A325033 -id:A325033 - cp's OEIS frontend

A379681 Sum plus product of the multiset of prime indices of n.

1, 2, 4, 3, 6, 5, 8, 4, 8, 7, 10, 6, 12, 9, 11, 5, 14, 9, 16, 8, 14, 11, 18, 7, 15, 13, 14, 10, 20, 12, 22, 6, 17, 15, 19, 10, 24, 17, 20, 9, 26, 15, 28, 12, 19, 19, 30, 8, 24, 16, 23, 14, 32, 15, 23, 11, 26, 21, 34, 13, 36, 23, 24, 7, 27, 18, 38, 16, 29, 20

Offset: 1

Gus Wiseman, Jan 05 2025

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.

Includes all positive integers.

For prime factors instead of indices we have A075254, subtracted A075255.
Positions of first appearances are A379682.
For minus instead of plus we have A325036, which takes the following values:
- zero: A301987, counted by A001055
- negative: A325037, counted by A114324
- positive: A325038, counted by A096276 shifted right
- negative one: A325041, counted by A028422
- one: A325042, counted by A001055 shifted right
- nonnegative: A325044, counted by A096276
- nonpositive: A379721, counted by A319005
A000040 lists the primes, differences A001223.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
Cf. A000720, A003963, A178503, A175508, A319000, A325032, A325033, A325034, A325035, A325040.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Plus@@prix[n]+Times@@prix[n],{n,30}]

a(n) = A056239(n) + A003963(n).

The last position of k is 2^(k-1).

A380956 Position of first appearance of n in A380955 (sum of prime indices minus sum of distinct prime indices).

Original entry on oeis.org

1, 4, 8, 16, 27, 64, 81, 256, 243, 529, 729, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 16129, 17161, 18769, 19321, 22201, 22801, 24649, 26569, 27889, 29929, 32041, 32761, 36481

Offset: 0

Gus Wiseman, Feb 12 2025

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.

Also the position of first appearance of n in A374248.

			The terms together with their prime indices begin:
     1: {}
     4: {1,1}
     8: {1,1,1}
    16: {1,1,1,1}
    27: {2,2,2}
    64: {1,1,1,1,1,1}
    81: {2,2,2,2}
   256: {1,1,1,1,1,1,1,1}
   243: {2,2,2,2,2}
   529: {9,9}
   729: {2,2,2,2,2,2}
   961: {11,11}
  1369: {12,12}
  1681: {13,13}
  1849: {14,14}
  2209: {15,15}

For length instead of sum we have A151821.
For factors instead of indices we have A280286 (sorted A381075), firsts of A280292.
Counting partitions by this statistic gives A364916.
Positions of first appearances in A380955.
The sorted version is A380957.
For product instead of sum we have firsts of A380986.
A multiplicative version is A380987 (sorted A380988), firsts of A290106.
For prime multiplicities instead of prime indices we have A380989, firsts of A380958.
A000040 lists the primes, differences A001223.
A005117 lists squarefree numbers, complement A013929.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, length A001222.
A304038 lists distinct prime indices, sum A066328, length A001221.
Cf. A000720, A046660, A071625, A075255, A116861, A136565, A156061, A178503, A175508, A325033, A366528, A366749, A374248.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
q=Table[Total[prix[n]]-Total[Union[prix[n]]],{n,1000}];
Table[Position[q,k][[1,1]],{k,0,mnrm[q+1]-1}]

After a(12) = 961, this appears to converge to prime(n)^2.

A380957 Sorted positions of first appearances in A380955 (sum of prime indices minus sum of distinct prime indices).

Original entry on oeis.org

1, 4, 8, 16, 27, 64, 81, 243, 256, 529, 729, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 16129, 17161, 18769, 19321, 22201, 22801, 24649, 26569, 27889, 29929, 32041, 32761, 36481

Offset: 1

Gus Wiseman, Feb 13 2025

nonn

Also appears to be sorted firsts of A374248.

