A094457 -id:A094457 - cp's OEIS frontend

A094521 a(n) = gcd(n, A094457(n)).

1, 1, 2, 1, 2, 1, 4, 3, 2, 1, 4, 1, 2, 5, 8, 1, 6, 1, 4, 3, 2, 1, 8, 5, 2, 9, 4, 1, 10, 1, 16, 11, 2, 5, 12, 1, 2, 3, 8, 1, 6, 1, 4, 15, 2, 1, 16, 7, 10, 3, 4, 1, 18, 5, 8, 3, 2, 1, 20, 1, 2, 9, 32, 5, 22, 1, 4, 3, 10, 1, 24, 1, 2, 25, 4, 11, 6, 1, 16, 27, 2, 1, 12, 5, 2, 3, 8, 1, 30, 7, 4, 3, 2, 5, 32, 1

Offset: 2

Marc LeBrun, May 06 2004

			a(18) = gcd(18,12) = 6.

Michael De Vlieger, Table of n, a(n) for n = 2..10000

Cf. A094457 A094522.

f[n_] := Block[{g, h}, g[x_] := Flatten[Table[#1, {#2}] & @@@ FactorInteger@ x]; h[x_] := If[x == 2, 1, NextPrime[x, -1]]; If[n == 1, 1, Max[Times @@ MapAt[h, g[n], #] & /@ Range[Length@ g[n]]]]]; Table[GCD[i, f@ i], {i, 2, 74}] (* Michael De Vlieger, Jan 31 2015 *)

A300273 Sorted list of Heinz numbers of collapsible integer partitions.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 36, 37, 40, 41, 43, 47, 48, 49, 53, 59, 61, 63, 64, 67, 71, 73, 79, 81, 83, 84, 89, 97, 101, 103, 107, 108, 109, 112, 113, 121, 125, 127, 128, 131, 137, 139, 144, 149, 151, 157, 163, 167, 169

Offset: 1

Gus Wiseman, Mar 01 2018

nonn

A positive integer is in this sequence iff it can be reduced to a prime number by a sequence of collapses, where a collapse is a replacement of prime(n)^k with prime(n*k) in a number's prime factorization (k > 1).

			A sequence of collapses is 84 -> 63 -> 49 -> 19 corresponding to the sequence of partitions (4211) -> (422) -> (44) -> (8). Hence 84 is in the sequence.

Cf. A000041, A001222, A002577, A005117, A018818, A056239, A094457, A112798, A145515, A215366, A275870, A289078, A289079, A291441, A296150, A299202.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
repcaps[q_]:=Union[{q},If[SquareFreeQ[q],{},Union@@repcaps/@Union[Times[q/#,Prime[Plus@@primeMS[#]]]&/@Select[Rest[Divisors[q]],!PrimeQ[#]&&PrimePowerQ[#]&]]]];
Select[Range[200],MemberQ[repcaps[#],_?PrimeQ]&]

A300272 Sorted list of Heinz numbers of odd partitions.

Original entry on oeis.org

2, 5, 8, 11, 17, 20, 23, 31, 32, 41, 44, 47, 50, 59, 67, 68, 73, 80, 83, 92, 97, 103, 109, 110, 124, 125, 127, 128, 137, 149, 157, 164, 167, 170, 176, 179, 188, 191, 197, 200, 211, 227, 230, 233, 236, 241, 242, 257, 268, 269, 272, 275, 277, 283, 292, 307, 310

Offset: 1

Gus Wiseman, Mar 01 2018

nonn

An odd partition is an integer partition of an odd number into an odd number of parts, all of which are odd.

Any product of three members of this sequence is also in the sequence.

			Sequence of odd partitions begins: (1), (3), (111), (5), (7), (311), (9), (11), (11111), (13), (511), (15), (331), (17), (19), (711), (21), (31111), (23), (911), (25), (27), (29), (531), (1111), (333), (31), (1111111).

Cf. A000009, A031368, A066208, A078408, A094457, A215366, A299757, A300063.

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

A064802 a(n) = Min { m > n | prime factorizations of m and n differ in one factor only}, a(1) = 1.

Original entry on oeis.org

1, 3, 5, 6, 7, 9, 11, 12, 15, 14, 13, 18, 17, 21, 21, 24, 19, 27, 23, 28, 33, 26, 29, 36, 35, 34, 45, 42, 31, 42, 37, 48, 39, 38, 49, 54, 41, 46, 51, 56, 43, 63, 47, 52, 63, 58, 53, 72, 77, 70, 57, 68, 59, 81, 65, 84, 69, 62, 61, 84, 67, 74, 99, 96, 85, 78, 71, 76, 87, 98, 73

Offset: 1

Reinhard Zumkeller, Oct 21 2001

a(A000040(k)) = A000040(k + 1).
A094457 gives next smaller comparable number, replacing the prime factor 2 with 1. - Michael De Vlieger, Jan 31 2015
From Peter Munn, Oct 13 2023: (Start)
For n > 1, a(n) is the smallest number m > n in the factorization neighborhood of n given by A127185(m, n) <= 2.
Usually, the minimum m is achieved by replacing the largest prime factor with the next prime. So through the first 60 terms about 1 term in 5 differs from the corresponding term of A253550, but this proportion drops to 611 of the first 10000 terms. Nevertheless, I see reasons (deriving from the distribution of the lengths of prime gaps) to doubt that the asymptotic density of {n : a(n) <> A253550(n)} is less than 611/10000.
(End)

			n = 20 = 2 * 2 * 5: as 2 * 3 * 5 > 2 * 2 * 7 = 28 we have a(20) = 28.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000040, A094457, A127185, A253550.

f[n_] := Block[{g}, g[x_] := Flatten[Table[#1, {#2}] & @@@ FactorInteger@ x]; If[n == 1, 1, Min[Times @@ MapAt[NextPrime, g[n], #] & /@ Range[Length@ g[n]]]]]; Array[f, 71] (* Michael De Vlieger, Jan 31 2015 *)

A300271 Smallest Heinz number of a partition obtained from y by removing one square from its Young diagram, where y is the integer partition with Heinz number n > 1.

Original entry on oeis.org

1, 2, 2, 3, 3, 5, 4, 6, 5, 7, 6, 11, 7, 9, 8, 13, 9, 17, 10, 14, 11, 19, 12, 15, 13, 18, 14, 23, 15, 29, 16, 21, 17, 21, 18, 31, 19, 26, 20, 37, 21, 41, 22, 27, 23, 43, 24, 35, 25, 34, 26, 47, 27, 33, 28, 38, 29, 53, 30, 59, 31, 42, 32, 39, 33, 61, 34, 46

Offset: 2

Gus Wiseman, Mar 01 2018

nonn

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

Cf. A000041, A000085, A001221, A005117, A056239, A094457, A112798, A122111, A153452, A215366, A238690, A296150, A299925.

dip[n_]:=Min@@Table[n/q*If[q===2,1,NextPrime[q,-1]],{q,Select[Divisors[n],PrimeQ]}];
Table[dip[n],{n,2,50}]

cp's OEIS Frontend

A094521 a(n) = gcd(n, A094457(n)).

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A300273 Sorted list of Heinz numbers of collapsible integer partitions.

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A300272 Sorted list of Heinz numbers of odd partitions.

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A064802 a(n) = Min { m > n | prime factorizations of m and n differ in one factor only}, a(1) = 1.

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A300271 Smallest Heinz number of a partition obtained from y by removing one square from its Young diagram, where y is the integer partition with Heinz number n > 1.

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Mathematica