A346698 -id:A346698 - cp's OEIS frontend

A346697 Sum of the odd-indexed parts (odd bisection) of the multiset of prime indices of n.

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

Offset: 1

Gus Wiseman, Aug 01 2021

nonn

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.

			The prime indices of 1100 are {1,1,3,3,5}, so a(1100) = 1 + 3 + 5 = 9.
The prime indices of 2100 are {1,1,2,3,3,4}, so a(2100) = 1 + 2 + 3 = 6.

The version for standard compositions is A209281(n+1) (even: A346633).
Subtracting the even version gives A316524 (reverse: A344616).
The even version is A346698.
The reverse version is A346699.
The even reverse version is A346700.
A000120 and A080791 count binary digits 1 and 0, with difference A145037.
A000302 counts compositions with odd alternating sum, ranked by A053738.
A001414 adds up prime factors, row sums of A027746.
A029837 adds up parts of standard compositions (alternating: A124754).
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, A341446, A344614, A344617, A344653, A344654, A345957, A345958, A345959.

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

a(n) = A056239(n) - A346698(n).
a(n) = A316524(n) + A346698(n).
a(n odd omega) = A346699(n).
a(n even omega) = A346700(n).
A344616(n) = A346699(n) - A346700(n).

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).

A365067 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n whose odd parts sum to k, for k ranging from mod(n,2) to n in steps of 2.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 3, 3, 2, 2, 4, 3, 4, 3, 5, 5, 3, 4, 4, 6, 5, 6, 6, 5, 8, 7, 5, 6, 8, 6, 10, 7, 10, 9, 10, 8, 12, 11, 7, 10, 12, 12, 10, 15, 11, 14, 15, 15, 16, 12, 18, 15, 11, 14, 20, 18, 20, 15, 22, 15, 22, 21, 25, 24, 24, 18, 27

Offset: 0

Gus Wiseman, Oct 16 2023

The version for all k = 0..n is A113685 (including zeros).

			Triangle begins:
   1
   1
   1  1
   1  2
   2  1  2
   2  2  3
   3  2  2  4
   3  4  3  5
   5  3  4  4  6
   5  6  6  5  8
   7  5  6  8  6 10
   7 10  9 10  8 12
  11  7 10 12 12 10 15
  11 14 15 15 16 12 18
  15 11 14 20 18 20 15 22
  15 22 21 25 24 24 18 27
Row n = 8 counts the following partitions:
  (8)     (611)    (431)     (521)      (71)
  (62)    (4211)   (41111)   (332)      (53)
  (44)    (22211)  (3221)    (32111)    (5111)
  (422)            (221111)  (2111111)  (3311)
  (2222)                                (311111)
                                        (11111111)
Row n = 9 counts the following partitions:
  (81)     (63)      (54)       (72)        (9)
  (621)    (6111)    (522)      (5211)      (711)
  (441)    (432)     (4311)     (3321)      (531)
  (4221)   (42111)   (411111)   (321111)    (51111)
  (22221)  (3222)    (32211)    (21111111)  (333)
           (222111)  (2211111)              (33111)
                                            (3111111)
                                            (111111111)

Row sums are A000041.
The version including all k is A113685, even version A113686.
Column k = 1 is A119620.
The even version and the reverse version are both A174713.
For odd-indexed instead of odd parts we have A346697, even version A346698.
The corresponding rank statistic is A366528, even version A366531.
A000009 counts partitions into odd parts, ranks A066208.
A086543 counts partitions with odd parts, ranks A366322.
A239261 counts partitions with (sum of odd parts) = (sum of even parts).
Cf. A035363, A045931, A053253, A066967, A130780, A171966, A241638, A268335, A325698, A366533.

Table[Length[Select[IntegerPartitions[n], Total[Select[#,OddQ]]==k&]],{n,0,15},{k,Mod[n,2],n,2}]

T(n,k) = A000009(k) * A000041((n-k)/2).

