A024619 -id:A024619 - cp's OEIS frontend

A375055 Nonsquarefree numbers k divisible by at least 3 distinct primes.

60, 84, 90, 120, 126, 132, 140, 150, 156, 168, 180, 198, 204, 220, 228, 234, 240, 252, 260, 264, 270, 276, 280, 294, 300, 306, 308, 312, 315, 336, 340, 342, 348, 350, 360, 364, 372, 378, 380, 396, 408, 414, 420, 440, 444, 450, 456, 460, 468, 476, 480, 490, 492, 495

Offset: 1

Michael De Vlieger, Oct 22 2024

Also, numbers k such that there exists a pair of necessarily composite divisors {d, k/d}, d < k/d, with quality Q, i.e., gcd(d, k/d) > 1 but there exists a prime p | d that does not divide k/d, and also a prime q | k/d that does not divide d.
A178212 is a proper subset.
This sequence is distinct from A123712 since 420 is here.
This sequence is distinct from A182855 since 360 is here.

			a(1) = 60 = 2^2 * 3 * 5, the smallest number such that bigomega(60) > omega(60) > 2. Bigomega(60) = 4, omega(60) = 3.
72 is not in the sequence because it is the product of 2 distinct prime factors.
a(2) = 84 = 2^2 * 3 * 7, since bigomega(84) = 4, omega(84) = 3.
a(3) = 90 = 2 * 3^2 * 5, since bigomega(90) = 4, omega(90) = 3.
a(4) = 120 = 2^3 * 3 * 5, since bigomega(120) = 5, omega(120) = 3.
210 is not in the sequence because it is squarefree.
a(35) = 360 = 2^3 * 3^2 * 5 since bigomega(360) = 6, omega(360) = 3.
a(43) = 420 = 2^2 * 3 * 5 * 7 since bigomega(420) = 5, omega(420) = 4, etc.
.
Table showing pairs of factors of a(n) for select n, such that the pair possesses quality Q (see comments).
    n    a(n)   pair of factors with quality Q.
  -------------------------------------------------------------------
    1     60    6 X 10;
    2     84    6 X 14;
    3     90    6 X 15;
    4    120    6 X 20,  10 X 12;
    5    126    6 X 21;
    6    132    6 X 22;
    7    140   10 X 14;
    8    150   10 X 15;
   17    240    6 X 40,  10 X 24, 12 X 20;
   51    480    6 X 80,  10 X 48, 12 X 40, 20 X 24;
  117    840    6 X 140, 10 X 84, 12 X 70, 14 X 60, 20 X 42, 28 X 30.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001221, A001222, A013929, A024619, A126706, A178212.

Select[Range[500], PrimeOmega[#] > PrimeNu[#] > 2 &]

{a(n)} = { k : bigomega(k) > omega(k) > 2 }, where bigomega = A001222 and omega = A001221.

A309639 Index of the least harmonic number H_i whose denominator (A002805) is divisible by n.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 4, 13, 7, 5, 16, 17, 9, 19, 5, 9, 11, 23, 9, 25, 13, 27, 7, 29, 5, 31, 32, 11, 17, 7, 9, 37, 19, 13, 8, 41, 9, 43, 11, 9, 23, 47, 16, 49, 25, 17, 13, 53, 27, 11, 8, 19, 29, 59, 5, 61, 31, 9, 64, 13, 11, 67, 17, 24, 7, 71, 9, 73, 37, 25

Offset: 1

Robert G. Wilson v, Aug 11 2019

nonn

a(n) is not a divisor of n for n = 21, 24, 42, 69, 84, 105, 115, 120, 138, 168, 171, ..., (A330736).

The sequence for the numerators only has terms for 1, 3, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 33, ..., .

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000961, A002805, A024619, A034699, A330691, A330692, A330741, A330753 (ordinal transform), A330734, A330735, A330736.