For length instead of sum we have A151821.
Counting partitions by this statistic (sum minus sum of distinct parts) gives A364916.
Sorted positions of first appearances in A380955.
The unsorted version is A380956.
For product instead of sum we have sorted firsts of A380986.
The multiplicative version is A380988, unsorted A380987, firsts of A290106.
For prime multiplicities instead of prime indices we have A380989, firsts of A380958.
For factors instead of indices we have A381075, see A280286, A280292.
A000040 lists the primes, differences A001223.
A005117 lists squarefree numbers, complement A013929.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
Cf. A000720, A046660, A071625, A075255, A116861, A136565, A156061, A178503, A175508, A325033, A366528, A366749, A374248.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
q=Table[Total[prix[n]]-Total[Union[prix[n]]],{n,1000}];
Select[Range[Length[q]],FreeQ[Take[q,#-1],q[[#]]]&]

A325035 Product of sums of the multisets of prime indices of each prime index of 2 * n + 1.

Original entry on oeis.org

1, 1, 2, 2, 1, 3, 3, 2, 4, 3, 2, 4, 4, 1, 4, 5, 3, 4, 4, 3, 6, 5, 2, 5, 4, 4, 4, 6, 3, 7, 5, 2, 6, 8, 4, 5, 6, 4, 6, 6, 1, 9, 8, 4, 5, 6, 5, 6, 6, 3, 7, 6, 4, 6, 10, 4, 6, 8, 3, 8, 9, 6, 8, 11, 5, 5, 6, 2, 7, 8, 5, 9, 8, 4, 7, 6, 4, 10, 12, 4, 8, 9, 6, 8, 9, 3

Offset: 0

Gus Wiseman, Mar 25 2019

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.

			91 has prime indices {4,6} with prime indices {{1,1},{1,2}} with sums {2,3} with product a(45) = 6.

Cf. A000720, A001055, A001222, A003963, A056239, A112798, A302242, A318949.

Cf. A324926, A325031, A325032, A325033, A325034.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@Plus@@@primeMS/@primeMS[n],{n,1,200,2}]

Fully multiplicative with a(prime(n)) = A056239(n), restricted to odd n.

A379682 Least number whose prime indices have sum + product = n.

Original entry on oeis.org

1, 2, 4, 3, 6, 5, 10, 7, 14, 11, 15, 13, 26, 17, 25, 19, 33, 23, 35, 29, 58, 31, 51, 37, 74, 41, 65, 43, 69, 47, 85, 53, 105, 59, 93, 61, 122, 67, 115, 71, 123, 73, 145, 79, 158, 83, 141, 89, 161, 97, 185, 101, 177, 103, 205, 107, 214, 109, 201, 113, 226, 127

Offset: 1

Gus Wiseman, Jan 05 2025

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 positions of 11 in A379681 are: 15, 22, 56, 72, 160, 384, 1024, so a(11) = 15.

Position of first appearance of n in A379681.
The subtraction A325036 takes the following values:
- zero: A301987, counted by A001055
- negative: A325037, counted by A114324
- positive: A325038, counted by A096276 shifted right
- negative one: A325041, counted by A028422
- one: A325042, counted by A001055 shifted right
- nonnegative: A325044, counted by A096276
- nonpositive: A379721, counted by A319005
A000040 lists the primes, differences A001223.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
Cf. A000720, A003963, A075254, A178503, A175508, A319000, A325032, A325033, A325034, A325035, A325040.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sp=Table[Plus@@prix[n]+Times@@prix[n],{n,1000}];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
Table[Position[sp,n][[1,1]],{n,mnrm[sp]}]

A357187 First differences A357186 = "Take the k-th composition in standard order for each part k of the n-th composition in standard order, then add up everything.".

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, -1, 1, 0, 0, -1, 1, 0, 0, 0, 1, 0, 0, -1, 0, 1, 0, -1, 1, 0, 0, -2, 1, 1, 0, -1, 1, 0, 0, 0, 0, 1, 0, -1, 1, 0, 0, -2, 1, 0, 0, 0, 1, 0, 0, -1, 0, 1, 0, -1, 1, 0, 0, -3, 1, 1, 0, 0, 1, 0, 0, -1, 0, 1, 0, -1, 1, 0, 0, -1, 1, 0

Offset: 0

Gus Wiseman, Sep 28 2022

sign

Are there any terms > 1?