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).

A366531 Sum of even prime indices of n.

Original entry on oeis.org

0, 0, 2, 0, 0, 2, 4, 0, 4, 0, 0, 2, 6, 4, 2, 0, 0, 4, 8, 0, 6, 0, 0, 2, 0, 6, 6, 4, 10, 2, 0, 0, 2, 0, 4, 4, 12, 8, 8, 0, 0, 6, 14, 0, 4, 0, 0, 2, 8, 0, 2, 6, 16, 6, 0, 4, 10, 10, 0, 2, 18, 0, 8, 0, 6, 2, 0, 0, 2, 4, 20, 4, 0, 12, 2, 8, 4, 8, 22, 0, 8, 0, 0, 6

Offset: 1

Gus Wiseman, Oct 22 2023

nonn

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.

			The prime indices of 198 are {1,2,2,5}, so a(198) = 2+2 = 4.

Zeros are A066208, counted by A000009.
The triangle for the odd version is A113685, without zeros A365067.
The triangle for this statistic is A113686, without zeros A174713.
The odd version is A366528.
The halved version is A366533.
A066207 lists numbers with all even prime indices, counted by A035363.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A239261 counts partitions with sum of odd parts = sum of even parts.
A257991 counts odd prime indices, even A257992.
A346697 adds up odd-indexed prime indices, even-indexed A346698.
A366322 lists numbers with not all prime indices even, counted by A086543.
Cf. A000720, A055396, A055922, A061395, A162641, A171966, A258117, A325698, A325700, A352140, A352141.

Table[Total[Cases[FactorInteger[n], {p_?(EvenQ@*PrimePi),k_}:>PrimePi[p]*k]],{n,100}]

a(n) = A056239(n) - A366528(n).

A366533 Sum of even prime indices of n divided by 2.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 2, 0, 2, 0, 0, 1, 3, 2, 1, 0, 0, 2, 4, 0, 3, 0, 0, 1, 0, 3, 3, 2, 5, 1, 0, 0, 1, 0, 2, 2, 6, 4, 4, 0, 0, 3, 7, 0, 2, 0, 0, 1, 4, 0, 1, 3, 8, 3, 0, 2, 5, 5, 0, 1, 9, 0, 4, 0, 3, 1, 0, 0, 1, 2, 10, 2, 0, 6, 1, 4, 2, 4, 11, 0, 4, 0, 0, 3, 0, 7

Offset: 1

Gus Wiseman, Oct 23 2023

nonn

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.

			The prime indices of 198 are {1,2,2,5}, so a(198) = (2+2)/2 = 2.

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

Zeros are A066208, counted by A000009.
The triangle for this statistic (without zeros) is A174713.
The un-halved odd version is A366528.
The un-halved version is A366531.
A066207 lists numbers with all even prime indices, counted by A035363.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A113685 counts partitions by sum of odd parts, even version A113686.
A239261 counts partitions with (sum of odd parts) = (sum of even parts).
A257991 counts odd prime indices, even A257992.
A346697 adds up odd-indexed prime indices, even-indexed A346698.
A365067 counts partitions by sum of odd parts (without zeros).
A366322 lists numbers with not all prime indices even, counted by A086543.
Cf. A000720, A055396, A055922, A061395, A162641, A171966, A258117, A325698, A325700, A352140, A352141.

f:= proc(n) local F,t;
  F:= map(t -> [numtheory:-Pi(t[1]),t[2]], ifactors(n)[2]);
  add(`if`(t[1]::even, t[1]*t[2]/2, 0), t=F)
end proc:
map(f, [$1..100]); # Robert Israel, Nov 22 2023

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[Select[prix[n],EvenQ]]/2,{n,100}]

a(n) = A366531(n)/2.

