cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Views

Author

Gus Wiseman, Jan 24 2025

Keywords

Comments

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.

Examples

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

Crossrefs

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.

Programs

  • Mathematica
    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}]

Formula

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

Views

Author

Gus Wiseman, Jan 25 2025

Keywords

Comments

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.

Examples

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

Crossrefs

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.

Programs

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

Formula

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

A380409 Product of prime indices plus sum of prime factors.

Original entry on oeis.org

1, 3, 5, 5, 8, 7, 11, 7, 10, 10, 16, 9, 19, 13, 14, 9, 24, 12, 27, 12, 18, 18, 32, 11, 19, 21, 17, 15, 39, 16, 42, 11, 24, 26, 24, 14, 49, 29, 28, 14, 54, 20, 57, 20, 23, 34, 62, 13, 30, 21, 34, 23, 69, 19, 31, 17, 38, 41, 76, 18, 79, 44, 29, 13, 36, 26, 86
Offset: 1

Views

Author

Gus Wiseman, Jan 25 2025

Keywords

Comments

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.

Examples

			72 has prime factors {2,2,2,3,3} and prime indices {1,1,1,2,2}, so a(72) = 12 + 4 = 16.

Crossrefs

For factors instead of indices we have A075254.
For indices instead of factors we have A379681.
For minus instead of plus we have A380344, zeros A331384.
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, A380410.

Programs

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

Formula

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

A380410 Numbers with greater product of prime indices than sum of prime factors.

Original entry on oeis.org

1, 45, 49, 63, 75, 77, 81, 91, 99, 105, 117, 119, 121, 125, 126, 133, 135, 143, 147, 150, 153, 161, 162, 165, 169, 171, 175, 182, 187, 189, 195, 198, 203, 207, 209, 210, 217, 221, 225, 231, 234, 238, 242, 243, 245, 247, 250, 253, 255, 259, 261, 266, 270, 273
Offset: 1

Views

Author

Gus Wiseman, Jan 25 2025

Keywords

Comments

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.

Examples

			126 has prime indices {1,2,2,4} and prime factors {2,3,3,7}, and 16 > 15, so 126 is in the sequence.
The terms together with their prime indices begin:
     1: {}
    45: {2,2,3}
    49: {4,4}
    63: {2,2,4}
    75: {2,3,3}
    77: {4,5}
    81: {2,2,2,2}
    91: {4,6}
    99: {2,2,5}
   105: {2,3,4}
   117: {2,2,6}
   119: {4,7}
   121: {5,5}
   125: {3,3,3}
   126: {1,2,2,4}
   133: {4,8}
   135: {2,2,2,3}

Crossrefs

For factors instead of indices we have A002808.
The case of prime powers is A244623.
For indices instead of factors we have A325037, see also A325038.
The version for equality is A331384, counted by A331383.
Positions of positive terms in A380344.
Partitions of this type are counted by A380411.
A000040 lists the primes, differences A001223.
A001222 counts prime factors with multiplicity.
A055396 gives least prime index, greatest A061395.
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
Cf. A000720, A175508, A301987, A319000, A325040, A325041, A325044, A379682, A380220.

Programs

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

Formula

A003963(a(n)) > A001414(a(n)).
Showing 1-4 of 4 results.

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