A102750 -id:A102750 - cp's OEIS frontend

A243058 Fixed points of A243057 and A243059.

1, 2, 3, 5, 6, 7, 11, 12, 13, 17, 19, 21, 23, 24, 29, 30, 31, 37, 41, 43, 47, 48, 53, 59, 61, 63, 65, 67, 70, 71, 73, 79, 83, 89, 96, 97, 101, 103, 107, 109, 113, 127, 131, 133, 137, 139, 149, 151, 154, 157, 163, 165, 167, 173, 179, 180, 181, 189, 191, 192, 193, 197, 199, 210

Offset: 1

Antti Karttunen, May 31 2014

nonn

Number n is present if its prime factorization n = p_a * p_b * p_c * ... * p_i * p_j * p_k (where a <= b <= c <= ... <= i <= j <= k are the indices of prime factors, not necessarily all distinct; sorted into nondescending order) satisfies the condition that the first differences of those prime indices (a-0, b-a, c-b, ..., j-i, k-j) form a palindrome.

The above condition implies that none of the terms of A070003 are present, as then at least the difference k-j would be zero, but on the other hand, a-0 is at least 1. Cf. also A243068.

			12 = 2*2*3 = p_1 * p_1 * p_2 is present, as the first differences (deltas) of the indices of its nondistinct prime factors (1-0, 1-1, 2-1) = (1,0,1) form a palindrome.
18 = 2*3*3 = p_1 * p_2 * p_2 is NOT present, as the deltas of the indices of its nondistinct prime factors (1-0, 2-1, 2-2) = (1,1,0) do NOT form a palindrome.
65 = 5*13 = p_3 * p_6 is present, as the deltas of the indices of its nondistinct prime factors (3-0, 6-3) = (3,3) form a palindrome.

Antti Karttunen, Table of n, a(n) for n = 1..2048

A subsequence of A243068.
Apart from 1 also a subsequence of A102750.
A000040 is a subsequence.
Cf. A242413, A242417, A243057, A243059, A242417.

A243059 If n = p_a * p_b * ... * p_h * p_i * p_j * p_k, with p_a <= p_b <= ... <= p_k, then a(n) = p_{k-j} * p_{k-i} * p_{k-h} * ... * p_{k-a} * p_k, where p_{0} = 0 and for k>=1, p_{k} = A000040(k). a(1)=1 by convention.

Original entry on oeis.org

1, 2, 3, 0, 5, 6, 7, 0, 0, 15, 11, 12, 13, 35, 10, 0, 17, 0, 19, 45, 21, 77, 23, 24, 0, 143, 0, 175, 29, 30, 31, 0, 55, 221, 14, 0, 37, 323, 91, 135, 41, 105, 43, 539, 20, 437, 47, 48, 0, 0, 187, 1573, 53, 0, 33, 875, 247, 667, 59, 90, 61, 899, 63, 0, 65, 385, 67, 2873, 391, 70, 71, 0

Offset: 1

Antti Karttunen, May 31 2014

nonn

A243058 gives all n such that a(n) = n (the fixed points of this sequence, which include primes).

Differs from A243057 in that the "degenerate cases" A070003 are here zeros, but is otherwise equal to it (at the points given by A102750), i.e. for all n, a(A102750(n)) = A243057(A102750(n)) = A242420(A102750(n)).

			For n = 9 = 3*3 = p_2 * p_2, we have a(n) = p_{3-3} * p_3 = 0*3 = 0. [Like all terms in A070003 this is an example of "degenerate case", where some p's in the product get index 0, and thus are set to 0 by the convention used here.]
For n = 10 = 2*5 = p_1 * p_3, we have a(n) = p_{3-1} * p_3 = 3*5 = 15.
For n = 12 = 2*2*3 = p_1 * p_1 * p_2, we have a(n) = p_{2-1} * p{2-1} * p_2 = p_1^2 * p_2 = 12.
For n = 15 = 3*5 = p_2 * p_3, we have a(n) = p_{3-2} * p_3 = 2*5 = 10.
For n = 2200 = 2*2*2*5*5*11 = p_1 * p_1 * p_1 * p_3 * p_3 * p_5, we have a(n) = p_{5-3} * p_{5-3} * p_{5-1} * p_{5-1} * p_{5-1} * p_5 = 3*3*7*7*7*11 = 33957.
For n = 33957 = 3*3*7*7*7*11 = p_2 * p_2 * p_4 * p_4 * p_4 * p_5, we have a(n) = p_{5-4} * p_{5-4} * p_{5-4} * p_{5-2} * p_{5-2} * p_5 = 2*2*2*5*5*11 = 2200.

