A325230 -id:A325230 - cp's OEIS frontend

A325226 Number of prime factors of n that are less than the largest, counted with multiplicity.

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 1, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 2, 0, 0, 1, 1, 1, 2, 0, 1, 1, 3, 0, 2, 0, 2, 2, 1, 0, 4, 0, 1, 1, 2, 0, 1, 1, 3, 1, 1, 0, 3, 0, 1, 2, 0, 1, 2, 0, 2, 1, 2, 0, 3, 0, 1, 1, 2, 1, 2, 0, 4, 0, 1, 0, 3, 1, 1, 1, 3, 0, 3, 1, 2, 1, 1, 1, 5, 0, 1, 2, 2, 0, 2, 0, 3, 2

Offset: 1

Gus Wiseman, Apr 12 2019

nonn

			The prime factors of 300 are {2,2,3,5,5} of which {2,2,3} are less than the largest, so a(300) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Positions of 0's are A000961. Positions of 1's are A325230. Positions of terms > 1 are A307517.

Cf. A000094, A000245, A001222, A052126, A053585, A061395, A062977, A064989, A071178, A105441, A108951, A256617.

Table[PrimeOmega[n/Power@@FactorInteger[n][[-1]]],{n,100}]

A071178(n) = if(1==n, 0, factor(n)[omega(n), 2]);
A325226(n) = (bigomega(n) - A071178(n)); \\ Antti Karttunen, Nov 17 2019

a(n) = A001222(n/A053585(n)).

a(n) = A001222(n) - A071178(n) = A062977(A108951(n)). - Antti Karttunen, Nov 17 2019

Data section extended up to term a(105) by Antti Karttunen, Nov 17 2019

A325229 Heinz numbers of integer partitions such that lesser of the maximum part and the number of parts is 2.

Original entry on oeis.org

6, 9, 10, 12, 14, 15, 18, 21, 22, 24, 25, 26, 27, 33, 34, 35, 36, 38, 39, 46, 48, 49, 51, 54, 55, 57, 58, 62, 65, 69, 72, 74, 77, 81, 82, 85, 86, 87, 91, 93, 94, 95, 96, 106, 108, 111, 115, 118, 119, 121, 122, 123, 129, 133, 134, 141, 142, 143, 144, 145, 146

Offset: 1

Gus Wiseman, Apr 12 2019

nonn

The enumeration of these partitions by sum is given by A265283.

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

			The sequence of terms together with their prime indices begins:
    6: {1,2}
    9: {2,2}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   15: {2,3}
   18: {1,2,2}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}
   25: {3,3}
   26: {1,6}
   27: {2,2,2}
   33: {2,5}
   34: {1,7}
   35: {3,4}
   36: {1,1,2,2}
   38: {1,8}
   39: {2,6}
   46: {1,9}

Cf. A056239, A061395, A093641, A112798, A252464, A257541, A263297, A265283, A325224, A325225, A325227, A325230, A325232.

Select[Range[300],Min[PrimeOmega[#],PrimePi[FactorInteger[#][[-1,1]]]]==2&]

A362983 Number of prime factors of n (with multiplicity) that are greater than the least.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 2, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 0, 1, 1, 1, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 1, 0, 2, 1, 1, 0, 3, 1, 1, 1, 1, 0, 2, 0, 1, 1, 0, 1, 2, 0, 1, 1, 2, 0, 2, 0, 1, 2, 1, 1, 2, 0, 1, 0, 1, 0, 2, 1, 1, 1

Offset: 1

Gus Wiseman, May 18 2023

nonn

			The prime factorization of 360 is 2*2*2*3*3*5, with factors greater than the least 3*3*5, so a(360) = 3.

Positions of 0's are A000961.
Positions of numbers > 0 are A024619.
Positions of first appearances appear to be A099856.
For "less than greatest" instead of "greater than least" we have A325226.
For multiplicities instead of parts we have A363131.
A027746 lists prime factors, A112798 indices, A124010 exponents.
A047966 counts uniform partitions, ranks A072774.
A363128 counts partitions with more than one non-mode, complement A363129.
Cf. A001221, A001222, A052126, A053585, A061395, A064989, A071178, A105441, A307517, A325230.

Table[PrimeOmega[n]-If[n==1,0,FactorInteger[n][[1,2]]],{n,30}]

a(n) = A001222(n) - A067029(n).

a(n) = A001222(A028234(n)).

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A325226 Number of prime factors of n that are less than the largest, counted with multiplicity.

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A325229 Heinz numbers of integer partitions such that lesser of the maximum part and the number of parts is 2.

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A362983 Number of prime factors of n (with multiplicity) that are greater than the least.

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