A346633 Sum of even-indexed parts (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, 3, 2, 1, 2, 0, 1, 2, 1, 3, 2, 1, 2, 4, 3, 2, 3, 1, 2, 3, 2, 0, 1, 2, 1, 3, 2, 1, 2, 4, 3, 2, 3, 1, 2, 3, 2, 5, 4, 3, 4, 2, 3, 4, 3, 1, 2, 3, 2, 4, 3, 2, 3, 0, 1, 2, 1, 3, 2, 1, 2, 4, 3, 2, 3, 1, 2, 3, 2, 5, 4, 3, 4, 2, 3, 4

Offset: 0

Gus Wiseman, Aug 01 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. This gives a bijective correspondence between nonnegative integers and integer compositions.

			Composition number 741 in standard order is (2,1,1,3,2,1), so a(741) = 1 + 3 + 1 = 5.

Including odd-indexed parts gives A029837.
Subtracting from the odd version gives A124754.
Positions of zeros are A131577.
The odd-indexed version is A209281(n+1).
The version for prime indices is A346698 (reverse: A346700).
A000120 and A080791 count binary digits 1 and 0, with difference A145037.
A011782 counts compositions.
A056239 adds up prime indices, row sums of A112798.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A345197 counts compositions by sum, length, and alternating sum.
Cf. A000070, A000097, A008549, A025047, A088218, A344606, A344653, A346697.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Total[Last/@Partition[Append[stc[n],0],2]],{n,0,100}]

a(n) = (A029837(n) - A124754(n))/2.
a(n) = A029837(n) - A209281(n + 1).
a(n) = A124754(n) + A209281(n + 1).

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).

A329888 a(n) = A329900(A329602(n)); Heinz number of the even bisection (even-indexed parts) of the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 4, 1, 3, 1, 2, 3, 2, 1, 4, 5, 2, 3, 2, 1, 3, 1, 4, 3, 2, 5, 6, 1, 2, 3, 4, 1, 3, 1, 2, 3, 2, 1, 4, 7, 5, 3, 2, 1, 6, 5, 4, 3, 2, 1, 6, 1, 2, 3, 8, 5, 3, 1, 2, 3, 5, 1, 6, 1, 2, 5, 2, 7, 3, 1, 4, 9, 2, 1, 6, 5, 2, 3, 4, 1, 6, 7, 2, 3, 2, 5, 8, 1, 7, 3, 10, 1, 3, 1, 4, 5

Offset: 1

Antti Karttunen, Dec 22 2019

nonn

From Gus Wiseman, Aug 05 2021 and Antti Karttunen, Oct 13 2021: (Start)
Also the product of primes at even positions in the weakly decreasing list (with multiplicity) of prime factors of n. For example, the prime factors of 108 are (3,3,3,2,2), with even bisection (3,2), with product 6, so a(108) = 6.
Proof: A108951(n) gives a number with the same largest prime factor (A006530) and its exponent (A071178) as in n, and with each smaller prime p = 2, 3, 5, 7, ... < A006530(n) having as its exponent the partial sum of the exponents of all prime factors >= p present in n (with primes not present in n having the exponent 0). Then applying A000188 replaces each such "partial sum exponent" k with floor(k/2). Finally, A319626 replaces those halved exponents with their first differences (here the exponent of the largest prime present stays intact, because the next larger prime's exponent is 0 in n). It should be easy to see that if prime q is not present in n (i.e., does not divide it), then neither it is present in a(n). Moreover, if the partial sum exponent of q is odd and only one larger than the partial sum exponent of the next larger prime factor of n, then q will not be present in a(n), while in all other cases q is present in a(n). See also the last example.
(End)

