A329888 -id:A329888 - cp's OEIS frontend

A329900 Primorial deflation of n: starting from x = n, repeatedly divide x by the largest primorial A002110(k) that divides it, until x is an odd number. Then a(n) = Product prime(k_i), for primorial indices k_1 >= k_2 >= ..., encountered in the process.

1, 2, 1, 4, 1, 3, 1, 8, 1, 2, 1, 6, 1, 2, 1, 16, 1, 3, 1, 4, 1, 2, 1, 12, 1, 2, 1, 4, 1, 5, 1, 32, 1, 2, 1, 9, 1, 2, 1, 8, 1, 3, 1, 4, 1, 2, 1, 24, 1, 2, 1, 4, 1, 3, 1, 8, 1, 2, 1, 10, 1, 2, 1, 64, 1, 3, 1, 4, 1, 2, 1, 18, 1, 2, 1, 4, 1, 3, 1, 16, 1, 2, 1, 6, 1, 2, 1, 8, 1, 5, 1, 4, 1, 2, 1, 48, 1, 2, 1, 4, 1, 3, 1, 8, 1

Offset: 1

Antti Karttunen, Dec 22 2019

nonn

When applied to arbitrary n, the "primorial deflation" (term coined by Matthew Vandermast in A181815) induces the splitting of n to two factors A328478(n)*A328479(n) = n, where we call A328478(n) the non-deflatable component of n (which is essentially discarded), while A328479(n) is the deflatable component. Only if n is in A025487, then the entire n is deflatable, i.e., A328478(n) = 1 and A328479(n) = n.

According to Daniel Suteu, also the ratio (A319626(n) / A319627(n)) can be viewed as a "primorial deflation". That definition coincides with this one when restricted to terms of A025487, as for all k in A025487, A319626(k) = a(k), and A319627(k) = 1. - Antti Karttunen, Dec 29 2019

Antti Karttunen, Table of n, a(n) for n = 1..65537

A left inverse of A108951. Coincides with A319626 on A025487.

Cf. A002110, A002182, A111701, A181815, A181817, A181819, A181821, A276084, A304886, A319626, A319627, A328478, A328479, A329889, A329902, A330685, A330686, A330689.

Array[If[OddQ@ #, 1, Times @@ Prime@ # &@ Rest@ NestWhile[Append[#1, {#3, Drop[#, -LengthWhile[Reverse@ #, # == 0 &]] &[#2 - PadRight[ConstantArray[1, #3], Length@ #2]]}] & @@ {#1, #2, LengthWhile[#2, # > 0 &]} & @@ {#, #[[-1, -1]]} &, {{0, TakeWhile[If[# == 1, {0}, Function[f, ReplacePart[Table[0, {PrimePi[f[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, f]]@ FactorInteger@ #], # > 0 &]}}, And[FreeQ[#[[-1, -1]], 0], Length[#[[-1, -1]] ] != 0] &][[All, 1]] ] &, 105] (* Michael De Vlieger, Dec 28 2019 *)
Array[Times @@ Prime@(TakeWhile[Reap[FixedPointList[Block[{k = 1}, While[Mod[#, Prime@ k] == 0, k++]; Sow[k - 1]; #/Product[Prime@ i, {i, k - 1}]] &, #]][[-1, 1]], # > 0 &]) &, 105] (* Michael De Vlieger, Jan 11 2020 *)

A329900(n) = { my(m=1, pp=1); while(1, forprime(p=2, ,if(n%p, if(2==p, return(m), break), n /= p; pp = p)); m *= pp); (m); };

A111701(n) = forprime(p=2, , if(n%p, return(n), n /= p));
A276084(n) = { for(i=1,oo,if(n%prime(i),return(i-1))); }
A329900(n) = if(n%2,1,prime(A276084(n))*A329900(A111701(n)));

