A117929 -id:A117929 - cp's OEIS frontend

A306433 Number of partitions of n into 2 distinct prime powers (not including 1).

0, 0, 0, 0, 0, 1, 1, 2, 1, 2, 2, 3, 3, 3, 2, 3, 3, 2, 3, 3, 4, 4, 3, 2, 4, 3, 3, 4, 4, 3, 5, 3, 5, 4, 6, 4, 7, 2, 4, 4, 6, 3, 5, 3, 5, 5, 5, 2, 7, 3, 6, 4, 6, 2, 7, 3, 7, 4, 5, 2, 7, 3, 5, 4, 6, 2, 9, 2, 7, 5, 7, 2, 9, 3, 6, 6, 7, 3, 9, 2, 8, 4, 5, 4, 10, 3, 8, 4, 7, 3, 11, 4, 8, 3, 6, 2

Offset: 0

Ilya Gutkovskiy, Apr 30 2019

nonn

			a(12) = 3 because we have [9, 3], [8, 4] and [7, 5].

Index entries for sequences related to partitions

Cf. A000961, A071330, A106244, A117929, A246655, A307726, A307825.

Table[Count[IntegerPartitions[n, {2}], _?(And[UnsameQ @@ #, AllTrue[#, PrimePowerQ[#] &]] &)], {n, 0, 95}]

a(n) = [x^n y^2] Product_{k>=1} (1 + y*x^A246655(k)).

A344989 Smallest number whose number of partitions into n distinct primes is n, or zero if there are no such partitions.

Original entry on oeis.org

2, 16, 26, 33, 55, 59, 0, 0, 124, 159, 233, 227, 276, 0, 372, 480, 0, 0, 0, 752, 0, 920, 0, 1011, 0, 1211, 1425, 0, 0, 0, 0, 0, 2050, 2336, 2495, 0, 0, 0, 0, 3340, 0, 3712, 0, 0, 4303, 0, 0, 0, 0, 5195, 0, 5669, 0, 6163, 6673, 0, 0, 0, 7504, 0, 0, 8670, 0, 9304, 9623, 0, 0, 0, 10638, 10981, 0, 12062, 0

Offset: 1

Metin Sariyar, Jun 04 2021

From David A. Corneth, Aug 21 2025: (Start)
How to prove a 0? I used the heuristic:
a(n) = 0 if 2*n consecutive integers can be written in strictly more than n ways as a sum of n distinct primes and up to that point no positive integer has exactly n such ways.
What other rules where used? (End)

			a(2) = 16 because 16 is the smallest number whose number of partitions into 2 distinct primes is 2; 16 = 3+13 = 5+11.

Chris K. Caldwell and G. L. Honaker, Jr., Prime Curios! 233

Cf. A364692 asks for the largest number with the same properties.

Cf. A000586, A077914, A117929, A125688, A219180, A219198, A219199, A219200, A219201, A219202, A219203, A219204.

a(12)-a(20) from Alois P. Heinz, Jun 04 2021

More terms from David A. Corneth, Aug 21 2025

A352305 a(n) is the (conjectured) largest even number that can be expressed as the sum of two distinct primes in exactly n ways.

Original entry on oeis.org

6, 38, 68, 128, 158, 188, 398, 362, 458, 542, 632, 692, 602, 992, 808, 908, 1112, 1238, 1412, 1418, 1718, 1544, 1574, 1622, 1682, 2048, 2252, 2018, 2672, 2042, 2558, 2936, 2504, 2978, 2966, 3092, 3218, 3242, 3272, 3506, 3632, 3754, 4022, 4058, 4052, 4412, 4448, 4478

Offset: 0

Ilya Gutkovskiy, Mar 11 2022

nonn

			a(5) = 188 because 188 = 7 + 181 = 31 + 157 = 37 + 151 = 61 + 127 = 79 + 109 and it is conjectured that 188 is the last term of A080854.

Cf. A000954, A038609, A061357, A077914, A077969, A078299, A080854, A080862, A087747, A117929, A212613.

More terms from Hugo Pfoertner, Dec 18 2024

A362039 Least number s such that there are 2 different sets of primes { a1, a2, ..., an } and { b1, b2, ..., bn } with the integers in each set having the same sum s, the same sum of squares, etc., up to and including the same sum of (n-1)-st powers.

