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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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

Views

Author

Gus Wiseman, Feb 14 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.
All terms are odd.

Examples

			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}

Crossrefs

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.

Programs

  • Mathematica
    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

Views

Author

Gus Wiseman, Feb 18 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.
All terms are odd.

Examples

			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}

Crossrefs

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.

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,1000}];
Select[Range[Length[q]],FreeQ[Take[q,#-1],q[[#]]]&]

A379845 Even squarefree numbers x such that the product of prime indices of x is a multiple of the sum of prime indices of x.

Original entry on oeis.org

2, 30, 154, 190, 390, 442, 506, 658, 714, 874, 1110, 1118, 1254, 1330, 1430, 1786, 1794, 1798, 1958, 2310, 2414, 2442, 2470, 2730, 2958, 3034, 3066, 3266, 3390, 3534, 3710, 3770, 3874, 3914, 4042, 4466, 4526, 4758, 4930, 5106, 5434, 5474, 5642, 6090, 6106
Offset: 1

Views

Author

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

Examples

			The terms together with their prime indices begin:
     2: {1}
    30: {1,2,3}
   154: {1,4,5}
   190: {1,3,8}
   390: {1,2,3,6}
   442: {1,6,7}
   506: {1,5,9}
   658: {1,4,15}
   714: {1,2,4,7}
   874: {1,8,9}
  1110: {1,2,3,12}

Crossrefs

Even squarefree case of A326149.
For nonprime instead of even we have A326158.
Squarefree case of A379319.
Even case of A379844.
Partitions of this type are counted by A380221, see A379733, A379735.
A003963 multiplies together prime indices.
A005117 lists the squarefree numbers.
A056239 adds up prime indices.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- multiple: A057567, ranks A326155
- divisor: A057568, ranks A326149
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029, A379720
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A000720, A001222, A112798, A175508, A324850, A324851, A326150, A326151, A326153/A326154, A326156, A326157.

Programs

  • Mathematica
    Select[Range[2,1000],EvenQ[#]&&SquareFreeQ[#]&&Divisible[Times@@prix[#],Plus@@prix[#]]&]

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).

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).

A381076 Sorted positions of first appearances in A066503 (n minus squarefree kernel of n).

Original entry on oeis.org

1, 4, 8, 16, 18, 20, 24, 25, 27, 32, 44, 48, 50, 52, 54, 64, 68, 72, 75, 76, 80, 81, 92, 96, 98, 108, 112, 116, 121, 125, 128, 144, 148, 152, 160, 162, 164, 172, 175, 176, 188, 189, 192, 196, 198, 200, 212, 216, 232, 236, 242, 243, 244, 256, 260, 264, 268, 272
Offset: 1

Views

Author

Gus Wiseman, Feb 18 2025

Keywords

Comments

In A066503, each value appears for the first time at one of these positions.

Crossrefs

For quotient instead of difference we have A001694, sorted firsts of A003557.
Sorted positions of first appearances in A066503.
For indices and sum we have A380957 (unsorted A380956), firsts of A380955.
For indices and quotient we have A380988 (unsorted A380987), firsts of A290106.
For sum instead of product we have A381075, sorted firsts of A280292, see A280286.
For indices instead of factors we have A381077, sorted firsts of A380986.
A000040 lists the primes, differences A001223.
A001414 adds up prime factors (indices A056239), row sums of A027746 (indices A112798).
A003963 gives product of prime indices, distinct A156061.
A005117 lists squarefree numbers, complement A013929.
A007947 gives squarefree kernel.
A020639 gives least prime factor (index A055396), greatest A061395 (index A006530).
Cf. A001221, A001222, A038838, A046660, A075255, A081770, A116861, A136565, A178503, A175508, A304038, A380989.

Programs

  • Mathematica
    prifacs[n_]:=If[n==1,{},Flatten[Apply[ConstantArray,FactorInteger[n],{1}]]];
q=Table[Times@@prifacs[n]-Times@@Union[prifacs[n]],{n,1000}];
Select[Range[Length[q]],FreeQ[Take[q,#-1],q[[#]]]&]

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)).

A380411 Number of integer partitions of n such that the product of parts is greater than the sum of primes indexed by the parts.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 4, 8, 14, 23, 39, 58, 85, 121, 168, 228, 308, 404, 533, 691, 892, 1136, 1449, 1820, 2291, 2857, 3553, 4387, 5418, 6646, 8144, 9931, 12086, 14649, 17733, 21379, 25747, 30905, 37049, 44282, 52863, 62936, 74841, 88792, 105202, 124387
Offset: 0

Views

Author

Gus Wiseman, Jan 26 2025

Keywords

Examples

			The partition y = (4,3,2) has product of parts 4*3*2 = 24 and sum of corresponding primes 7+5+3 = 15, so y is counted under a(9).
The a(0) = 1 through a(10) = 14 partitions:
  ()  .  .  .  .  .  .  (322)  (44)    (54)     (55)
                               (332)   (333)    (64)
                               (422)   (432)    (433)
                               (2222)  (522)    (442)
                                       (3222)   (532)
                                       (3321)   (622)
                                       (4221)   (3322)
                                       (22221)  (3331)
                                                (4222)
                                                (4321)
                                                (5221)
                                                (22222)
                                                (32221)
                                                (33211)

Crossrefs

For parts instead of primes on the RHS we have A114324.
The version for divisibility instead of inequality is A330954.
The version for equality is A331383, ranks A331384.
These partitions are ranked by A380410.
A000040 lists the primes, differences A001223.
A000041 counts integer partitions, strict A000009.
A001414 gives sum of prime factors.
A003963 gives product of prime indices
A379666 counts partitions by sum and product.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- multiple: A057567, ranks A326155
- divisor: A057568 (strict A379733), ranks A326149, see A379319, A380217.
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029, A379720
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A175508, A318950, A319000, A319916, A326152, A379320, A379734, A380218, A380219, A380344.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Times@@#>Plus@@Prime/@#&]],{n,0,30}]
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