A360614 -id:A360614 - cp's OEIS frontend

A360555 Two times the median of the first differences of the 0-prepended prime indices of n > 1.

2, 4, 1, 6, 2, 8, 0, 2, 3, 10, 2, 12, 4, 3, 0, 14, 2, 16, 2, 4, 5, 18, 1, 3, 6, 0, 2, 20, 2, 22, 0, 5, 7, 4, 1, 24, 8, 6, 1, 26, 2, 28, 2, 2, 9, 30, 0, 4, 2, 7, 2, 32, 1, 5, 1, 8, 10, 34, 2, 36, 11, 4, 0, 6, 2, 38, 2, 9, 2, 40, 0, 42, 12, 2, 2, 5, 2, 44, 0, 0

Offset: 2

Gus Wiseman, Feb 14 2023

nonn

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length). Since the denominator is always 1 or 2, the median can be represented as an integer by multiplying by 2.

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 0-prepended prime indices of 1617 are {0,2,4,4,5}, with sorted differences {0,1,2,2}, with median 3/2, so a(1617) = 3.

The version for divisors is A063655.
Differences of 0-prepended prime indices are listed by A287352.
The version for prime indices is A360005.
The version for distinct prime indices is A360457.
The version for distinct prime factors is A360458.
The version for prime factors is A360459.
The version for prime multiplicities is A360460.
Positions of even terms are A360556
Positions of odd terms are A360557
Positions of 0's are A360558, counted by A360254.
For mean instead of two times median we have A360614/A360615.
A112798 lists prime indices, length A001222, sum A056239.
A325347 counts partitions with integer median, complement A307683.
A326567/A326568 gives mean of prime indices.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A026424, A359907, A359908, A359912, A360550.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[2*Median[Differences[Prepend[prix[n],0]]],{n,2,100}]

A360556 Numbers > 1 whose first differences of 0-prepended prime indices have integer median.

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 23, 26, 27, 28, 29, 30, 31, 32, 35, 37, 38, 39, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 83, 84, 86, 87, 89

Offset: 1

Gus Wiseman, Feb 16 2023

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 median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The 0-prepended prime indices of 1617 are {0,2,4,4,5}, with sorted differences {0,1,2,2}, with median 3/2, so 1617 is not in the sequence.

For mean instead of median we have A340610.
Positions of even terms in A360555.
The complement is A360557 (without 1).
These partitions are counted by A360688.
- For divisors (A063655) we have A139711, complement A139710.
- For prime indices (A360005) we have A359908, complement A359912.
- For distinct prime indices (A360457) we have A360550, complement A360551.
- For distinct prime factors (A360458) we have A360552, complement A100367.
- For prime factors (A360459) we have A359913, complement A072978.
- For prime multiplicities (A360460) we have A360553, complement A360554.
- For 0-prepended differences (A360555) we have A360556, complement A360557.
A112798 lists prime indices, length A001222, sum A056239.
A325347 = partitions w/ integer median, complement A307683, strict A359907.
A359893 and A359901 count partitions by median, odd-length A359902.
A360614/A360615 = mean of first differences of 0-prepended prime indices.
Cf. A000975, A026424, A078175, A316413, A360009, A360558, A360669, A360681.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],IntegerQ[Median[Differences[Prepend[prix[#],0]]]]&]

A360557 Numbers > 1 whose sorted first differences of 0-prepended prime indices have non-integer median.

Original entry on oeis.org

4, 10, 15, 22, 24, 25, 33, 34, 36, 40, 46, 51, 54, 55, 56, 62, 69, 77, 82, 85, 88, 93, 94, 100, 104, 115, 118, 119, 121, 123, 134, 135, 136, 141, 146, 152, 155, 161, 166, 177, 184, 187, 194, 196, 201, 205, 206, 217, 218, 219, 220, 221, 225, 232, 235, 240, 248

Offset: 1

Gus Wiseman, Feb 17 2023

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 median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The 0-prepended prime indices of 1617 are {0,2,4,4,5}, with sorted differences {0,1,2,2}, with median 3/2, so 1617 is in the sequence.