Original entry on oeis.org

16, 55, 120, 433, 378

Offset: 2

Jean-Marc Rebert, Apr 15 2023

We are to find the least number s such that there is a solution in primes to the system of equations:
a1^k + a2^k + ... + an^k = b1^k + b2^k + ... + bn^k, (k = 1, 2, ..., n-1) and {a1, ..., an} != {b1, ..., bn}.
a(7), a(8) are respectively <= 2399, 348592.

			a(2) = 16, because 3 + 13 = 16 = 5 + 11 and no lesser sum of 2 distinct primes has this property.
a(3) = 55, because 7 + 19 + 29 = 55 = 11 + 13 + 31 and 7^2 + 19^2 + 29^2 = 1251 = 11^2 + 13^2 + 31^2, and no lesser sum of 3 distinct primes has this property.
a(4) = 120, because with u = [13, 29, 31, 47] and v = [17, 19, 41, 43], Sum_{i=1..4} u(i) = 120  = Sum_{i=1..4} v(i) and Sum_{i=1..4} u(i)^2 = 4100 = Sum_{i=1..4} v(i)^2 and Sum_{i=1..4} u(i)^3 = 1602000 = Sum_{i=1..4} v(i)^3 and no lesser sum of 4 distinct primes has this property.
From _Andrew Howroyd_, Apr 18 2023: (Start)
a(5) = 433 with {13, 59, 67, 131, 163} and {23, 31, 103, 109, 167}.
a(6) = 378 with {17, 37, 43, 83, 89, 109} and {19, 29, 53, 73, 97, 107}.
(End)

Carlos Rivera, Problem 67. Multigrade Prime Solutions, The Prime Puzzles & Problems Connection.
Chen Shuwen, The Prouhet-Tarry-Escott Problem, Equal Sums of Like Powers.

Cf. A087747, A117929, A239066, A239067, A239068, A323395.

\\ Call with pr=1 to also print solution sets.
a(n, pr=0)={
  forstep(s=3*n, oo, 2, my(P=vector(s,i,primepi(i)), X=primes(P[s]));
    local(found=0, M=Map(), V=vector(n));
    my(onSet()=my(key=vector(n-2, j, sum(i=1, n, V[i]^(j+1))), z);
      if(mapisdefined(M,key,&z), found++; if(pr, print(V, z)), mapput(M,key,V)));
    my(recurse(r,m,k)=if(k==0, onSet(), for(m=max(k,P[(r-1)\k])+1, min(m, P[r-3*(k-1)]), V[k]=X[m]; self()(r-X[m], m-1, k-1)) ));
    recurse(s, #X, n);
    if(found, return(s));
  )
} \\ Andrew Howroyd, Apr 18 2023

a(2) = min({k >= 1 : A117929(k) >= 2}) = Min_{m >= 2} A087747(m) = A087747(2). - Peter Munn, May 01 2023

a(5)-a(6) from Andrew Howroyd, Apr 18 2023

Edited by Peter Munn, May 01 2023

A370090 Numbers that can be expressed in exactly one way as the unordered sum of two distinct primes.

Original entry on oeis.org

5, 7, 8, 9, 10, 12, 13, 14, 15, 19, 21, 25, 31, 33, 38, 39, 43, 45, 49, 55, 61, 63, 69, 73, 75, 81, 85, 91, 99, 103, 105, 109, 111, 115, 129, 133, 139, 141, 151, 153, 159, 165, 169, 175, 181, 183, 193, 195, 199, 201, 213, 225, 229, 231, 235, 241, 243, 253, 259, 265

Offset: 1

Wesley Ivan Hurt, Feb 11 2024

nonn

Apparently, a number that is the predecessor or successor of a prime number does not have a sum as defined here, except for a finite number of primes, which may be {7, 11, 13, 37}. - Peter Luschny, Feb 16 2024

			5 = 2+3; 7 = 2+5; 8 = 3+5; 9 = 2+7; 10 = 3+7 (10 = 5+5 is not considered).

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Goldbach conjecture
Index entries for sequences related to partitions

Cf. A117929, A048974, A065091, A067187 (not necessarily distinct).

If we change 1 way (this sequence) we get A077914 (2 ways), A077969 (3 ways), A078299 (4 ways), A080854 (5 ways), and A080862 (6 ways).

select(n -> A117929(n) = 1, [seq(1..265)]);  # Peter Luschny, Feb 16 2024

tdpQ[{a_,b_}]:=AllTrue[{a,b},PrimeQ]&&a!=b; Select[Range[300],Count[IntegerPartitions[#,{2}],?tdpQ]==1&] (* _Harvey P. Dale, Dec 30 2024 *)

from sympy import sieve
from collections import Counter
from itertools import combinations
def aupton(max):
    sieve.extend(max)
    a = Counter(c[0]+c[1] for c in combinations(sieve._list, 2))
    return [n for n in range(1, max+1) if a[n] == 1]
print(aupton(265)) # Michael S. Branicky, Feb 16 2024

A140955 Even integers that do not have at least two partitions into 2 distinct primes.

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 12, 14, 38

Offset: 1

Gil Broussard, Jul 25 2008

If A056636(3) = 128 (as is conjectured), then 38 is the last term in the sequence. - Charles R Greathouse IV, Sep 07 2022

			8 is a term because 3+5 is the only sum of primes = 8.
16 is not in the sequence because 16 = 3+13 and 5+11.
The only ways to express 10 as a sum of two unordered primes are 3+7 and 5+5. In one of the sums the primes are distinct. Thus, 10 is in this sequence. - _Tanya Khovanova_, Sep 07 2022

Cf. A056636, A061357, A077914, A117929.