For odd n, a(n) = 1, for even n, a(n) = A000040(A276084(n)) * a(A111701(n)).
For even n, a(n) = A000040(A276084(n)) * a(n/A002110(A276084(n))).
A108951(a(n)) = A328479(n), for n >= 1.
a(A108951(n)) = n, for n >= 1.
a(A328479(n)) = a(n),  for n >= 1.
a(A328478(n)) = 1, for n >= 1.
a(A002110(n)) = A000040(n), for n >= 1.
a(A000142(n)) = A307035(n), for n >= 0.
a(A283477(n)) = A019565(n), for n >= 0.
a(A329886(n)) = A005940(1+n), for n >= 0.
a(A329887(n)) = A163511(n), for n >= 0.
a(A329602(n)) = A329888(n), for n >= 1.
a(A025487(n)) = A181815(n), for n >= 1.
a(A124859(n)) = A181819(n), for n >= 1.
a(A181817(n)) = A025487(n), for n >= 1.
a(A181821(n)) = A122111(n), for n >= 1.
a(A002182(n)) = A329902(n), for n >= 1.
a(A260633(n)) = A329889(n), for n >= 1.
a(A033833(n)) = A330685(n), for n >= 1.
a(A307866(1+n)) = A330686(n), for n >= 1.
a(A330687(n)) = A330689(n), for n >= 1.

A346704 Product of primes at even positions in the weakly increasing list (with multiplicity) of prime factors of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 2, 3, 5, 1, 2, 1, 7, 5, 4, 1, 3, 1, 2, 7, 11, 1, 6, 5, 13, 3, 2, 1, 3, 1, 4, 11, 17, 7, 6, 1, 19, 13, 10, 1, 3, 1, 2, 3, 23, 1, 4, 7, 5, 17, 2, 1, 9, 11, 14, 19, 29, 1, 10, 1, 31, 3, 8, 13, 3, 1, 2, 23, 5, 1, 6, 1, 37, 5, 2, 11, 3, 1, 4, 9

Offset: 1

Gus Wiseman, Aug 08 2021

			The prime factors of 108 are (2,2,3,3,3), with even bisection (2,3), with product 6, so a(108) = 6.
The prime factors of 720 are (2,2,2,2,3,3,5), with even bisection (2,2,3), with product 12, so a(720) = 12.

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

Positions of first appearances are A129597.
Positions of 1's are A008578.
Positions of primes are A168645.
The sum of prime indices of a(n) is A346698(n).
The odd version is A346703 (sum: A346697).
The odd reverse version is A346701 (sum: A346699).
The reverse version appears to be A329888 (sum: A346700).
A001221 counts distinct prime factors.
A001222 counts all prime factors.
A027187 counts partitions of even length, ranked by A028260.
A056239 adds up prime indices, row sums of A112798.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A316524 gives the alternating sum of prime indices (reverse: A344616).
A335433/A335448 rank separable/inseparable partitions.
A344606 counts alternating permutations of prime indices.
A344617 gives the sign of the alternating sum of prime indices.
A346633 adds up the even bisection of standard compositions.
Cf. A026424, A035363, A209281, A236913, A342768, A344653, A345957, A345958, A345960, A345961, A345962.

f:= proc(n) local F,i;
  F:= ifactors(n)[2];
  F:= sort(map(t -> t[1]$t[2],F));
  mul(F[i],i=2..nops(F),2)
end proc:
map(f, [$1..100]); # Robert Israel, Aug 12 2024

Table[Times@@Last/@Partition[Flatten[Apply[ConstantArray,FactorInteger[n],{1}]],2],{n,100}]

a(n) * A346703(n) = n.

A056239(a(n)) = A346698(n).

A346703 Product of primes at odd positions in the weakly increasing list (with multiplicity) of prime factors of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 2, 7, 4, 3, 2, 11, 6, 13, 2, 3, 4, 17, 6, 19, 10, 3, 2, 23, 4, 5, 2, 9, 14, 29, 10, 31, 8, 3, 2, 5, 6, 37, 2, 3, 4, 41, 14, 43, 22, 15, 2, 47, 12, 7, 10, 3, 26, 53, 6, 5, 4, 3, 2, 59, 6, 61, 2, 21, 8, 5, 22, 67, 34, 3, 14, 71, 12, 73, 2, 15, 38

Offset: 1

Gus Wiseman, Aug 08 2021

nonn

			The prime factors of 108 are (2,2,3,3,3), with odd bisection (2,3,3), with product 18, so a(108) = 18.
The prime factors of 720 are (2,2,2,2,3,3,5), with odd bisection (2,2,3,5), with product 60, so a(720) = 60.

