, 15, 15, 16, 16, 17, 18, 19, 19, 20, 25, 26, 29, 30, 30, 31, 31, 33, 34, 35, 36, 37, 37, 38, 39, 40, 40, 41, 41, 42, 43, 44, 44, 45, 52, 53, 54, 55, 55, 56, 57, 58, 59, 60, 60, 61, 61, 62, 63, 65, 66, 67, 67, 68, 69, 70

Offset: 1

Peter Luschny, Jan 30 2025

nonn

Cf. A014963, A061397, A048671, A010051, A182936, A034387, A072107, A380117.

pSum := L -> ListTools:-PartialSums(L): h := n -> n/A048671(n) - n*A010051(n):
aList := upto -> pSum([seq(h(k), k = 1..upto)]): aList(70);

Accumulate[Table[Exp[MangoldtLambda[n]] - If[PrimeQ[n], n, 0] , {n, 1, 70}]]

a(n) = A072107(n) - A034387(n). - Amiram Eldar, Jan 30 2025

A089895 Prime numbers p for which there exists an integer q > p such that the sum of all primes <= p equals the sum of all primes between p+1 and q.

Original entry on oeis.org

3, 3833, 468872968241

Offset: 1

Randy L. Ekl, Jan 10 2004

Primes p such that 2*A034387(p) is a term of A034387. - Max Alekseyev, Aug 24 2023

No other terms below 10^13. - Max Alekseyev, Aug 25 2023

			2+3+5+...+3833 = 3847+...+5557 and therefore 3833 is in the sequence.

Carlos Rivera, Puzzle 18.- Some special sums of consecutive primes, The Prime Puzzles & Problems Connection. See specifically the large solution by Giovanni Resta.

a[m_] := Module[{pLst, cumsum, p, q, k, target, idx}, pLst = Prime[Range[PrimePi[m]]]; cumsum = Accumulate[pLst]; pairs = {}; For[k = 1, k <= Length[pLst], k++, p = pLst[[k]]; target = 2*cumsum[[k]]; idx = FirstPosition[Drop[cumsum, k], target]; If[idx =!= Missing["NotFound"], q = pLst[[k + First[idx]]]; If[q > p, AppendTo[pairs, p];]]]; pairs]; a[10000] (* Robert P. P. McKone, Aug 25 2023 *)

p=2;s=2;q=3;t=3;while(p<512345678900, while(s<=t,p=nextprime(p+1);s=s+p;t=t-p);if (s==t,print1(p,", "),);while(t

Better definition from Adam M. Kalman (mocha(AT)clarityconnect.com), Jun 16 2005

Edited by Max Alekseyev, Aug 24 2023

A139390 Sum of primes <= 3^n.

Original entry on oeis.org

0, 5, 17, 100, 791, 5830, 42468, 327198, 2575838, 20476640, 166554645, 1353822880, 11150031169, 92258920888, 769310640408, 6447635236133, 54292816788848, 459112338326268, 3896226837717653, 33172345145637461, 283258796052356059, 2425130743589880412, 20812174068479995267

Offset: 0

Cino Hilliard, Jun 09 2008

nonn

For large n, these numbers can be closely approximated by the number of primes < (3^n)^2. For example, the sum of primes < 3^12 = 11150031169. The number of primes < (3^12)^2 = 3^24 = 11152818693. The error here 0.000250.

The second term, 5, is the addition of the primes 2 and 3 since we defined the sequence as less than or equal.

Amiram Eldar, Table of n, a(n) for n = 0..43 (calculated using Kim Walisch's primesum program)
Cino Hilliard, Sum of Primes. [broken link]
Cino Hilliard, Sumprimesgmp program. [broken link]
Kim Walisch, Sum of the primes below x (primesum).

Cf. A000244, A034387, A130739.

a(n) = vecsum(primes([1, 3^n])); \\ Michel Marcus, Jul 02 2024

a(n) = A034387(A000244(n)). - Amiram Eldar, Jul 02 2024

Duplicated term removed and a(20)-a(22) added by Amiram Eldar, Jul 02 2024

A140234 Sum of the semiprimes <= n.

Original entry on oeis.org

0, 0, 0, 0, 4, 4, 10, 10, 10, 19, 29, 29, 29, 29, 43, 58, 58, 58, 58, 58, 58, 79, 101, 101, 101, 126, 152, 152, 152, 152, 152, 152, 152, 185, 219, 254, 254, 254, 292, 331, 331, 331, 331, 331, 331, 331, 377, 377, 377, 426, 426, 477, 477, 477, 477, 532, 532, 589

Offset: 0

Jonathan Vos Post, May 13 2008

This is to semiprimes A001358 as A034387 is to primes A000040. From the prime number theorem A034387(n) has the asymptotic expression: a(n) ~ n^2 / (2 log n), so what is the asymptotic expression for a(n)?

Cf. A001358, A034387, A051352, A062198, A072000, A100959, A140235.

a[n_]:=Total[Select[Range[n],PrimeOmega[#]==2&]];Array[a,58,0] (* James C. McMahon, Jul 06 2025 *)

a(n) = Sum_{j such that j is in A001358 and j<=n} = A062198(A072000(n)).

cp's OEIS Frontend

A063959 Sum of the primes from 1 to n!.

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A063960 Sum of non-unitary prime divisors of n!: sum of those prime divisors for which the exponent in the prime factorization exceeds 1.

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A074793 Sum of prime powers less than or equal to n.

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A143537 Triangle read by rows: T(n,k) = number of primes in the interval [k..n], n >= 1, 1 <= k <= n.

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A294113 Sum of the smaller parts of the partitions of 2n into two parts with larger part prime.

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A341700 Sum of the primes p satisfying n < p <= 2n.

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A380118 a(n) = Sum_{k=1..n} (A014963(k) - A061397(k)).

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