A054845 -id:A054845 - cp's OEIS frontend

A055000 Integers that can be expressed as the sum of consecutive primes in exactly 5 ways.

311, 863, 14369, 14699, 15329, 16277, 19717, 20272, 25416, 28500, 29033, 36467, 37607, 40433, 41074, 42463, 45101, 46660, 48731, 49253, 49499, 50560, 53424, 55813, 59068, 67141, 68787, 70104, 70429, 70692, 71548, 76423, 78756, 78791

Offset: 1

Jud McCranie, May 30 2000

nonn

R. K. Guy, Unsolved Problems in Number Theory, section C2.

Ray Chandler, Table of n, a(n) for n = 1..10000
Carlos Rivera, Puzzle 46. Primes expressible as sum of consecutive primes in K ways, The Prime Puzzles and Problems Connection.

Cf. A054845, A054859, A054996, A054997, A054998, A054999, A055001.

A054845(a(n)) = 5. - Ray Chandler, Sep 20 2023

A060436 Numerator of Sum_{k=1..n} d(k)/k, where d() = A000005().

Original entry on oeis.org

1, 2, 8, 41, 229, 269, 2003, 2213, 2353, 2521, 28571, 30881, 410693, 427853, 443869, 1850551, 31939847, 33301207, 640891093, 664170349, 226316943, 231019823, 5365187609, 16690477147, 84523231511, 85896110711, 784963282799, 802173304199, 23423652688171

Offset: 1

N. J. A. Sloane, Nov 02 2008

The old entry with this sequence number was a duplicate of A054845.

			1, 2, 8/3, 41/12, 229/60, 269/60, 2003/420, 2213/420, 2353/420, 2521/420, 28571/4620, 30881/4620, ...

M. N. Huxley, Area, Lattice Points and Exponential Sums, Oxford, 1996; p. 237.

Robert Israel, Table of n, a(n) for n = 1..2296
Vaclav Kotesovec, Graph - The asymptotic ratio of Sum_{k=1..n} d(k)/k
Mathematics.StackExchange, The asymptotic expansion for the weighted sum of divisors, Aug 19 2013.

Cf. A000005, A065080.

t:= 0:
for n from 1 to 50 do
  t:= t + numtheory:-tau(n)/n;
  A[n]:= numer(t);
od:
seq(A[n],n=1..50); # Robert Israel, Mar 20 2018

Sum_{k=1..n} A000005(k)/k = a(n)/A065080(n) ~ log(n)^2/2 + 2*gamma*log(n) + gamma^2 - 2*gamma_1, where gamma is the Euler-Mascheroni constant A001620 and gamma_1 is the first Stieltjes constant A082633. - Vaclav Kotesovec, Aug 30 2018

A272041 Smallest integer that can be expressed as the sum of n primes in at least n distinct ways.

Original entry on oeis.org

2, 10, 15, 18, 19, 22, 25, 27, 29, 32, 34, 36, 39, 42, 44, 46, 49, 51, 53, 55, 58, 60, 63, 65, 67, 69, 72, 74, 76, 78, 80, 83, 85, 87, 90, 92, 94, 96, 98, 100, 102, 105, 107, 109, 111, 113, 115, 117, 120, 122, 124, 126, 128, 131, 133, 135, 137, 139, 141, 143

Offset: 1

Matthew Ryan, Apr 21 2016

nonn

Initial terms found by exhaustive search in Excel.

			The sequence is defined here as starting at n=1 to avoid the term a(0). Even though there cannot be exactly zero ways to add zero primes, there is always at least one way to add 0 primes to get any n (i.e., the sum of itself for any nonprime or (1+..+1) for any prime), and zero would be the lowest such number.
Sum of 1 prime in 1 way: 2.
Sum of 2 primes in 2 ways: 3+7 = 5+5 = 10.
Sum of 3 primes in 3 ways: 3+5+7 = 5+5+5 = 2+2+11 = 15.
Sum of 4 primes in 4 ways: 2+2+3+11 = 2+2+7+7 = 3+3+5+7 = 3+5+5+5 = 18.
Sum of 60 primes in 61 ways, e.g.: 57*2 + 3 + 7 + 19 = 37*2 + 23*3 = 143. - _Lars Blomberg_, Jul 18 2017

Lars Blomberg and Giovanni Resta, Table of n, a(n) for n = 1..5000 (first 99 terms from Lars Blomberg)

Cf. A054845, A054859.

a[n_] := Block[{k = 1}, While[Length@ Quiet@ IntegerPartitions[k,{n}, Prime@ Range@ PrimePi@ k, n] < n, k++]; k]; Array[a, 50]

a(36)-a(60) from Lars Blomberg, Jul 18 2017

A309770 Numbers that are sums of one or more consecutive primes in more than one way.

Original entry on oeis.org

5, 17, 23, 31, 36, 41, 53, 59, 60, 67, 71, 72, 83, 90, 97, 100, 101, 109, 112, 119, 120, 127, 131, 138, 139, 143, 152, 173, 180, 181, 187, 197, 199, 204, 210, 211, 221, 223, 228, 233, 240, 251, 258, 263, 269, 271, 276, 281, 287, 300, 304, 311, 323, 330, 331, 340, 349

Offset: 1

Ilya Gutkovskiy, Aug 16 2019

nonn

Contains A067372 as a subsequence.

			5 is in the sequence because it can be written as either 5 or 2 + 3.
36 is the sequence because it can be written as either 5 + 7 + 11 + 13 or 17 + 19.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Sums

Cf. A000040, A034707, A050936, A054845, A054996, A067372, A067377.

N:= 1000: # for terms <= N
P:= select(isprime, [2, seq(i,i=3..N,2)]):
S:= [0,op(ListTools:-PartialSums(P))]:
V:= Vector(N):
for i from 1 to nops(S) do
  for j from i-1 to 1 by -1 do
    v:= S[i]-S[j];
    if v > N then break fi;
    V[v]:= V[v]+1;
od od:
select(t -> V[t]>1, [$1..N]); # Robert Israel, Aug 22 2019

A054845(a(n)) > 1.

cp's OEIS Frontend

A055000 Integers that can be expressed as the sum of consecutive primes in exactly 5 ways.

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A060436 Numerator of Sum_{k=1..n} d(k)/k, where d() = A000005().

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A272041 Smallest integer that can be expressed as the sum of n primes in at least n distinct ways.

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A309770 Numbers that are sums of one or more consecutive primes in more than one way.

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