Positions of 2's are A001747.
Positions of primes are A037143 (complement: A033942).
The even reverse version appears to be A329888.
Positions of first appearances are A342768.
The sum of prime indices of a(n) is A346697(n), reverse: A346699.
The reverse version is A346701.
The even version is A346704.
A001221 counts distinct prime factors.
A001222 counts all prime factors.
A056239 adds up prime indices, row sums of A112798.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A209281 (shifted) adds up the odd bisection of standard compositions.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A335433/A335448 rank separable/inseparable partitions.
A344606 counts alternating permutations of prime indices.
A344617 gives the sign of the alternating sum of prime indices.
A346633 adds up the even bisection of standard compositions.
A346698 gives the sum of the even bisection of prime indices.
A346700 gives the sum of the even bisection of reversed prime indices.
Cf. A025047, A027187, A027193, A053738, A097805, A106356, A341446, A344653, A345957, A345958, A345959.

Table[Times@@First/@Partition[Append[Flatten[Apply[ConstantArray,FactorInteger[n],{1}]],0],2],{n,100}]

a(n) * A346704(n) = n.

A056239(a(n)) = A346697(n).

A346700 Sum of the even bisection (even-indexed parts) of the integer partition with Heinz number n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 03 2021

nonn

First differs from A334107 at a(64) = 3, A334107(64) = 2.

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The partition with Heinz number 1100 is (5,3,3,1,1), so a(1100) = 3 + 1 = 4.
The partition with Heinz number 2100 is (4,3,3,2,1,1), so a(2100) = 3 + 2 + 1 = 6.

Antti Karttunen, Table of n, a(n) for n = 1..10201
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to Heinz numbers

Sum of prime indices of A329888(n).
Subtracting from the odd version gives A344616 (non-reverse: A316524).
The unreversed version for standard compositions is A346633.
The odd non-reverse version is A346697.
The non-reverse version (multisets instead of partitions) is A346698.
The odd version is A346699.
A001414 adds up prime factors, row sums of A027746.
A056239 adds up prime indices, row sums of A112798.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344606 counts alternating permutations of prime indices.
Cf. A000070, A025047, A124754, A209281, A329888, A341446, A344617, A344653, A345957, A345958, A346701, A346702, A346704.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[Last/@Partition[Reverse[primeMS[n]],2]],{n,100}]

A346700(n) = if(1==n,0,my(f=factor(n),s=0,p=0); forstep(k=#f~,1,-1,while(f[k,2], s += (p%2)*primepi(f[k,1]); f[k,2]--; p++)); (s)); \\ Antti Karttunen, Sep 21 2021

a(n) = A056239(n) - A346699(n).
a(n) = A346699(n) - A344616(n).
a(n even omega) = A346697(n).
a(n odd omega) = A346698(n).
A316524(n) = A346697(n) - A346698(n).
a(n) = A056239(A329888(n)). - Gus Wiseman and Antti Karttunen, Oct 13 2021

Data section extended up to 105 terms by Antti Karttunen, Sep 21 2021

A346699 Sum of the odd bisection (odd-indexed parts) of the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 2, 2, 3, 5, 3, 6, 4, 3, 2, 7, 3, 8, 4, 4, 5, 9, 3, 3, 6, 4, 5, 10, 4, 11, 3, 5, 7, 4, 3, 12, 8, 6, 4, 13, 5, 14, 6, 5, 9, 15, 4, 4, 4, 7, 7, 16, 4, 5, 5, 8, 10, 17, 4, 18, 11, 6, 3, 6, 6, 19, 8, 9, 5, 20, 4, 21, 12, 5, 9, 5, 7, 22, 5, 4

Offset: 1

Gus Wiseman, Aug 03 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The partition with Heinz number 1100 is (5,3,3,1,1), so a(1100) = 5 + 3 + 1 = 9.
The partition with Heinz number 2100 is (4,3,3,2,1,1), so a(2100) = 4 + 3 + 1 = 8.

