A301988 -id:A301988 - cp's OEIS frontend

A379318 Odd numbers whose product of prime indices is a multiple of their sum of prime indices.

3, 5, 7, 9, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 63, 65, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 125, 127, 131, 137, 139, 149, 151, 157, 163, 165, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239

Offset: 1

Gus Wiseman, Jan 16 2025

nonn

Contains all odd primes.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The sum and product of prime indices are A056239 and A003963 respectively.

			The terms together with their prime indices begin:
     2: {1}         53: {16}           109: {29}
     3: {2}         59: {17}           113: {30}
     5: {3}         61: {18}           125: {3,3,3}
     7: {4}         63: {2,2,4}        127: {31}
     9: {2,2}       65: {3,6}          131: {32}
    11: {5}         67: {19}           137: {33}
    13: {6}         71: {20}           139: {34}
    17: {7}         73: {21}           149: {35}
    19: {8}         79: {22}           150: {1,2,3,3}
    23: {9}         81: {2,2,2,2}      151: {36}
    29: {10}        83: {23}           154: {1,4,5}
    30: {1,2,3}     84: {1,1,2,4}      157: {37}
    31: {11}        89: {24}           163: {38}
    37: {12}        97: {25}           165: {2,3,5}
    41: {13}       101: {26}           167: {39}
    43: {14}       103: {27}           169: {6,6}
    47: {15}       107: {28}           173: {40}
    49: {4,4}      108: {1,1,2,2,2}    179: {41}

Including evens gives A326149, counted by A057568.
For nonprime instead of odd we get A326150.
For even instead of odd we get A379319, counted by A379320.
Partitions of this type are counted by A379734, strict A379735, see A379733.
For squarefree instead of odd we get A379844, even case A379845.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- divisible: A057567, ranks A326155
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029, A379720
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A069016, A301988, A318950, A319000, A319916, A324851, A325041, A326152, A379671, A379678.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],OddQ[#]&&Divisible[Times@@prix[#],Total[prix[#]]]&]

A325043 Heinz numbers of integer partitions, with at least three parts, whose product of parts is one fewer than their sum.

Original entry on oeis.org

18, 60, 168, 216, 400, 528, 1248, 2240, 2880, 3264, 7296, 14080, 17664, 25088, 32256, 41472, 44544, 66560, 95232, 153600, 227328, 315392, 348160, 405504, 503808, 1056768, 1556480, 2310144, 2981888, 3833856, 5210112, 6881280, 7536640, 7929856, 8847360, 11599872

Offset: 1

Gus Wiseman, Mar 25 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are numbers with at least three prime factors (counted with multiplicity) whose product of prime indices (A003963) is one fewer than their sum of prime indices (A056239).

			The sequence of terms together with their prime indices begins:
     18: {1,2,2}
     60: {1,1,2,3}
    168: {1,1,1,2,4}
    216: {1,1,1,2,2,2}
    400: {1,1,1,1,3,3}
    528: {1,1,1,1,2,5}
   1248: {1,1,1,1,1,2,6}
   2240: {1,1,1,1,1,1,3,4}
   2880: {1,1,1,1,1,1,2,2,3}
   3264: {1,1,1,1,1,1,2,7}
   7296: {1,1,1,1,1,1,1,2,8}
  14080: {1,1,1,1,1,1,1,1,3,5}
  17664: {1,1,1,1,1,1,1,1,2,9}
  25088: {1,1,1,1,1,1,1,1,1,4,4}
  32256: {1,1,1,1,1,1,1,1,1,2,2,4}
  41472: {1,1,1,1,1,1,1,1,1,2,2,2,2}
  44544: {1,1,1,1,1,1,1,1,1,2,10}
  66560: {1,1,1,1,1,1,1,1,1,1,3,6}
  95232: {1,1,1,1,1,1,1,1,1,1,2,11}

Cf. A000720, A003963, A056239, A112798, A178503, A175508, A301987, A319000.

Cf. A325032, A325033, A325036, A325037, A325038, A325041, A325042, A325044.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[10000],And[PrimeOmega[#]>2,Times@@primeMS[#]==Total[primeMS[#]]-1]&]

a(n) = 2 * A301988(n).

More terms from Jinyuan Wang, Jun 27 2020

cp's OEIS Frontend

A379318 Odd numbers whose product of prime indices is a multiple of their sum of prime indices.

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