H:= 1: B[1]:= 1:
for n from 2 to 200 do H:= H + 1/n; B[n]:= denom(H) od:
f:= proc(n) local F, t0, t;
  t0:= max(seq(t[1]^t[2],t=ifactors(n)[2]));
  for t from t0 do if B[t] mod n = 0 then return t fi od
end proc:
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Aug 11 2019

s = 0; k = 1; t[_] := 0; While[k < 101, s = s + 1/k; lst = Select[ Range@ 100, Mod[Denominator@ s, #] == 0 &]; If[t[#] == 0, t[#] = k] & /@ lst; k++]; t@# & /@ Range@75

f(n) = denominator(sum(k=2, n, 1/k)); \\ A002805
a(n) = my(k=1); while(f(k) % n, k++); k; \\ Michel Marcus, Aug 11 2019

A309639list(up_to) = { my(s=0,v002805=vector(up_to),v309639=vector(up_to)); v002805[1] = 1; for(k=2,up_to,s += 1/k; v002805[k] = denominator(s)); for(n=1,up_to,for(j=1,up_to,if(!(v002805[j]%n),v309639[n] = j; break))); (v309639); }; \\ Antti Karttunen, Dec 29 2019

a(n) = n iff n is a power of a prime (A000961).
a(n) < n iff n is a member of A024619.
a(n) >= A034699(n). - Robert Israel, Aug 11 2019
gcd(a(n), n) = A330691(n). - Antti Karttunen, Dec 29 2019

A375734 Indices of consecutive prime-powers (exclusive) differing by 1. Positions of 1's in A057820.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 17, 43, 70, 1077, 6635, 12369, 43578, 105102700

Offset: 1

Gus Wiseman, Sep 04 2024

The corresponding prime-powers A246655(a(n)) are given by A006549.

From A006549, it is not known whether this sequence is infinite.

			The fifth prime-power is 7 and the sixth is 8, so 5 is in the sequence.

For nonprime numbers (A002808) we have A375926, differences A373403.
Positions of 1's in A057820.
First differences are A373671.
For nonsquarefree numbers we have A375709, differences A373409.
For non-prime-powers we have A375713.
For non-perfect-powers we have A375740.
For squarefree numbers we have A375927, differences A373127.
Prime-powers:
- terms: A000961, complement A024619.
- differences: A057820.
- runs: A373675, A373673, A373674, A174965
- anti-runs: A373576, A120430, A006549, A373671
Non-prime-powers:
- terms: A361102
- differences: A375708
- runs: A373678, A373676, A373677, A110969 (A373669, sorted A373670).
- anti-runs: A373679, A373575, A255346, A373672
A000040 lists all of the primes, differences A001223.
A025528 counts prime-powers up to n.
Cf. A007916, A014963, A027833, A046933, A053289, A093555, A375714.

Join@@Position[Differences[Select[Range[100],PrimePowerQ]],1]

Numbers k such that A246655(k+1) - A246655(k) = 1.

The inclusive version is a(n) + 1 shifted.

a(14) from Amiram Eldar, Sep 24 2024

A376248 Irregular triangle where row n lists m such that rad(m) | n and bigomega(m) <= bigomega(n), where rad = A007947 and bigomega = A001222.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 2, 4, 1, 5, 1, 2, 3, 4, 6, 9, 1, 7, 1, 2, 4, 8, 1, 3, 9, 1, 2, 4, 5, 10, 25, 1, 11, 1, 2, 3, 4, 6, 8, 9, 12, 18, 27, 1, 13, 1, 2, 4, 7, 14, 49, 1, 3, 5, 9, 15, 25, 1, 2, 4, 8, 16, 1, 17, 1, 2, 3, 4, 6, 8, 9, 12, 18, 27, 1, 19, 1, 2, 4, 5, 8, 10, 20, 25, 50, 125

Offset: 1

Michael De Vlieger, Oct 09 2024

Analogous to A162306 regarding m such that rad(m) | n, but instead of taking m <= n, we take m such that bigomega(m) <= bigomega(n).
Row n is a finite set of products of prime power factors p^k (i.e., p^k | n) such that Sum_{p|n} k <= bigomega(n).
For prime power n = p^k, k >= 0 (i.e., n in A000961), row p^k of this sequence is the same as row p^k of A027750 and A162306. Therefore, for prime p, row p of this sequence is the same as row p of A027750 and A162306: {1, p}.
For n in A024619, row n of this sequence does not match row n of A162306, since the former contains gpf(n)^bigomega(n) = A006530(n)^A001222(n), which is larger than n.

