A038499 -id:A038499 - 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[#]]&]

A360661 Number of factorizations of n into a prime number of factors > 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 3, 0, 1, 1, 3, 0, 3, 0, 3, 1, 1, 0, 5, 1, 1, 2, 3, 0, 4, 0, 5, 1, 1, 1, 7, 0, 1, 1, 5, 0, 4, 0, 3, 3, 1, 0, 9, 1, 3, 1, 3, 0, 5, 1, 5, 1, 1, 0, 9, 0, 1, 3, 7, 1, 4, 0, 3, 1, 4, 0, 12, 0, 1, 3, 3, 1, 4, 0, 9, 3, 1, 0, 9, 1, 1, 1, 5, 0, 9, 1, 3, 1, 1, 1, 13, 0, 3, 3, 7

Offset: 1

Ilya Gutkovskiy, Feb 15 2023

nonn

From Bernard Schott, Mar 25 2023: (Start)
a(n) depends only on the prime signature of n.
a(n) = 0 iff n is in A008578 (1 with primes).
a(n) = 1 iff n is in A001358 (semiprimes).
a(n) = 2 iff n is in A030078 (p^3).
a(n) = 3 iff n is in A080258 (p^4 or p*q^2).
a(n) = 4 iff n is in A007304 (p*q*r). (End)

			a(2) = 0 since 2 = 2 is the unique factorization of 2.
a(4) = 1 since 4 = 2^2 = 2 * 2.
a(6) = 1 since 6 = 2 * 3.
a(8) = 2 since 8 = 2^3 = 2 * 4 = 2 * 2 * 2.
a(12) = 3 since 12 = 3 * 2^2 = 2 * 6 = 3 * 4 = 2 * 2 * 3.
a(16) = 3 since 16 = 2^4 = 2 * 8 = 4 * 4 = 2 * 2 * 4.
a(30) = 4 since 30 = 2 * 3 * 5 = 2 * 15 = 3 * 10 = 5 * 6.

Sean A. Irvine, Java program (github)
Eric Weisstein's World of Mathematics, Unordered Factorization

Cf. A001055, A038499, A339890.

Cf. A001248, A001358, A006881, A007304, A008578, A030078, A030514, A054753, A080258.

From Bernard Schott, Mar 25 2023: (Start)
a(A000040(n)) = 0.
a(A001248(n)) = a(A006881(n)) = 1.
a(A030514(n)) = a(A054753(n)) = 3. (End)

A376348 a(n) is the number of multisets with n primes with which an n-gon with perimeter prime(n) can be formed.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 7, 7, 12, 19, 19, 25, 44, 72, 72, 119, 147, 152, 234, 292, 435, 777, 920, 946, 1135, 1161, 1377, 3702, 4293, 5942, 5942, 10741, 10741, 14483, 18953, 22091, 28658, 37686, 37686, 63053, 63053, 72389, 72389, 132732, 233773, 265312, 265312, 300443, 373266

Offset: 3

Felix Huber, Oct 13 2024

nonn

a(n) is the number of partitions of prime(n) into n prime parts < prime(n)/2.

First differs from A259254 at n=31: a(31) = 3702 but A259254(31) = 3703.

			a(7) = 2 because exactly the 2 partitions (2, 2, 2, 2, 3, 3, 3) and (2, 2, 2, 2, 2, 2, 5) have 7 prime parts and their sum is p(7) = 17.

Eric Weisstein's World of Mathematics, Polygon

Cf. A005044, A007042, A038499, A062890, A069906, A069907, A259254, A288253, A288254, A288255, A288256, A318604, A366398.

A376348:=proc(n)
   local a,p,x,i;
   a:=0;
   p:=ithprime(n);
   for x from NumberTheory:-pi(p/n)+1 to NumberTheory:-pi(p/2) do
      a:=a+numelems(select(i->nops(i)=n-1 and andmap(isprime,i),combinat:-partition(ithprime(n)-ithprime(x),ithprime(x))))
   od;
   return a
end proc;
seq(A376348(n),n=3..42);

a(n)={my(m=prime(n), p=primes(primepi((m-1)\2))); polcoef(polcoef(1/prod(i=1, #p, 1 - y*x^p[i], 1 + O(x*x^m)), m),n)} \\ Andrew Howroyd, Oct 13 2024

a(43) onwards from Andrew Howroyd, Oct 13 2024

A095700 Expansion of 1 + Sum_{i>=1} (x^prime(i)/Product_{j=1..i} (1-x^j)).

Original entry on oeis.org

1, 0, 1, 2, 2, 4, 4, 7, 8, 11, 13, 19, 21, 29, 34, 44, 52, 67, 78, 100, 117, 145, 171, 212, 248, 303, 356, 430, 504, 606, 707, 846, 986, 1168, 1360, 1605, 1861, 2186, 2531, 2957, 3415, 3976, 4578, 5311, 6103, 7050, 8085, 9315, 10654, 12238, 13971, 16000, 18230

Offset: 0

Jon Perry, Jul 06 2004

nonn

Cf. A038499.

cp's OEIS Frontend

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

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A360661 Number of factorizations of n into a prime number of factors > 1.

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A376348 a(n) is the number of multisets with n primes with which an n-gon with perimeter prime(n) can be formed.

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A095700 Expansion of 1 + Sum_{i>=1} (x^prime(i)/Product_{j=1..i} (1-x^j)).

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