A325034 -id:A325034 - cp's OEIS frontend

A379721 Numbers whose prime indices have sum <= product.

1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 30, 31, 33, 35, 37, 39, 41, 42, 43, 45, 47, 49, 50, 51, 53, 54, 55, 57, 59, 61, 63, 65, 66, 67, 69, 70, 71, 73, 75, 77, 78, 79, 81, 83, 84, 85, 87, 89, 90, 91, 93, 95, 97, 98, 99, 100, 101, 102, 103

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.
Partitions of this type are counted by A319005.
The complement is A325038.

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    7: {4}
    9: {2,2}
   11: {5}
   13: {6}
   15: {2,3}
   17: {7}
   19: {8}
   21: {2,4}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   30: {1,2,3}

The case of equality is A301987, inequality A325037.
Nonpositive positions in A325036.
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.
A379681 gives sum plus product of prime indices, firsts A379682.
Counting and ranking multisets by comparing sum and product:
- same: A001055 (strict A045778), ranks A301987
- divisible: A057567, ranks A326155
- divisor: A057568, ranks A326149, see A326156, A326172, A379733
- greater: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less: A114324, ranks A325037, see A318029
- less or equal: A319005, ranks A379721 (this)
- different: A379736, ranks A379722, see A111133
Cf. A000720, A003963, A046022, A075254, A075255, A178503, A175508, A319000, A325034, A325035, A379720.

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

Number k such that A056239(k) <= A003963(k).

A325036 Difference between product and sum of prime indices of n.

Original entry on oeis.org

1, 0, 0, -1, 0, -1, 0, -2, 0, -1, 0, -2, 0, -1, 1, -3, 0, -1, 0, -2, 2, -1, 0, -3, 3, -1, 2, -2, 0, 0, 0, -4, 3, -1, 5, -2, 0, -1, 4, -3, 0, 1, 0, -2, 5, -1, 0, -4, 8, 2, 5, -2, 0, 1, 7, -3, 6, -1, 0, -1, 0, -1, 8, -5, 9, 2, 0, -2, 7, 4, 0, -3, 0, -1, 10, -2, 11, 3, 0, -4, 8, -1, 0, 0, 11, -1, 8, -3, 0, 4, 14, -2, 9

Offset: 1

Gus Wiseman, Mar 25 2019

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.

			The prime indices of 45 are {2,2,3}, with product 12 and sum 7, so a(45) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Positions of zeros are A301987. Positions of ones are A325041. Positions of negative ones are A325042.
Cf. A000720, A001222, A003963, A056239, A112798, A122111, A178503, A175508, A319000.
Cf. A325032, A325033, A325034, A325035, A325037, A325038, A325040, A325044.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@primeMS[n]-Total[primeMS[n]],{n,100}]
dps[n_]:=Module[{pi=Flatten[Table[PrimePi[#[[1]]],#[[2]]]&/@FactorInteger[n]]},Times@@pi-Total[pi]]; Join[{1},Array[dps,100,2]] (* Harvey P. Dale, May 26 2023 *)

A003963(n) = { n=factor(n); n[, 1]=apply(primepi, n[, 1]); factorback(n) }; \\ From A003963
A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); };
A325036(n) = (A003963(n) - A056239(n)); \\ Antti Karttunen, May 08 2022

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

For all n >= 1, a(A325040(n)) = a(A122111(A325040(n))). - Antti Karttunen, May 08 2022

Data section extended up to a(93) by Antti Karttunen, May 08 2022

A325033 Sum of sums of the multisets of prime indices of each prime index of n.

Original entry on oeis.org

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

Offset: 1

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 sum a(91) = 5.

Cf. A000720, A056239, A109082, A112798, A302242.

Cf. A324926, A325032, A325034, A325035.

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

Totally additive with a(prime(n)) = A056239(n).

A325032 Product of products of the multisets of prime indices of each prime index of n.

Original entry on oeis.org

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

Offset: 1

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.

			94 has prime indices {1,15} with prime indices {{},{2,3}} with product a(94) = 6.

Cf. A000720, A001055, A001222, A003963, A056239, A112798, A302242, A320325.

Cf. A324850, A324926, A324930, A325031, A325033, A325034, A325035.

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

Fully multiplicative with a(prime(n)) = A003963(n).

A380955 Sum of prime indices of n (with multiplicity) minus sum of distinct prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Feb 11 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 prime indices of 96 are {1,1,1,1,1,2}, with sum 7, and with distinct prime indices {1,2}, with sum 3, so a(96) = 7 - 3 = 4.