			Triangle begins:
   n    row n of this sequence:
  -------------------------------------------
   1:   1;
   2:   1,  2;
   3:   1,  3;
   4:   1,  2   4;
   5:   1,  5;
   6:   1,  2,  3,  4,  6,  9;
   7:   1,  7;
   8:   1,  2,  4,  8;
   9:   1,  3,  9;
  10:   1,  2,  4,  5, 10, 25;
  11:   1, 11;
  12:   1,  2,  3,  4,  6,  8, 9, 12, 18, 27;
        ...
Row n = 10 of this sequence, presented according to 2^k, k = 0..bigomega(n) by columns, 5^i, i = 0..bigomega(n) by rows, showing terms m > n with an asterisk. The remaining m and the parenthetic 8 are in row 10 of A162306:
   1   2   4  (8)
   5  10
  25*
Row n = 12 of this sequence, presented according to 2^k, k = 0..bigomega(n) by columns, 3^i, i = 0..bigomega(n) by rows, showing terms m > n with an asterisk. The remaining m are in row 12 of A162306:
   1   2   4   8
   3   6  12
   9  18*
  27*

Michael De Vlieger, Table of n, a(n) for n = 1..17475 (rows n = 1..1000, flattened)
Michael De Vlieger, Log log scatterplot of rows n = 1..2^16 of this sequence, flattened, (3153752 terms).

Cf. A000961, A001221, A001222, A006530, A007947, A010846, A024619, A027750, A162306, A376567.

Table[Clear[p]; MapIndexed[Set[p[First[#2]], #1] &, FactorInteger[n][[All, 1]]]; k = PrimeOmega[n]; w = PrimeNu[n]; Union@ Map[Times @@ MapIndexed[p[First[#2]]^#1 &, #] &, Select[Tuples[Range[0, k], w], Total[#] <= k &] ], {n, 120}]

Row n of this sequence is { m : rad(m) | n, bigomega(m) <= bigomega(n) }.

A376567(n) = binomial(bigomega(n) + omega(n)) = Length of row n, where omega = A001221.

A377781 First differences of A065514(n) = greatest number < prime(n) that is 1 or a prime-power.

Original entry on oeis.org

1, 2, 1, 4, 2, 5, 1, 2, 8, 2, 3, 5, 4, 2, 6, 4, 6, 5, 3, 4, 2, 8, 2, 6, 8, 4, 2, 4, 2, 16, 3, 3, 6, 2, 10, 2, 6, 6, 6, 4, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 4, 13, 1, 6, 6, 2, 6, 4, 8, 4, 14, 4, 2, 4, 14, 12, 4, 2, 4, 8, 6, 6, 6, 4, 6, 8, 4, 8, 10, 2, 10

Offset: 1

Gus Wiseman, Nov 14 2024

nonn

Note 1 is a power of a prime but not a prime-power.

Differences of A065514, which is the restriction of A031218 (differences A377782).
The opposite is A377703 (restriction of A000015), differences of A345531.
The opposite for nonsquarefree is A377784, differences of A377783.
For nonsquarefree we have A378034, differences of A378032 (restriction of A378033).
The opposite for squarefree is A378037, differences of A112926 (restriction of A067535).
For squarefree we have A378038, differences of A112925 (restriction of A070321).
A000040 lists the primes, differences A001223.
A000961 and A246655 list the prime-powers, differences A057820.
A024619 lists the non-prime-powers, differences A375735, seconds A376599.
A361102 lists the non-powers of primes, differences A375708.
Prime-powers between primes:
- A053607 primes
- A080101 count (exclusive)
- A304521 by bits
- A366833 count
- A377057 positive
- A377286 zero
- A377287 one
- A377288 two
Cf. A001597, A007916, A008864, A053289, A120327, A343249, A375706, A377281, A377282, A377289.

Differences[Table[NestWhile[#-1&,Prime[n]-1,#>1&&!PrimePowerQ[#]&],{n,100}]]

A378371 Distance between n and the least non prime power >= n, allowing 1.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Nov 28 2024

nonn

Non prime powers allowing 1 (A361102) are numbers that are not a prime power (A246655), namely 1, 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, ...

			The least non prime power >= 4 is 6, so a(4) = 2.

Sequences obtained by adding n to each term are placed in parentheses below.
For prime we have A007920 (A151800), strict A013632.
For composite we have A010051 (A113646 except initial terms).
For perfect power we have A074984 (A377468)
For squarefree we have A081221 (A067535).
For nonsquarefree we have (A120327).
For non perfect power we have A378357 (A378358).
The opposite version is A378366 (A378367).
For prime power we have A378370, strict A377282 (A000015).
This sequence is A378371 (A378372).
A000040 lists the primes, differences A001223.
A000961 and A246655 list the prime powers, differences A057820.
A024619 and A361102 list the non prime powers, differences A375708 and A375735.
Prime powers between primes: A053607, A080101, A304521, A366833, A377057.
Cf. A007916, A053707, A065514, A276781 (A031218), A343249, A345531, A376596, A376597, A377051, A377054, A377289, A378457.

