A070215 -id:A070215 - cp's OEIS frontend

A316151 Heinz numbers of strict integer partitions of prime numbers into prime parts.

3, 5, 11, 15, 17, 31, 33, 41, 59, 67, 83, 93, 109, 127, 157, 177, 179, 191, 211, 241, 277, 283, 327, 331, 353, 367, 401, 431, 461, 509, 537, 547, 563, 587, 599, 617, 709, 739, 773, 797, 831, 859, 877, 919, 967, 991, 1031, 1059, 1063, 1087, 1153, 1171, 1201

Offset: 1

Gus Wiseman, Jun 25 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			Sequence of strict integer partitions of prime numbers into prime parts, preceded by their Heinz numbers, begins:
   3: (2)
   5: (3)
  11: (5)
  15: (3,2)
  17: (7)
  31: (11)
  33: (5,2)
  41: (13)
  59: (17)
  67: (19)
  83: (23)
  93: (11,2)

Cf. A000586, A000607, A038499, A056239, A056768, A064688, A070215, A085755, A302590, A316092.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],And[SquareFreeQ[#],PrimeQ[Total[primeMS[#]]],And@@PrimeQ/@primeMS[#]]&]

A331901 Number of compositions (ordered partitions) of the n-th prime into distinct prime parts.

Original entry on oeis.org

1, 1, 3, 3, 1, 3, 25, 9, 61, 91, 99, 151, 901, 303, 1759, 3379, 5239, 4713, 8227, 12901, 12537, 23059, 65239, 159421, 232369, 489817, 351237, 726295, 564363, 1101883, 2517865, 6916027, 11825821, 4942227, 27166753, 21280053, 39547957, 52630273, 113638975

Offset: 1

Ilya Gutkovskiy, Jan 31 2020

nonn

			a(4) = 3 because we have [7], [5, 2] and [2, 5].

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Index entries for sequences related to compositions

Cf. A000040, A023360, A056768, A070215, A219107, A265112, A299168.

s:= proc(n) option remember; `if`(n<1, 0, ithprime(n)+s(n-1)) end:
b:= proc(n, i, t) option remember; `if`(s(i)`if`(p>n, 0, b(n-p, i-1, t+1)))(ithprime(i))+b(n, i-1, t)))
    end:
a:= n-> b(ithprime(n), n, 0):
seq(a(n), n=1..42);  # Alois P. Heinz, Jan 31 2020

s[n_] := s[n] = If[n < 1, 0, Prime[n] + s[n - 1]];
b[n_, i_, t_] := b[n, i, t] = If[s[i] < n, 0, If[n == 0, t!, Function[p, If[p > n, 0, b[n - p, i - 1, t + 1]]][Prime[i]] + b[n, i - 1, t]]];
a[n_] := b[Prime[n], n, 0];
Array[a, 42] (* Jean-François Alcover, Nov 26 2020, after Alois P. Heinz *)

a(n) = A219107(A000040(n)).

A334292 Number of sets of primes less than the n-th prime whose sum is the n-th prime.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 1, 2, 4, 6, 8, 10, 13, 14, 18, 25, 34, 38, 49, 60, 66, 86, 101, 129, 177, 203, 223, 256, 277, 319, 521, 594, 723, 775, 1063, 1135, 1363, 1633, 1835, 2191, 2600, 2760, 3644, 3862, 4293, 4548, 6261, 8557, 9452, 9963, 11000, 12773, 13437, 17121, 19774, 22799

Offset: 1

Gil Broussard, Apr 21 2020

nonn

			a(5) = 0 because 11 is the 5th prime and there are 0 sets of primes < 11 whose sum = 11.
a(9) = 4 because 23 is the 9th prime and there are 4 sets of primes < 23 whose sums = 23: 13+7+3, 13+5+3+2, 11+7+5, 11+7+3+2.

David A. Corneth, Table of n, a(n) for n = 1..10001

Cf. A000586, A070215, A111133.

lista(nn) = {my(v, w=primes(nn)); v=Vec(prod(i=1, nn, 1+'x^w[i]) + O('x^(w[nn]+1))); for(i=1, nn, print1(v[w[i]+1]-1, ", ")); } \\ Jinyuan Wang, May 04 2020

Same generating function as A111133, but on the domain of prime numbers.

a(n) = A070215(n) - 1. - Jinyuan Wang, May 04 2020

More terms from David A. Corneth, Apr 22 2020

A335306 a(n) is the smallest composite number whose sum of distinct prime divisors is prime(n).

Original entry on oeis.org

4, 9, 6, 10, 121, 22, 210, 34, 273, 399, 58, 435, 651, 82, 777, 903, 1645, 118, 885, 1281, 142, 1065, 1533, 1659, 1335, 3115, 202, 2037, 214, 2163, 3729, 6213, 2667, 274, 2919, 298, 2235, 4917, 3297, 3423, 5845, 358, 3801, 382, 7059, 394, 6501, 7385, 8229, 454, 4683

Offset: 1

David James Sycamore, May 31 2020

nonn

a(n) <= prime(n)^2 for all n, the equality applies to n = 1,2,5 since 2,3,11 are the only primes which cannot be expressed as the sum of distinct smaller primes. For n other than 1,2,5, a(n) is squarefree, and corresponds to the partition (q_1, q_2,....q_k) of n into distinct primes whose product is the least possible value compared with the product of all distinct prime partitions of n. The intersection of this sequence with A261023 corresponds to primes in A133225.

a(n) >= max(4,2*prime(n)-4) with equality if and only if n = 1 or n is in A107770. - Chai Wah Wu, Jun 01 2020

			a(7) = 10 since (2,5) is the only prime partition of 7 into distinct smaller parts, and 2*5 = 10. a(11) = 11^2 = 121 because the prime partitions of 11 into smaller parts are: (2,2,7), (2,2,2,5), (2,2,2,2,3), (3,3,5), (2,3,3,3), none of which have only distinct primes.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000040, A000586, A008472, A107770, A070215, A133225, A261023.

a[n_] := Block[{k = 4, p = Prime@ n}, While[PrimeQ[k] || p != Total[First /@ FactorInteger[k]], k++]; k]; Array[a, 50] (* Giovanni Resta, May 31 2020 *)

a(n) = {my(p=prime(n)); forcomposite(k=1, p^2, if (vecsum(factor(k)[, 1]) == p, return(k)););} \\ Michel Marcus, May 31 2020

from sympy import prime, primefactors
def A335306(n):
    p = prime(n)
    for m in range(max(4,2*p-4),p**2+1):
        if sum(primefactors(m)) == p:
            return m # Chai Wah Wu, Jun 01 2020

More terms from Michel Marcus, May 31 2020

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A316151 Heinz numbers of strict integer partitions of prime numbers into prime parts.

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A331901 Number of compositions (ordered partitions) of the n-th prime into distinct prime parts.

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A334292 Number of sets of primes less than the n-th prime whose sum is the n-th prime.

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A335306 a(n) is the smallest composite number whose sum of distinct prime divisors is prime(n).

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