Positions of 0's are A005117, complement A013929.
For length instead of sum we have A046660.
Positions of 1's are A081770.
For factors instead of indices we have A280292, firsts A280286 (sorted A381075).
A multiplicative version is A290106.
Counting partitions by this statistic gives A364916.
Dominates A374248.
Positions of first appearances are A380956, sorted A380957.
For prime multiplicities instead of prime indices we have A380958.
For product instead of sum we have A380986.
A000040 lists the primes, differences A001223.
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, A071625, A075254, A075255, A116861, A124010, A136565, A178503, A325033, A325034, A366528, A366749, A379681.

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

a(n) = A056239(n) - A066328(n).

Additive: a(m*n) = a(m) + a(n) if gcd(m,n) = 1.

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

Original entry on oeis.org

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).

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]}]

A380987 Position of first appearance of n in A290106 (product of prime indices divided by product of distinct prime indices).

Original entry on oeis.org

1, 9, 25, 27, 121, 169, 289, 81, 125, 841, 961, 675, 1681, 1849, 2209, 243, 3481, 1125, 4489, 3267, 5329, 6241, 6889, 2025, 1331, 10201, 625, 7803, 11881, 12769, 16129, 729, 18769, 19321, 22201, 2197, 24649, 26569, 27889, 9801, 32041, 32761, 36481, 25947

Offset: 1

Gus Wiseman, Feb 14 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.

All terms are odd.

			The first position of 12 in A290106 is 675, with prime indices {2,2,2,3,3}, so a(12) = 675.
The terms together with their prime indices begin:
      1: {}
      9: {2,2}
     25: {3,3}
     27: {2,2,2}
    121: {5,5}
    169: {6,6}
    289: {7,7}
     81: {2,2,2,2}
    125: {3,3,3}
    841: {10,10}
    961: {11,11}
    675: {2,2,2,3,3}
   1681: {13,13}
   1849: {14,14}
   2209: {15,15}
    243: {2,2,2,2,2}
   3481: {17,17}
   1125: {2,2,3,3,3}

For factors instead of indices we have A064549 (sorted A001694), firsts of A003557.
The additive version for factors is A280286 (sorted A381075), firsts of A280292.
Position of first appearance of n in A290106.
The additive version is A380956 (sorted A380957), firsts of A380955.
For difference instead of quotient see A380986.
The sorted version is A380988.
A000040 lists the primes, differences A001223.
A003963 gives product of prime indices, distinct A156061.
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, A007947, A046660, A066503, A071625, A136565, A178503, A175508, A325034, A379681.

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[Times@@prix[n]/Times@@Union[prix[n]],{n,10000}];
Table[Position[q,k][[1,1]],{k,mnrm[q]}]

A380988 Sorted positions of first appearances in A290106 (product of prime indices divided by product of distinct prime indices).

Original entry on oeis.org

1, 9, 25, 27, 81, 121, 125, 169, 243, 289, 625, 675, 729, 841, 961, 1125, 1331, 1681, 1849, 2025, 2187, 2197, 2209, 3125, 3267, 3481, 4489, 4913, 5329, 5625, 6075, 6241, 6561, 6889, 7803, 9801, 10125, 10201, 11881, 11979, 12769, 14641, 15125, 15625, 16129

Offset: 1

Gus Wiseman, Feb 18 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.

All terms are odd.

			The prime indices of 225 are {2,2,3,3}, with image A290106(225) = 6. The prime indices of 169 are {6,6}, also with image 6. Since the latter is the first with image 6, 169 is in the sequence, and 225 is not.
The terms together with their prime indices begin:
     1: {}
     9: {2,2}
    25: {3,3}
    27: {2,2,2}
    81: {2,2,2,2}
   121: {5,5}
   125: {3,3,3}
   169: {6,6}
   243: {2,2,2,2,2}
   289: {7,7}
   625: {3,3,3,3}
   675: {2,2,2,3,3}
   729: {2,2,2,2,2,2}
   841: {10,10}
   961: {11,11}
  1125: {2,2,3,3,3}
  1331: {5,5,5}
  1681: {13,13}
  1849: {14,14}
  2025: {2,2,2,2,3,3}

For factors instead of indices we have A001694 (unsorted A064549), firsts of A003557.
Sorted firsts of A290106.
The additive version is A380957 (sorted A380956), firsts of A380955.
For difference instead of quotient see A380986.
The unsorted version is A380987.
The additive version for factors is A381075 (unsorted A280286), firsts of A280292.
A000040 lists the primes, differences A001223.
A003963 gives product of prime indices, distinct A156061.
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, A007947, A046660, A066503, A071625, A136565, A178503, A175508, A325034, A379681.

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

cp's OEIS Frontend

A379721 Numbers whose prime indices have sum <= product.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A325036 Difference between product and sum of prime indices of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A325033 Sum of sums of the multisets of prime indices of each prime index of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A325032 Product of products of the multisets of prime indices of each prime index of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A380955 Sum of prime indices of n (with multiplicity) minus sum of distinct prime indices of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs