login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Thanks to everyone who made a donation during our annual appeal!
To see the list of donors, or make a donation, see the OEIS Foundation home page.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A325395 Heinz numbers of integer partitions whose augmented differences are strictly increasing. 13
1, 2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 91, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 209, 211, 221 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

The augmented differences aug(y) of an integer partition y of length k are given by aug(y)_i = y_i - y_{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).

The enumeration of these partitions by sum is given by A325357.

LINKS

Table of n, a(n) for n=1..60.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

EXAMPLE

The sequence of terms together with their prime indices begins:

    1: {}

    2: {1}

    3: {2}

    5: {3}

    7: {4}

    9: {2,2}

   11: {5}

   13: {6}

   17: {7}

   19: {8}

   23: {9}

   25: {3,3}

   29: {10}

   31: {11}

   35: {3,4}

   37: {12}

   41: {13}

   43: {14}

   47: {15}

   49: {4,4}

MATHEMATICA

primeptn[n_]:=If[n==1, {}, Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]];

aug[y_]:=Table[If[i<Length[y], y[[i]]-y[[i+1]]+1, y[[i]]], {i, Length[y]}];

Select[Range[100], Less@@aug[primeptn[#]]&]

CROSSREFS

Cf. A056239, A093641, A112798, A240027, A325351, A325357, A325366, A325389, A325394, A325396, A325398, A325456, A325460.

Sequence in context: A319151 A180934 A090332 * A070566 A325623 A136327

Adjacent sequences:  A325392 A325393 A325394 * A325396 A325397 A325398

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 02 2019

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 20 08:18 EST 2020. Contains 331081 sequences. (Running on oeis4.)