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

1,1

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).

Links

Table of n, a(n) for n=1..46.

Example

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}

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A095611 A335272 A358255 * A200070 A324210 A146081

Adjacent sequences: A307531 A307532 A307533 * A307535 A307536 A307537

Keyword

nonn,changed

Author

Gus Wiseman, Apr 13 2019

Status

approved