The version for standard compositions is A209281(n+1) (even: A346633).
Subtracting the even version gives A344616 (non-reverse: A316524).
The even version is A346700.
The non-reverse version (multisets instead of partitions) is A346697.
The even non-reverse version is A346698.
A001414 adds up prime factors, row sums of A027746.
A056239 adds up prime indices, row sums of A112798.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344606 counts alternating permutations of prime indices.
Cf. A000070, A025047, A120452, A124754, A329888, A341446, A344617, A345957, A345958, A346701, A346703.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[First/@Partition[Append[Reverse[primeMS[n]],0],2]],{n,100}]

a(n) = A056239(A346701(n)).
a(n) = A056239(n) - A346700(n).
a(n) = A344616(n) + A346700(n).
a(n odd omega) = A346697(n).
a(n even omega) = A346698(n).
A316524(n) = A346697(n) - A346698(n).

A346701 Heinz number of the odd bisection (odd-indexed parts) of the integer partition with Heinz number n.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 4, 3, 5, 11, 6, 13, 7, 5, 4, 17, 6, 19, 10, 7, 11, 23, 6, 5, 13, 9, 14, 29, 10, 31, 8, 11, 17, 7, 6, 37, 19, 13, 10, 41, 14, 43, 22, 15, 23, 47, 12, 7, 10, 17, 26, 53, 9, 11, 14, 19, 29, 59, 10, 61, 31, 21, 8, 13, 22, 67, 34, 23, 14, 71

Offset: 1

Gus Wiseman, Aug 03 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The partition (2,2,2,1,1) has Heinz number 108 and odd bisection (2,2,1) with Heinz number 18, so a(108) = 18.
The partitions (3,2,2,1,1), (3,2,2,2,1), (3,3,2,1,1) have Heinz numbers 180, 270, 300 and all have odd bisection (3,2,1) with Heinz number 30, so a(180) = a(270) = a(300) = 30.

Positions of last appearances are A000290 without the first term 0.
Positions of primes are A037143 (complement: A033942).
The even version is A329888.
Positions of first appearances are A342768.
The sum of prime indices of a(n) is A346699(n), non-reverse: A346697.
The non-reverse version is A346703.
The even non-reverse version is A346704.
A001221 counts distinct prime factors.
A001222 counts all prime factors.
A056239 adds up prime indices, row sums of A112798.
A103919 counts partitions by sum and alternating sum, reverse A344612.
A209281 (shifted) adds up the odd bisection of standard compositions.
A316524 gives the alternating sum of prime indices, reverse A344616.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344606 counts alternating permutations of prime indices.
A344617 gives the sign of the alternating sum of prime indices.
A346700 gives the sum of the even bisection of reversed prime indices.
Cf. A025047, A027187, A027193, A053738, A106356, A277103, A341446, A344653, A345957, A345958, A345959, A346698, A346702.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@Prime/@First/@Partition[Append[Reverse[primeMS[n]],0],2],{n,100}]

a(n) * A329888(n) = n.

A056239(a(n)) = A346699(n).

A129597 Central diagonal of array A129595.

Original entry on oeis.org

1, 4, 6, 16, 10, 24, 14, 64, 54, 40, 22, 96, 26, 56, 90, 256, 34, 216, 38, 160, 126, 88, 46, 384, 250, 104, 486, 224, 58, 360, 62, 1024, 198, 136, 350, 864, 74, 152, 234, 640, 82, 504, 86, 352, 810, 184, 94, 1536, 686, 1000, 306, 416, 106, 1944, 550, 896, 342

Offset: 1

Antti Karttunen, May 01 2007, based on Marc LeBrun's Jan 11 2006 message on SeqFan mailing list

nonn

These are the positions of first appearances of each positive integer in A346704. - Gus Wiseman, Oct 16 2021

Antti Karttunen, Table of n, a(n) for n = 1..10000

a(n) = A129595(n,n).
The sum of prime indices of a(n) is 2*A056239(n) - A061395(n) + 1 for n > 1.
The version for odd indices is A342768(n) = a(n)/2 for n > 1.
Except the first term, the sorted version is 2*A346635.
These are the positions of first appearances in A346704.
A001221 counts distinct prime factors.
A001222 counts prime factors with multiplicity.
A027187 counts partitions of even length, ranked by A028260.
A346633 adds up the even bisection of standard compositions (odd: A209281).
A346698 adds up the even bisection of prime indices (reverse: A346699).
Cf. A000290, A006530, A037143, A329888, A344606, A345957, A346697, A346700, A346701.

