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.

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A380220 Least positive integer whose prime indices satisfy (product) - (sum) = n. Position of first appearance of n in A325036.

Original entry on oeis.org

2, 1, 21, 25, 39, 35, 57, 55, 49, 65, 75, 77, 129, 95, 91, 105, 183, 119, 125, 143, 133, 185, 147, 161, 169, 195, 175, 209, 339, 217, 255, 253, 259, 305, 247, 285, 273, 245, 301, 299, 345, 323, 325, 357, 371, 435, 669, 391, 361, 403, 399, 473, 343, 469, 481
Offset: 0

Views

Author

Gus Wiseman, Jan 21 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. The sum and product of prime indices are A056239 and A003963 respectively.

Examples

			The least number whose prime indices satisfy (product) - (sum) = 3 is 25 (prime indices {3,3}), so a(3) = 25.

Crossrefs

Position of first appearance of n in A325036.
For sum instead of difference we have A379682, firsts of A379681.
A000040 lists the primes, differences A001223.
A003963 multiplies together prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
The subtraction A325036 takes the following values:
- zero: A301987, counted by A001055 (strict A045778).
- negative: A325037, counted by A114324, see A318029
- 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
Cf. A000720, A178503, A175508, A318950, A319000, A325034, A326151, A326153/A326154, A379319, A379722.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
pp=Table[Total[prix[n]]-Times@@prix[n],{n,100}];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
Table[Position[pp,-i][[1,1]],{i,0,mnrm[-DeleteCases[pp,0|_?Positive]]}]

Formula

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

A381077 Sorted positions of first appearances in A380986 (product of prime indices minus product of distinct prime indices).

Original entry on oeis.org

1, 9, 25, 49, 63, 81, 99, 121, 125, 135, 169, 171, 245, 279, 289, 343, 361, 363, 369, 375, 387, 477, 529, 531, 575, 603, 625, 675, 711, 729, 747, 833, 841, 847, 873, 875, 891, 909, 961, 981, 1029, 1083, 1125, 1127, 1179, 1225, 1251, 1377, 1413, 1445, 1467
Offset: 1

Views

Author

Gus Wiseman, Feb 20 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. A position of first appearance in a sequence q is an index k such that q(k) is different from q(j) for all j < k.
All terms are odd.

Examples

			The terms together with their prime indices begin:
     1: {}
     9: {2,2}
    25: {3,3}
    49: {4,4}
    63: {2,2,4}
    81: {2,2,2,2}
    99: {2,2,5}
   121: {5,5}
   125: {3,3,3}
   135: {2,2,2,3}
   169: {6,6}
   171: {2,2,8}
   245: {3,4,4}
   279: {2,2,11}

Crossrefs

For length instead of product we have A151821, firsts of A046660.
For factors instead of indices we have A381076, sorted firsts of A066503.
For sum of factors instead of product of indices we have A381075 (unsorted A280286), A280292.
For quotient instead of difference we have A380988 (unsorted A380987), firsts of A290106.
For quotient and factors we have A001694 (unsorted A064549), firsts of A003557.
For sum instead of product we have A380957 (unsorted A380956), firsts of A380955.
Sorted firsts of A380986, which has nonzero terms at positions A038838.
A000040 lists the primes, differences A001223.
A003963 gives product of prime indices, distinct A156061.
A005117 lists the squarefree numbers, complement A013929.
A007947 gives squarefree kernel.
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. A000079, A000720, A001222, A081770, A075255, A116861, A178503, A374248, A379681, A380958.

Programs

  • Mathematica
    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,10000}];
Select[Range[Length[q]],FreeQ[Take[q,#-1],q[[#]]]&]
Previous Showing 11-15 of 15 results.

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