Antti Karttunen, Table of n, a(n) for n = 1..10001

Fixed points: A243058 (includes primes).
Positions of zeros: A070003.
Cf. A243057, A000040, A032742, A243056, A242419, A242420, A243286.

a(1)=1, and for n>1, a(n) = q_{A243056(n)} * a(A032742(n)). Here q_{k} stands for 0 when k=0, and otherwise for the k-th prime, A000040(k).

If n = p_a * p_b * ... * p_h * p_i * p_j * p_k, where p_a <= p_b <= ... <= p_k are (not necessarily distinct) primes (sorted into nondescending order) in the prime factorization of n, then a(n) = p_{k-j} * p_{k-i} * p_{k-h} * ... * p_{k-a} * p_k, where p_{0} = 0 and for k>=1, p_{k} = A000040(k).

A253561 Square array read by antidiagonals: A(row,col) = A122111(A246278(row,col)).

Original entry on oeis.org

2, 3, 4, 6, 9, 8, 5, 18, 27, 16, 12, 25, 54, 81, 32, 10, 36, 125, 162, 243, 64, 24, 50, 108, 625, 486, 729, 128, 7, 72, 250, 324, 3125, 1458, 2187, 256, 15, 49, 216, 1250, 972, 15625, 4374, 6561, 512, 20, 75, 343, 648, 6250, 2916, 78125, 13122, 19683, 1024, 48, 100, 375, 2401, 1944, 31250, 8748, 390625, 39366, 59049, 2048, 14, 144, 500, 1875, 16807, 5832, 156250, 26244, 1953125, 118098, 177147, 4096

Offset: 2

Antti Karttunen, Jan 03 2015

If we assume here that a(1) = 1 (but which is not explicitly included because outside of the array), then A253562 gives the inverse permutation.

The top row A253568 contains the same terms as A102750, but in different order.

			The top left corner of the array:
   2,  3,   6,   5,   12,   10,   24,    7,   15,   20,  48,   14,  96,   40,
   4,  9,  18,  25,   36,   50,   72,   49,   75,  100, 144,   98, 288,  200,
   8, 27,  54, 125,  108,  250,  216,  343,  375,  500, 432,  686, 864, 1000,
  16, 81, 162, 625,  324, 1250,  648, 2401, 1875, 2500,1296, 4802,2592, 5000,
  32,243, 486,3125,  972, 6250, 1944,16807, 9375,12500,3888,33614,7776,25000,
...

Antti Karttunen, Table of n, a(n) for n = 2..466; the first 30 antidiagonals of the array

Inverse: A253562.
The leftmost column: A000079. Topmost row: A253568.
Cf. A071178, A122111, A246278, A102750, A253560, A253563.

(define (A253561 n) (A122111 (A246278 n)))

a(n) = A122111(A246278(n)). [As a linear sequence].
Other identities.
A071178(A(row,col)) = row for all col. [All terms on row k have k as the exponent of their largest prime factor.]
A253560(A(row,col)) = A(row+1,col). [For any n >= 2, A253560(n) gives the term which is immediately below n in the same column of this array.]

A336353 Numbers k such that sigma(k) does not have any prime factor larger than the largest prime factor of k.

Original entry on oeis.org

1, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95

Offset: 1

Antti Karttunen, Jul 19 2020

nonn

Note that the terms after 1 do not form a subsequence of A102750: the first counterexample is 343 = 7^3. See A336354.

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Positions of zeros in A336352.
Cf. A102750, A336317.
Subsequences: A000396, A001599, A065091, A105402, A333646 (see comment there), A336354.

PARI
```
isA336353(n) = !A336352(n);
```

A336317 Numbers k such that A122111(k) [conjugated prime factorization of k] is one of Ore's Harmonic numbers (in A001599).