Table[If[n==1,1,2*n^2/FactorInteger[n][[-1,1]]],{n,100}] (* Gus Wiseman, Aug 10 2021 *)

A129597(n) = if(1==n, n, my(f=factor(n)); (2*n*n)/f[#f~, 1]); \\ Antti Karttunen, Oct 16 2021

From Gus Wiseman, Aug 10 2021: (Start)
For n > 1, A001221(a(n)) = A099812(n).
If g = A006530(n) is the greatest prime factor of n > 1, then a(n) = 2n^2/g.
a(n) = A100484(A000720(n)) = 2n iff n is prime.
a(n > 1) = 2*A342768(n).
(End)

A346705 The a(n)-th composition in standard order is the even bisection of the n-th composition in standard order.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 2, 1, 0, 1, 2, 1, 4, 2, 1, 3, 0, 1, 2, 1, 4, 2, 1, 3, 8, 4, 2, 5, 1, 3, 6, 3, 0, 1, 2, 1, 4, 2, 1, 3, 8, 4, 2, 5, 1, 3, 6, 3, 16, 8, 4, 9, 2, 5, 10, 5, 1, 3, 6, 3, 12, 6, 3, 7, 0, 1, 2, 1, 4, 2, 1, 3, 8, 4, 2, 5, 1, 3, 6, 3, 16, 8, 4, 9, 2, 5

Offset: 0

Gus Wiseman, Aug 19 2021

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again.

a(n) is the row number in A066099 of the even bisection (even-indexed terms) of the n-th row of A066099.

			Composition number 741 in standard order is (2,1,1,3,2,1), with even bisection (1,3,1), which is composition number 25 in standard order, so a(741) = 25.

Length of the a(n)-th standard composition is A000120(n)/2 rounded down.
Positions of first appearances appear to be A088698, sorted: A277335.
The version for reversed prime indices appears to be A329888, sums A346700.
Sum of the a(n)-th standard composition is A346633.
An unordered reverse version for odd bisection is A346701, sums A346699.
The version for odd bisection is A346702, sums A209281(n+1).
An unordered version for odd bisection is A346703, sums A346697.
An unordered version is A346704, sums A346698.
A011782 counts compositions.
A029837 gives length of binary expansion, or sometimes A070939.
A066099 lists compositions in standard order.
A097805 counts compositions by alternating sum.
Cf. A000302, A025047, A088218, A124754, A124767, A228351, A290258, A290259, A344618, A345197.

Table[Total[2^Accumulate[Reverse[Last/@Partition[ Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse,2]]]]/2,{n,0,100}]

A029837(a(n)) = A346633(n).

cp's OEIS Frontend

A329900 Primorial deflation of n: starting from x = n, repeatedly divide x by the largest primorial A002110(k) that divides it, until x is an odd number. Then a(n) = Product prime(k_i), for primorial indices k_1 >= k_2 >= ..., encountered in the process.

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A346704 Product of primes at even positions in the weakly increasing list (with multiplicity) of prime factors of n.

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A346703 Product of primes at odd positions in the weakly increasing list (with multiplicity) of prime factors of n.

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A346700 Sum of the even bisection (even-indexed parts) of the integer partition with Heinz number n.

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A346699 Sum of the odd bisection (odd-indexed parts) of the integer partition with Heinz number n.

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A346701 Heinz number of the odd bisection (odd-indexed parts) of the integer partition with Heinz number n.

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A129597 Central diagonal of array A129595.

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A346705 The a(n)-th composition in standard order is the even bisection of the n-th composition in standard order.

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