A340610 - OEIS

A340610

Numbers whose number of prime factors (A001222) divides their greatest prime index (A061395).

2, 3, 5, 6, 7, 9, 11, 13, 14, 17, 19, 20, 21, 23, 26, 29, 30, 31, 35, 37, 38, 39, 41, 43, 45, 47, 49, 50, 52, 53, 56, 57, 58, 59, 61, 65, 67, 71, 73, 74, 75, 78, 79, 83, 84, 86, 87, 89, 91, 92, 95, 97, 101, 103, 106, 107, 109, 111, 113, 117, 122, 125, 126, 127

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OFFSET

1,1

COMMENTS

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.

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

FORMULA

A001222(a(n)) divides A061395(a(n)).

EXAMPLE

The sequence of terms together with their prime indices begins:

2: {1} 29: {10} 56: {1,1,1,4}

3: {2} 30: {1,2,3} 57: {2,8}

5: {3} 31: {11} 58: {1,10}

6: {1,2} 35: {3,4} 59: {17}

7: {4} 37: {12} 61: {18}

9: {2,2} 38: {1,8} 65: {3,6}

11: {5} 39: {2,6} 67: {19}

13: {6} 41: {13} 71: {20}

14: {1,4} 43: {14} 73: {21}

17: {7} 45: {2,2,3} 74: {1,12}

19: {8} 47: {15} 75: {2,3,3}

20: {1,1,3} 49: {4,4} 78: {1,2,6}

21: {2,4} 50: {1,3,3} 79: {22}

23: {9} 52: {1,1,6} 83: {23}

26: {1,6} 53: {16} 84: {1,1,2,4}

MAPLE

filter:= proc(n) local F, m, g, t;

F:= ifactors(n)[2];

m:= add(t[2], t=F);

g:= numtheory:-pi(max(seq(t[1], t=F)));

g mod m = 0;

end proc:

select(filter, [$2..1000]); # Robert Israel, Feb 08 2021

MATHEMATICA

Select[Range[2, 100], Divisible[PrimePi[FactorInteger[#][[-1, 1]]], PrimeOmega[#]]&]

CROSSREFS

Note: Heinz numbers are given in parentheses below.

The case of equality is A047993 (A106529).

The case where all parts are multiples, not just the maximum part, is A143773 (A316428), with strict case A340830, while the case of factorizations is A340853.

These are the Heinz numbers of certain partitions counted by A168659.

The reciprocal version is A340609.

The squarefree case is A340828 (A340856).

A001222 counts prime factors.

A006141 counts partitions whose length equals their minimum (A324522).

A056239 adds up prime indices.

A061395 selects the maximum prime index.

A067538 counts partitions whose length divides their sum (A316413).

A067538 counts partitions whose maximum divides their sum (A326836).

A112798 lists the prime indices of each positive integer.

A200750 counts partitions with length coprime to maximum (A340608).

A325134 = A001222 + A061395.