Table[NestWhile[#+1&,n,PrimePowerQ[#]&]-n,{n,100}]

a(n) = A378372(n) - n.

A378372 Least non prime power >= n, allowing 1.

Original entry on oeis.org

1, 6, 6, 6, 6, 6, 10, 10, 10, 10, 12, 12, 14, 14, 15, 18, 18, 18, 20, 20, 21, 22, 24, 24, 26, 26, 28, 28, 30, 30, 33, 33, 33, 34, 35, 36, 38, 38, 39, 40, 42, 42, 44, 44, 45, 46, 48, 48, 50, 50, 51, 52, 54, 54, 55, 56, 57, 58, 60, 60, 62, 62, 63, 65, 65, 66, 68

Offset: 1

Gus Wiseman, Nov 29 2024

nonn

Non prime powers allowing 1 (A361102) are numbers that are not a prime power (A246655), namely 1, 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, ...

			The least non prime power >= 4 is 6, so a(4) = 6.

Sequences obtained by subtracting n from each term are placed in parentheses below.
For prime power we have A000015 (A378370).
For squarefree we have A067535 (A081221).
For composite we have A113646 (A010051).
For nonsquarefree we have A120327.
For prime we have A151800 (A007920), strict (A013632).
Run-lengths are 1 and A375708.
For perfect power we have A377468 (A074984).
For non-perfect power we have A378358 (A378357).
The opposite is A378367, distance A378366.
This sequence is A378372 (A378371).
A000040 lists the primes, differences A001223.
A000961 and A246655 list the prime powers, differences A057820.
A024619 and A361102 list the non prime powers, differences A375708 and A375735.
Prime powers between primes: A053607, A080101, A304521, A366833, A377057.
Cf. A001597, A007916, A031218 (A276781), A053707, A065514, A345531 (A377281), A377051, A377282, A377289, A378457.

Table[NestWhile[#+1&,n,PrimePowerQ[#]&],{n,100}]

a(n) = A378371(n) + n.

A306927 a(n) = A001615(n) - n.

Original entry on oeis.org

0, 1, 1, 2, 1, 6, 1, 4, 3, 8, 1, 12, 1, 10, 9, 8, 1, 18, 1, 16, 11, 14, 1, 24, 5, 16, 9, 20, 1, 42, 1, 16, 15, 20, 13, 36, 1, 22, 17, 32, 1, 54, 1, 28, 27, 26, 1, 48, 7, 40, 21, 32, 1, 54, 17, 40, 23, 32, 1, 84, 1, 34, 33, 32, 19, 78, 1, 40, 27, 74, 1, 72

Offset: 1

Torlach Rush, Mar 16 2019

Analogous to A051953.
a(n) = A051953(n) if n is an element of A000961.
a(n) > A051953(n) if n is an element of A024619.
The sum of the proper divisors d of n such that n/d is squarefree. - Amiram Eldar, Sep 06 2019

			0 is a term because A001615(1) - 1 = 0.
1 is a term because A001615(2) - 2 = 1.
3 is a term because A001615(9) - 9 = 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wikipedia, Dedekind psi function.

Cf. A000961, A001615, A024619, A051953, A082020.

a[1] = 0; a[n_] := n * (Times @@ (1 + 1/FactorInteger[n][[;; , 1]]) - 1); Array[a, 100] (* Amiram Eldar, Sep 06 2019 *)

a(n) = n*(sumdivmult(n, d, issquarefree(d)/d) - 1); \\ Michel Marcus, Mar 18 2019

a(n) = A001615(n) - n.
a(n) = Sum_{d|n, dAmiram Eldar, Sep 06 2019
Sum_{k=1..n} a(k) = c * n^2 / 2 + O(n*log(n)), where c = 15/Pi^2 - 1 = 0.519817... . - Amiram Eldar, Dec 08 2023

A363728 Number of integer partitions of n that are not constant but satisfy (mean) = (median) = (mode), assuming there is a unique mode.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 4, 0, 3, 3, 7, 0, 12, 0, 18, 12, 9, 0, 50, 12, 14, 33, 54, 0, 115, 0, 92, 75, 31, 99, 323, 0, 45, 162, 443, 0, 507, 0, 467, 732, 88, 0, 1551, 274, 833, 627, 1228, 0, 2035, 1556, 2859, 1152, 221, 0, 9008, 0, 295, 4835, 5358

Offset: 1

Gus Wiseman, Jun 23 2023

nonn

A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.