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.

			We have A357186(5) - A357186(4) = 3 - 2 = 1, so a(4) = 1.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
Positions of first appearances appear to all belong to A052955.
Differences of A357186 (row-sums of A357135).
The version for partitions is A357458, differences of A325033.
Cf. A000120, A029837, A029931, A048896, A058891, A070939, A133494, A333766, A357134, A357137.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Differences[Table[stc/@stc[n]/.List->Plus,{n,0,100}]]

a(n) = A357186(n + 1) - A357186(n).

A380344 Product of prime indices minus sum of prime factors of n.

Original entry on oeis.org

1, -1, -1, -3, -2, -3, -3, -5, -2, -4, -6, -5, -7, -5, -2, -7, -10, -4, -11, -6, -2, -8, -14, -7, -1, -9, -1, -7, -19, -4, -20, -9, -4, -12, 0, -6, -25, -13, -4, -8, -28, -4, -29, -10, 1, -16, -32, -9, 2, -3, -6, -11, -37, -3, -1, -9, -6, -21, -42, -6, -43

Offset: 1

Gus Wiseman, Jan 24 2025

sign

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, with product A003963.

			72 has prime factors {2,2,2,3,3} and prime indices {1,1,1,2,2}, so a(72) = 4 - 12 = -8.

Positions of 0 are A331384.
For plus instead of minus we have A380409.
Positions of positives are A380410.
Triangles:
- A027746 = prime factors
- A112798 = prime indices
Statistics:
- A000027 = product of prime factors = row products of A027746
- A001414 = sum of prime factors = row sums of A027746
- A003963 = product of prime indices = row products of A112798
- A056239 = sum of prime indices = row sums of A112798
Combinations:
- A075254 = product of factors + sum of factors = A000027 + A001414
- A075255 = product of factors - sum of factors = A000027 - A001414
- A178503 = product of factors - sum of indices = A000027 - A056239
- A325036 = product of indices - sum of indices = A003963 - A056239
- A379681 = product of indices + sum of indices = A003963 + A056239
- A380344 = product of indices - sum of factors = A003963 - A001414
- A380345 = product of factors + sum of indices = A000027 + A056239
- A380409 = product of indices + sum of factors = A003963 + A001414
A000040 lists the primes, differences A001223.
A001222 counts prime factors with multiplicity.
A055396 gives least prime index, greatest A061395.
Cf. A000720, A175508, A319000, A325032, A325033, A325034, A325035, A325040, A379682, A380220.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@prix[n]-Plus@@Prime/@prix[n],{n,100}]

a(n) = A003963(n) - A001414(n).

A357139 Take the weakly increasing prime indices of each prime index of n, then concatenate.

Original entry on oeis.org

1, 2, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 2, 4, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 2, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 2, 5, 1, 3, 4, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 6, 1, 1, 1, 1, 4, 3, 1, 1, 2, 2, 2, 2, 3, 1, 1, 1, 1, 1, 2, 2, 1, 4, 1, 2

Offset: 1

Gus Wiseman, Sep 29 2022

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.

			Triangle begins:
   1:
   2:
   3:  1
   4:
   5:  2
   6:  1
   7:  1 1
   8:
   9:  1 1
  10:  2
  11:  3
  12:  1
  13:  1 2
For example, the weakly increasing prime indices of 105 are (2,3,4), with prime indices ((1),(2),(1,1)), so row 105 is (1,2,1,1).

Row lengths are A302242.
Positions of strict rows are A302505.
Positions of constant rows are A302593.
Row sums are A325033, products A325032.
The version for standard compositions is A357135, rank A357134.
A000961 lists prime powers.
A003963 multiples prime indices.
A056239 adds up prime indices.
Cf. A000720, A001221, A001222, A007716, A058891, A109082, A275024, A302243, A324926, A325034.

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

A357186 Take the k-th composition in standard order for each part k of the n-th composition in standard order, then add up everything.