For mean instead of median complement we have A340610, counted by A168659.
For mean instead of median we have A360668, counted by A200727.
Positions of odd terms in A360555.
The complement is A360556 (without 1), counted by A360688.
These partitions are counted by A360691.
- For divisors (A063655) we have A139710, complement A139711.
- For prime indices (A360005) we have A359912, complement A359908.
- For distinct prime indices (A360457) we have A360551, complement A360550.
- For distinct prime factors (A360458) we have A100367, complement A360552.
- For prime factors (A360459) we have A072978, complement A359913.
- For prime multiplicities (A360460) we have A360554, complement A360553.
- For 0-prepended differences (A360555) we have A360557, complement A360556.
A112798 lists prime indices, length A001222, sum A056239.
A287352 lists 0-prepended first differences of prime indices.
A325347 counts partitions with integer median, complement A307683.
A355536 lists first differences of prime indices.
A359893 and A359901 count partitions by median, odd-length A359902.
A360614/A360615 = mean of first differences of 0-prepended prime indices.
Cf. A000975, A026424, A078175, A316413, A360009, A360558, A360669, A360681.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],!IntegerQ[Median[Differences[Prepend[prix[#],0]]]]&]

A360615 Denominator of the average distance between consecutive 0-prepended prime indices of n; a(1) = 0.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Feb 19 2023

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 0-prepended prime indices of 100 are {0,1,1,3,3}, with differences (1,0,2,0), with mean 3/4, so a(100) = 4.

Winston de Greef, Table of n, a(n) for n = 1..10000

Positions of 1's are A340610
The numerator is A360614.
A112798 lists prime indices, length A001222, sum A056239, max A061395.
A124010 gives prime signature, mean A088529/A088530.
A316413 lists numbers with integer mean prime index, complement A348551.
A326567/A326568 gives mean of prime indices.
Cf. A106529, A334201, A340609, A360008, A360555, A360556, A360558.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[n==1,0,Denominator[Mean[Differences[Prepend[prix[n],0]]]]],{n,100}]

a(n) = if (n==1, 0, my(f=factor(n)); denominator(primepi(vecmax(f[, 1]))/ bigomega(f))); \\ Michel Marcus, Feb 20 2023

Denominator of A061395(n)/A001222(n), for n>1.

A360669 Nonprime numbers > 1 for which the prime indices have the same mean as their first differences.

Original entry on oeis.org

10, 39, 68, 115, 138, 259, 310, 328, 387, 517, 574, 636, 793, 795, 1034, 1168, 1206, 1241, 1281, 1340, 1534, 1691, 1825, 2212, 2278, 2328, 2343, 2369, 2370, 2727, 2774, 2905, 3081, 3277, 3818, 3924, 4064, 4074, 4247, 4268, 4360, 4539, 4850, 4905, 5243, 5335

Offset: 1

Gus Wiseman, Feb 18 2023

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 terms together with their prime indices begin:
     1: {}
    10: {1,3}
    39: {2,6}
    68: {1,1,7}
   115: {3,9}
   138: {1,2,9}
   259: {4,12}
   310: {1,3,11}
   328: {1,1,1,13}
   387: {2,2,14}
   517: {5,15}
   574: {1,4,13}
   636: {1,1,2,16}
For example, the prime indices of 138 are {1,2,9}, with mean 4, and with first differences (1,7), with mean also 4, so 138 is in the sequence.

These partitions are counted by A360670.
A058398 counts partitions by mean, see also A008284, A327482.
A112798 = prime indices, length A001222, sum A056239, mean A326567/A326568.
A124010 gives prime signature, mean A088529/A088530.
A301987 lists numbers whose sum of prime indices equals their product.
A316413 lists numbers whose prime indices have integer mean.
A334201 adds up all prime indices except the greatest.
A360614/A360615 = mean of first differences of 0-prepended prime indices.
Cf. A340610, A344415, A348551, A359904, A360008, A360068, A360681.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,1000],Mean[prix[#]]==Mean[Differences[prix[#]]]&]

A366785 Greatest common divisor of the number of prime factors (counted with multiplicity, A001222) and the greatest prime index (A061395) of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 2, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 3, 1, 3, 1, 2, 1, 4, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 3, 1, 1, 2, 1, 1, 4, 1, 2, 2, 1, 1, 1, 2, 3, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1

Offset: 1

Antti Karttunen, Oct 23 2023

nonn

Cf. A001222, A061395, A340608 (positions of 1's), A360614, A360615.