			From _Gus Wiseman_, Aug 15 2021: (Start)
The list of all numbers with image 12 and their corresponding prime factors begins:
  144: (3,3,2,2,2,2)
  216: (3,3,3,2,2,2)
  240: (5,3,2,2,2,2)
  288: (3,3,2,2,2,2,2)
  336: (7,3,2,2,2,2)
  360: (5,3,3,2,2,2)
(End)
The positions from the left are indexed as 1, 2, 3, ..., etc, so e.g., for 240 we pick the second, the fourth and the sixth prime factor, 3, 2 and 2, to obtain a(240) = 3*2*2 = 12. For 288, we similarly pick the second (3), the fourth (2) and the sixth (2) to obtain a(288) = 3*2*2 = 12. - _Antti Karttunen_, Oct 13 2021
Consider n = 11945934 = 2*3*3*3*7*11*13*13*17. Its primorial inflation is A108951(11945934) = 96478365991115908800000 = 2^9 * 3^8 * 5^5 * 7^5 * 11^4 * 13^3 * 17^1. Applying A000188 to this halves each exponent (floored down if the exponent is odd), leaving the factors 2^4 * 3^4 * 5^2 * 7^2 * 11^2 * 13^1 = 2497294800. Then applying A319626 to this number retains the largest prime factor (and its exponent), and subtracts from the exponent of each of the rest of primes the exponent of the next larger prime, so from 2^4 * 3^4 * 5^2 * 7^2 * 11^2 * 13^1 we get 2^(4-4) * 3^(4-2) * 5^(2-2) * 7^(2-2) * 11^(2-1) * 13^1 = 3^2 * 11^1 * 13^1 = 1287 = a(11945934), which is obtained also by selecting every second prime from the list [17, 13, 13, 11, 7, 3, 3, 3, 2] and taking their product. - _Antti Karttunen_, Oct 15 2021

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to Heinz numbers
Index entries for sequences related to primorial numbers

Cf. A000188, A108951, A319626, A329602, A329900.
A left inverse of A000290.
Positions of 1's are A008578.
Positions of primes are A168645.
The sum of prime indices of a(n) is A346700(n).
The odd version is A346701.
The odd non-reverse version is A346703.
The non-reverse version is A346704.
The version for standard compositions is A346705, odd A346702.
A001221 counts distinct prime factors.
A001222 counts all prime factors.
A001414 adds up prime factors, row sums of A027746.
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).
A346633 adds up the even bisection of standard compositions.
A346698 adds up the even bisection of prime indices.
Cf. A000097, A035363, A102750, A236913, A253553, A344606, A344617, A344653, A345957, A345958, A345959.

Table[Times@@Last/@Partition[Reverse[Flatten[Apply[ConstantArray,FactorInteger[n],{1}]]],2],{n,100}] (* Gus Wiseman, Oct 13 2021 *)

PARI
```
A329888(n) = A329900(A329602(n));
```

A329888(n) = if(1==n,n,my(f=factor(n),m=1,p=0); forstep(k=#f~,1,-1,while(f[k,2], m *= f[k,1]^(p%2); f[k,2]--; p++)); (m)); \\ (After Wiseman's new interpretation) - Antti Karttunen, Sep 21 2021

a(n) = A329900(A329602(n)) = A329900(A000188(A108951(n))).
A108951(a(n)) = A329602(n).
a(n^2) = n for all n >= 1.
a(n) * A346701(n) = n. - Gus Wiseman, Aug 07 2021
A056239(a(n)) = A346700(n). - Gus Wiseman, Aug 07 2021
Antti Karttunen, Sep 21 2021
From Antti Karttunen, Oct 13 2021: (Start)
a(n) = A319626(A329602(n)) = A319626(A000188(A108951(n))).
For all x in A102750, a(x) = a(A253553(x)). (End)

Name amended with Gus Wiseman's new interpretation - Antti Karttunen, Oct 13 2021

cp's OEIS Frontend

A346697 Sum of the odd-indexed parts (odd bisection) of the multiset of prime indices of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A365067 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n whose odd parts sum to k, for k ranging from mod(n,2) to n in steps of 2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A366531 Sum of even prime indices of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A366533 Sum of even prime indices of n divided by 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A346633 Sum of even-indexed parts (even bisection) of the n-th composition in standard order.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links