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 a(8) = 1 through a(18) = 12 partitions:
  3221  .  32221  .  4332    .  3222221  43332  5443      .  433332
                     5331       3322211  53331  6442         443331
                     322221     4222211  63321  7441         533322
                     422211                     32222221     533331
                                                33222211     543321
                                                42222211     633321
                                                52222111     733311
                                                             322222221
                                                             332222211
                                                             422222211
                                                             432222111
                                                             522222111

Non-constant partitions are counted by A144300, ranks A024619.
This is the non-constant case of A363719, ranks A363727.
These partitions have ranks A363729.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length (or decreasing mean), strict A008289.
A359893 and A359901 count partitions by median, odd-length A359902.
A362608 counts partitions with a unique mode.
Cf. A237984, A240219, A325347, A326567/A326568, A327472, A359894, A363720, A363721.

modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n],!SameQ@@#&&{Mean[#]}=={Median[#]}==modes[#]&]],{n,30}]

A368748 a(n) is the number of numbers between prime(n) and prime(n+1) that are not prime powers.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 1, 3, 3, 1, 4, 3, 1, 3, 4, 5, 1, 4, 3, 1, 5, 2, 5, 7, 3, 1, 3, 1, 3, 11, 2, 5, 1, 9, 1, 5, 5, 3, 4, 5, 1, 9, 1, 3, 1, 11, 11, 3, 1, 3, 5, 1, 8, 4, 5, 5, 1, 5, 3, 1, 8, 13, 3, 1, 3, 13, 5, 8, 1, 3, 5, 6, 5, 5, 3, 5, 7, 3, 7, 9, 1, 9, 1, 5, 3, 5, 7, 3, 1, 3

Offset: 1

David James Sycamore, Jan 04 2024

nonn

Here "between" refers to numbers in the range [prime(n) + 1, prime(n+1) - 1], all of which are composite, and the sequence counts the numbers in each such range which are not prime powers. Whereas the corresponding number of prime powers seems bounded (see A080101), the number of numbers which are not prime powers is unbounded (see A014963). Conjecture: Every nonnegative integer appears in this sequence (at least once).

			Between 2 and 3 there are no other numbers so a(1) = 0.
Between 3 and 5 there is only one number (4) and it is a prime power, so a(2) = 0.
Between 5 and 7 the only number is 6 and it is not a prime power, so a(3) = 1.
Between 47 and 53 there are 5 composite numbers, but one of them (49) is a prime power, so since 47 = prime(15), a(15) = 4.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Annotated log scatterplot of a(n), n = 1..2^20, showing records in red.

Cf. A000040, A000961, A001223, A014963, A024619, A080101, A085970, A093555, A368749.

N:= 101: # for a(1) .. a(N-1)
P:= [seq(ithprime(i),i=1..N)]:
PP:= {seq(seq(P[i]^j, j = 2 .. ilog[P[i]](P[N])),i=1..N)}:
seq(nops({$P[i]+1 .. P[i+1]-1} minus PP), i=1 .. N-1); # Robert Israel, Jan 04 2024

Map[Count[Range[#1, #2 - 1], ?(Not@*PrimePowerQ)] & @@ # &, Partition[Prime@ Range[120], 2, 1]] (* _Michael De Vlieger, Jan 04 2024 *)

a(n) = sum(k=prime(n)+1, prime(n+1)-1, !isprimepower(k)); \\ Michel Marcus, Jan 04 2024

a(n) = A001223(n) - A080101(n) - 1. - Michael De Vlieger, Jan 04 2024

More terms from Michel Marcus, Jan 04 2024

cp's OEIS Frontend

A375055 Nonsquarefree numbers k divisible by at least 3 distinct primes.

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A309639 Index of the least harmonic number H_i whose denominator (A002805) is divisible by n.

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A375734 Indices of consecutive prime-powers (exclusive) differing by 1. Positions of 1's in A057820.

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A376248 Irregular triangle where row n lists m such that rad(m) | n and bigomega(m) <= bigomega(n), where rad = A007947 and bigomega = A001222.

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A377781 First differences of A065514(n) = greatest number < prime(n) that is 1 or a prime-power.

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A378371 Distance between n and the least non prime power >= n, allowing 1.

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A378372 Least non prime power >= n, allowing 1.

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A306927 a(n) = A001615(n) - n.

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