Select[Range[0,100,2],Length[Select[Union/@IntegerPartitions[#,{2}],AllTrue[#,PrimeQ]&&Length[#]==2&]]<2&] (* James C. McMahon, Jul 15 2025 *)

is(n)=if(n%2, return(0)); my(t); forprime(p=3, n\2-1, if(isprime(n-p) && t++>1, return(0))); 1 \\ Charles R Greathouse IV, Sep 07 2022

Offset changed to 1 by Alois P. Heinz, Sep 07 2022

A286424 Number of partitions of p_n# into parts (q, k) both coprime to p_n#, with q prime and k nonprime, where p_n# = A002110(n).

Original entry on oeis.org

0, 0, 1, 1, 4, 110, 1432, 23338, 397661, 8193828, 212858328, 5941706227

Offset: 0

Michael De Vlieger, May 08 2017

Number of totative pairs (q, k) such that prime q + k nonprime = p_n# and both gcd(q, p_n#) = 1 and gcd(k, p_n#) = 1, with p_n < q <= pi(p_n#), where pi(p_n#) = A000849(n) - n = A048862(n).
Primes p_n < q <= pi(p_n#) are greater than the greatest prime factor of p_n# = p_n, and are thus coprime to p_n#. By the definition of primorial, we need not consider p >= p_n, as these p are divisors of p_n#, i.e., gcd(p, p_n#) = p. Since the totatives of m can be paired such that a + b = m, we need only determine if (p_n# - q) is not prime in order to count pairs (q, k).
a(n) < floor(A005867(n)/2).
a(n) <= A048862(n).
The totative pair (q,1) = (p_n# - 1, 1) is counted by a(n) for n in A057704, with (p_n# - 1) appearing in A057705.

			a(0) = 0 by definition. A002110(0) = 1; 1 is coprime to all numbers; the only possible totative pair is (1,1) and this does not include both a prime and a nonprime.
a(1) = 0 since, of the floor(A005867(1)/2) = 1 totative pair (1,1) of A002110(1) = 2, none include a both a prime and a nonprime.
a(2) = 1 since, the only totative pair (1,5) of A002110(1) = 6 includes both a prime and a nonprime.
a(3) = 1 since only (1,29) includes both a prime and a nonprime.
a(4) = 4 since (23,187), (41,169), (67,143), (89,121) include a both a prime and a nonprime.

C. K. Caldwell, The Prime Glossary, Primorial.
Eric Weisstein's World of Mathematics, Totative.

Cf. A000849, A002110, A005867, A048862, A048863, A057704, A057705, A116979, A117929.

Table[Function[P, Count[Prime@ Range[n + 1, PrimePi[P]], q_ /; ! PrimeQ[P - q]]]@ Product[Prime@ i, {i, n}], {n, 0, 9}] (* Michael De Vlieger, May 08 2017 *)

a(n) = (A000010(A002110(n)) - A048863(n)) - 2*A117929(A002110(n))
     = (A005867(n) - A048863(n)) - 2*A117929(A002110(n))
     = A048862(n) - 2*A117929(A002110(n)).

a(11) from Giovanni Resta, May 09 2017

A352596 Conjecturally the number of positive even integers that can be expressed as the sum of two distinct primes in exactly n ways.

Original entry on oeis.org

3, 5, 9, 12, 12, 16, 19, 13, 24, 19, 21, 25, 15, 29, 28, 16, 31, 22, 34, 32, 20, 29, 26, 24, 28, 36, 34, 35, 37, 22, 29, 37, 36, 34, 39, 32, 39, 35, 28, 31, 28

Offset: 0

Ilya Gutkovskiy, Mar 22 2022

Cf. A000974, A038609, A061357, A077914, A077969, A078299, A080854, A080862, A117929, A229492, A352305.

cp's OEIS Frontend

A306433 Number of partitions of n into 2 distinct prime powers (not including 1).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A344989 Smallest number whose number of partitions into n distinct primes is n, or zero if there are no such partitions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A352305 a(n) is the (conjectured) largest even number that can be expressed as the sum of two distinct primes in exactly n ways.

Views

Author

Keywords

Examples

Crossrefs

Extensions

A362039 Least number s such that there are 2 different sets of primes { a1, a2, ..., an } and { b1, b2, ..., bn } with the integers in each set having the same sum s, the same sum of squares, etc., up to and including the same sum of (n-1)-st powers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A370090 Numbers that can be expressed in exactly one way as the unordered sum of two distinct primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

A140955 Even integers that do not have at least two partitions into 2 distinct primes.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A286424 Number of partitions of p_n# into parts (q, k) both coprime to p_n#, with q prime and k nonprime, where p_n# = A002110(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A352596 Conjecturally the number of positive even integers that can be expressed as the sum of two distinct primes in exactly n ways.

Views

Author

Keywords

Crossrefs