A061395(n) = if(n>1, primepi(vecmax(factor(n)[, 1])), 0);
A366785(n) = gcd(bigomega(n), A061395(n));

a(n) = gcd(A001222(n), A061395(n)).

For n >= 2, a(n) = A001222(n) / A360615(n) = A061395(n) / A360614(n).

A360668 Numbers > 1 whose greatest prime index is not divisible by their number of prime factors (bigomega).

Original entry on oeis.org

4, 8, 10, 12, 15, 16, 18, 22, 24, 25, 27, 28, 32, 33, 34, 36, 40, 42, 44, 46, 48, 51, 54, 55, 60, 62, 63, 64, 66, 68, 69, 70, 72, 76, 77, 80, 81, 82, 85, 88, 90, 93, 94, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 115, 116, 118, 119, 120, 121, 123, 124

Offset: 1

Gus Wiseman, Feb 17 2023

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 numbers > 1 whose first differences of 0-prepended prime indices have non-integer mean.

			The prime indices of 1617 are {2,4,4,5}, and 5 is not divisible by 4, so 1617 is in the sequence.

These partitions are counted by A200727.
The complement is A340610 (without 1), counted by A168659.
For median instead of mean we have A360557, counted by A360691.
Positions of terms > 1 in A360615 (numerator: A360614).
A058398 counts partitions by mean, see also A008284, A327482.
A067340 lists numbers whose prime signature has integer mean.
A112798 = prime indices, length A001222, sum A056239, mean A326567/A326568.
A334201 adds up all prime indices except the greatest.
A348551 = numbers w/ non-integer mean of prime indices, complement A316413.
Cf. A072978, A078175, A139710, A307683, A359912, A360551, A360554, A360556, A360669.

Select[Range[2,100],!Divisible[PrimePi[FactorInteger[#][[-1,1]]],PrimeOmega[#]]&]

A360670 Number of integer partitions of n whose parts have the same mean as their negated first differences.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 1, 2, 0, 2, 0, 2, 2, 0, 0, 5, 1, 0, 3, 3, 0, 4, 0, 5, 3, 0, 2, 10, 0, 0, 4, 10, 0, 5, 0, 7, 9, 0, 0, 17, 1, 6, 5, 10, 0, 9, 8, 14, 6, 0, 0, 34, 0, 0, 9, 18, 13, 13, 0, 17, 7, 19, 0, 40, 0, 0, 28

Offset: 0

Gus Wiseman, Feb 18 2023

nonn

			The a(n) partitions for n = 4, 12, 24, 27, 30, 44:
  (3,1)  (9,3)    (18,6)      (19,7,1)  (21,8,1)      (33,11)
         (9,2,1)  (17,6,1)    (20,5,2)  (22,6,2)      (34,5,4,1)
                  (18,4,2)    (21,3,3)  (23,4,3)      (34,6,3,1)
                  (19,2,2,1)            (25,2,1,1,1)  (34,7,2,1)
                  (19,3,1,1)                          (34,8,1,1)
                                                      (35,4,3,2)
                                                      (35,5,2,2)
For example, the partition y = (28,4,3,1), with mean 9, has negated first differences (24,2,1), with mean 9, so y is counted under a(36).

These partitions have ranks A360669.
A000041 counts integer partitions, strict A000009.
A058398 counts partitions by mean, see also A008284, A327482.
A067538 counts partitions with integer mean, strict A102627, ranks A316413.
A326567/A326568 gives mean of prime indices.
A360614/A360615 = mean of first differences of 0-prepended prime indices.
Cf. A060016, A240219, A116608, A316313, A340610, A360068.

Table[Length[Select[IntegerPartitions[n], Mean[#]==Mean[Differences[Reverse[#]]]&]],{n,0,30}]

A360680 Numbers for which the prime signature has the same mean as the first differences of 0-prepended prime indices.

Original entry on oeis.org

1, 2, 6, 30, 49, 152, 210, 513, 1444, 1776, 1952, 2310, 2375, 2664, 2760, 2960, 3249, 3864, 3996, 4140, 4144, 5796, 5994, 6072, 6210, 6440, 6512, 6517, 6900, 7176, 7400, 7696, 8694, 9025, 9108, 9384, 10064, 10120, 10350, 10488, 10764, 11248, 11960, 12167

Offset: 1

Gus Wiseman, Feb 19 2023

nonn

A number's (unordered) prime signature (row n of A118914) is the multiset of positive exponents in its prime factorization.

