

A325351


Heinz number of the augmented differences of the integer partition with Heinz number n.


21



1, 2, 3, 4, 5, 6, 7, 8, 6, 10, 11, 12, 13, 14, 9, 16, 17, 12, 19, 20, 15, 22, 23, 24, 10, 26, 12, 28, 29, 18, 31, 32, 21, 34, 15, 24, 37, 38, 33, 40, 41, 30, 43, 44, 18, 46, 47, 48, 14, 20, 39, 52, 53, 24, 25, 56, 51, 58, 59, 36, 61, 62, 30, 64, 35, 42, 67, 68, 57, 30, 71, 48, 73, 74, 18, 76, 21, 66, 79, 80, 24, 82, 83, 60, 55, 86, 69, 88, 89, 36, 35
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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). Note that aug preserves length so this sequence preserves omega (number of prime factors counted with multiplicity).


LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to Heinz numbers


EXAMPLE

The partition (3,2,2,1) with Heinz number 90 has augmented differences (2,1,2,1) with Heinz number 36, so a(90) = 36.


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]}];
Table[Times@@Prime/@aug[primeptn[n]], {n, 100}]


PROG

(PARI)
augdiffs(n) = { my(diffs=List([]), f=factor(n), prevpi, pi=0, i=#f~); while(i, prevpi=pi; pi = primepi(f[i, 1]); if(prevpi, listput(diffs, 1+(prevpipi))); if(f[i, 2]>1, f[i, 2], i)); if(pi, listput(diffs, pi)); Vec(diffs); };
A325351(n) = factorback(apply(prime, augdiffs(n))); \\ Antti Karttunen, Nov 16 2019


CROSSREFS

Number of appearances of n is A008480(n).
Cf. A056239, A093641 (fixed points), A112798, A325350, A325352, A325355, A325366, A325389, A325394, A325395, A325396.
Sequence in context: A063917 A234344 A331298 * A279319 A171890 A287793
Adjacent sequences: A325348 A325349 A325350 * A325352 A325353 A325354


KEYWORD

nonn,look


AUTHOR

Gus Wiseman, Apr 23 2019


EXTENSIONS

More terms from Antti Karttunen, Nov 16 2019


STATUS

approved