Original entry on oeis.org

0, 1, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 3, 4, 4, 4, 3, 4, 4, 4, 4, 5, 5, 5, 4, 4, 5, 5, 4, 5, 5, 5, 3, 4, 5, 5, 4, 5, 5, 5, 5, 5, 6, 6, 5, 6, 6, 6, 4, 5, 5, 5, 5, 6, 6, 6, 5, 5, 6, 6, 5, 6, 6, 6, 3, 4, 5, 5, 5, 6, 6, 6, 5, 5, 6, 6, 5, 6, 6, 6, 5, 6, 6, 6, 6, 7, 7

Offset: 0

Gus Wiseman, Sep 28 2022

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 92 in standard order is (2,1,1,3), with compositions ((2),(1),(1),(1,1)) so a(92) = 2 + 1 + 1 + 1 + 1 = 6.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
This is the sum of A029837 over the n-th composition in standard order.
Vertex degrees are A133494.
The version for Heinz numbers of partitions is A325033.
Row sums of A357135.
First differences are A357187.
Cf. A000120, A001511, A003963, A029931, A048896, A058891, A070939, A096111, A329395, A333766, A335404, A357137.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[stc/@stc[n]/.List->Plus,{n,0,100}]

a(n) = A029837(A357134(n)).

A380345 a(n) = n + sum of prime indices of n.

Original entry on oeis.org

1, 3, 5, 6, 8, 9, 11, 11, 13, 14, 16, 16, 19, 19, 20, 20, 24, 23, 27, 25, 27, 28, 32, 29, 31, 33, 33, 34, 39, 36, 42, 37, 40, 42, 42, 42, 49, 47, 47, 46, 54, 49, 57, 51, 52, 56, 62, 54, 57, 57, 60, 60, 69, 61, 63, 63, 67, 69, 76, 67, 79, 74, 71, 70, 74, 74, 86

Offset: 1

Gus Wiseman, Jan 25 2025

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, with sum A056239.

			72 has prime indices {1,1,1,2,2}, so a(72) = 72 + 7 = 79.

For factors instead of indices we have A075254.
For minus instead of plus we have A178503.
Triangles:
- A027746 = prime factors
- A112798 = prime indices
Statistics:
- A000027 = product of prime factors = row products of A027746
- A001414 = sum of prime factors = row sums of A027746
- A003963 = product of prime indices = row products of A112798
- A056239 = sum of prime indices = row sums of A112798
Combinations:
- A075254 = product of factors + sum of factors = A000027 + A001414
- A075255 = product of factors - sum of factors = A000027 - A001414
- A178503 = product of factors - sum of indices = A000027 - A056239
- A325036 = product of indices - sum of indices = A003963 - A056239
- A379681 = product of indices + sum of indices = A003963 + A056239
- A380344 = product of indices - sum of factors = A003963 - A001414
- A380345 = product of factors + sum of indices = A000027 + A056239
- A380409 = product of indices + sum of factors = A003963 + A001414
A000040 lists the primes, differences A001223.
A001222 counts prime factors with multiplicity.
A055396 gives least prime index, greatest A061395.
Cf. A000720, A175508, A319000, A325032, A325033, A325034, A325035, A325040, A331384, A379682, A380220.

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

a(n) = n + A056239(n).

cp's OEIS Frontend

A379681 Sum plus product of the multiset of prime indices of n.

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A380956 Position of first appearance of n in A380955 (sum of prime indices minus sum of distinct prime indices).

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A380957 Sorted positions of first appearances in A380955 (sum of prime indices minus sum of distinct prime indices).

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A325035 Product of sums of the multisets of prime indices of each prime index of 2 * n + 1.

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A379682 Least number whose prime indices have sum + product = n.

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A357187 First differences A357186 = "Take the k-th composition in standard order for each part k of the n-th composition in standard order, then add up everything.".

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A380344 Product of prime indices minus sum of prime factors of n.

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A357139 Take the weakly increasing prime indices of each prime index of n, then concatenate.

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A357186 Take the k-th composition in standard order for each part k of the n-th composition in standard order, then add up everything.

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