			The terms together with their prime indices begin:
      1: {}
      2: {1}
      6: {1,2}
     30: {1,2,3}
     49: {4,4}
    152: {1,1,1,8}
    210: {1,2,3,4}
    513: {2,2,2,8}
   1444: {1,1,8,8}
   1776: {1,1,1,1,2,12}
   1952: {1,1,1,1,1,18}
   2310: {1,2,3,4,5}
   2375: {3,3,3,8}
   2664: {1,1,1,2,2,12}
   2760: {1,1,1,2,3,9}
   2960: {1,1,1,1,3,12}
For example, the prime indices of 2760 are {1,1,1,2,3,9}. The signature is (3,1,1,1), with mean 3/2. The first differences of 0-prepended prime indices are (1,0,0,1,1,6), with mean also 3/2. So 2760 is in the sequence.

For indices instead of 0-prepended differences: A359903, counted by A360068.
For median instead of mean we have A360681.
A112798 = prime indices, length A001222, sum A056239, mean A326567/A326568.
A124010 gives prime signature, mean A088529/A088530.
A316413 = numbers whose prime indices have integer mean, complement A348551.
A326619/A326620 gives mean of distinct prime indices.
A360614/A360615 = mean of first differences of 0-prepended prime indices.
Cf. A340610, A359904, A359905, A360008, A360460, A360555, A360556.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Mean[Length/@Split[prix[#]]] == Mean[Differences[Prepend[prix[#],0]]]&]

A361392 Number of integer partitions of n whose first differences have mean -1.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 1, 3, 2, 5, 4, 8, 7, 12, 12, 19, 19, 29, 31, 43, 48, 65, 73, 97, 110, 142, 164, 208, 240, 301, 350, 432, 504, 617, 719, 874, 1019, 1228, 1434, 1717, 2001, 2385, 2778, 3292, 3831, 4522, 5252, 6177, 7164, 8392, 9722, 11352, 13125, 15283, 17643

Offset: 0

Gus Wiseman, Mar 13 2023

nonn

These are partitions where the first part minus the last part is the number of parts minus 1.

			The a(3) = 1 through a(11) = 8 partitions:
  (21)  .  (32)   (321)  (43)    (422)   (54)     (442)    (65)
           (311)         (331)   (4211)  (432)    (4321)   (533)
                         (4111)          (4221)   (4411)   (4331)
                                         (4311)   (52111)  (4421)
                                         (51111)           (5222)
                                                           (52211)
                                                           (53111)
                                                           (611111)
For example, the partition y = (4,2,2,1) has first differences (-2,0,-1), with mean -1, so y is counted under a(9).

For mean 0 we have A032741.
The 0-appended version is A047993.
For any negative mean we have A144300.
A000041 counts integer partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean.
A067538 counts partitions with integer mean, ranks A316413.
A326567/A326568 gives mean of prime indices, conjugate A326839/A326840.
A360614/A360615 gives mean of 0-appended first differences of prime indices.
Cf. A102627, A237363, A237832, A348551, A360688.

Table[Length[Select[IntegerPartitions[n],Mean[Differences[#]]==-1&]],{n,0,30}]

cp's OEIS Frontend

A360555 Two times the median of the first differences of the 0-prepended prime indices of n > 1.

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A360556 Numbers > 1 whose first differences of 0-prepended prime indices have integer median.

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A360557 Numbers > 1 whose sorted first differences of 0-prepended prime indices have non-integer median.

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A360615 Denominator of the average distance between consecutive 0-prepended prime indices of n; a(1) = 0.

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A360669 Nonprime numbers > 1 for which the prime indices have the same mean as their first differences.

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A366785 Greatest common divisor of the number of prime factors (counted with multiplicity, A001222) and the greatest prime index (A061395) of n.

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A360668 Numbers > 1 whose greatest prime index is not divisible by their number of prime factors (bigomega).

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A360670 Number of integer partitions of n whose parts have the same mean as their negated first differences.

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A360680 Numbers for which the prime signature has the same mean as the first differences of 0-prepended prime indices.

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