A307534 Heinz numbers of strict integer partitions with 3 parts, all of which are odd.

110, 170, 230, 310, 374, 410, 470, 506, 590, 670, 682, 730, 782, 830, 902, 935, 970, 1030, 1034, 1054, 1090, 1265, 1270, 1298, 1370, 1394, 1426, 1474, 1490, 1570, 1598, 1606, 1670, 1705, 1790, 1826, 1886, 1910, 1955, 1970, 2006, 2110, 2134, 2162, 2255, 2266

Offset: 1

Author

Gus Wiseman, Apr 13 2019

Keywords

• nonn

Comments

The enumeration of these partitions by sum is given by A001399.

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

Examples

			The sequence of terms together with their prime indices begins:
   110: {1,3,5}
   170: {1,3,7}
   230: {1,3,9}
   310: {1,3,11}
   374: {1,5,7}
   410: {1,3,13}
   470: {1,3,15}
   506: {1,5,9}
   590: {1,3,17}
   670: {1,3,19}
   682: {1,5,11}
   730: {1,3,21}
   782: {1,7,9}
   830: {1,3,23}
   902: {1,5,13}
   935: {3,5,7}
   970: {1,3,25}
  1030: {1,3,27}
  1034: {1,5,15}
  1054: {1,7,11}
		

Crossrefs

Cf. A001221, A001222, A001399, A005117, A007304, A014612, A037144, A051037, A056239, A080193, A080257, A112798, A143207, A304636.

Programs

Mathematica

Select[Range[1000],SquareFreeQ[#]&&PrimeNu[#]==3&&OddQ[Times@@PrimePi/@First/@FactorInteger[#]]&]

Python

from math import isqrt
from sympy import primepi, primerange, integer_nthroot, nextprime
def A307534(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x-sum((primepi(x//(k*m))+1>>1)-(b+1>>1) for a,k in filter(lambda x:x[0]&1,enumerate(primerange(2,integer_nthroot(x,3)[0]+1),1)) for b,m in filter(lambda x:x[0]&1,enumerate(primerange(nextprime(k)+1,isqrt(x//k)+1),a+2))))
    return bisection(f,n,n) # Chai Wah Wu, Oct 20 2024