Original entry on oeis.org

1, 6, 40, 126, 154, 204, 1716, 1914, 2772, 8580, 11264, 12090, 12540, 50960, 62790, 64350, 77748, 83200, 104720, 152320, 186116, 193440, 331890, 382720, 432768, 518364, 648788, 684684, 753480, 817344, 895356, 1083852, 1113840, 1619352, 1675044, 1743588, 1759680, 1991340, 2060322, 2360484, 2492028, 2621080, 2932800

Offset: 1

Antti Karttunen, Jul 19 2020

nonn

Numbers k for which A336314(k) = A323173(k).
Sequence A122111(A001599(n)), n >= 1, sorted into ascending order. Positions of zeros in A323174 (corresponding to perfect numbers similarly mapped) is a subsequence.
Note that all terms after 1 seem to be present in A102750. This observation is equal to Ore's conjecture that there are no odd Harmonic numbers larger than one.
Also, all terms after 1 seem to be even, which would imply that apart from its initial 1, A001599 were a subsequence of A102750. However, this is false, as there are terms of A001599 not in A102750, for example 8011798098793361832960 found by David A. Corneth. Note that A122111(8011798098793361832960) = 96922193555635754403846044921625, which is thus an odd term of this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..61
Index entries for sequences computed from indices in prime factorization

Cf. A001599, A088902, A102750, A122111, A323173, A323174, A336314, A336315, A336353, A336397 (the intersection with A001599).

isA001599(n) = !((sigma(n,0)*n)%sigma(n,1));
isA336317(n) = isA001599(A122111(n)); \\ Program for A122111 given under that entry.

\\ Standalone program:
isA336317(n) = if(1==n,1,my(f=factor(n),es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,d=1,s=1,x=1,p,e); for(i=1, #es, pri += es[i]; p = prime(pri); e = 1+is[i]-is[1+i]; d *= e; s *= ((p^e)-1)/(p-1); x *= (p^(e-1))); !((x*d)%s));

A348044 The nearest common ancestor of n and n^2 in the Doudna tree (A005940).

Original entry on oeis.org

1, 2, 2, 4, 2, 2, 2, 8, 4, 2, 2, 2, 2, 2, 2, 16, 2, 4, 2, 2, 2, 2, 2, 2, 4, 2, 8, 2, 2, 2, 2, 32, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 2, 2, 2, 8, 2, 2, 2, 2, 2, 2, 2, 2, 2, 64, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 4, 2, 2, 2, 2, 2, 16, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 4, 2, 2, 2, 2, 2

Offset: 1

Antti Karttunen, Sep 27 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Main diagonal of A348042 and A348043.
Cf. A102750 (the positions of 2's).
Cf. A005940, A156552, A348041.

A348044(n) = A348041sq(n,n*n); \\ Uses also code from A348041.

a(n) = A348041(n, n^2) = A348042(n, n) = A348043(n, n).

A243074 a(1) = 1, a(n) = n/p^(k-1), where p = largest prime dividing n and p^k = highest power of p dividing n.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 12, 13, 14, 15, 2, 17, 6, 19, 20, 21, 22, 23, 24, 5, 26, 3, 28, 29, 30, 31, 2, 33, 34, 35, 12, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 7, 10, 51, 52, 53, 6, 55, 56, 57, 58, 59, 60, 61, 62, 63, 2, 65, 66, 67, 68, 69, 70, 71, 24, 73, 74, 15, 76, 77, 78, 79, 80, 3

Offset: 1

Antti Karttunen, May 31 2014

nonn

After 1, A102750 gives such k that a(k) = k, which are also the positions of the records as for all n, a(n) <= n. After 1, only terms of A102750 occur, each an infinite number of times.

			For n = 18 = 2*3*3, we discard all instances of the highest prime factor 3 except one, thus a(18) = 2*3 = 6.
For n = 54 = 2*3*3*3, we discard two copies of 3, and thus also the value of a(54) is 2*3 = 6.
For n = 20 = 2*5, the highest prime factor 5 occurs only once, so nothing is cast off, and a(20) = 20.

Antti Karttunen, Table of n, a(n) for n = 1..10001

Differs from A052410 for the first time at n=18.

Cf. A006530, A051119, A102750, A070003, A243057.

(define (A243074 n) (* (A051119 n) (A006530 n)))

a(n) = A006530(n) * A051119(n).

A253568 Even bisection of A122111: a(n) = A122111(2*n).

Original entry on oeis.org

2, 3, 6, 5, 12, 10, 24, 7, 15, 20, 48, 14, 96, 40, 30, 11, 192, 21, 384, 28, 60, 80, 768, 22, 45, 160, 35, 56, 1536, 42, 3072, 13, 120, 320, 90, 33, 6144, 640, 240, 44, 12288, 84, 24576, 112, 70, 1280, 49152, 26, 135, 63, 480, 224, 98304, 55, 180, 88, 960, 2560, 196608, 66, 393216, 5120, 140, 17, 360, 168, 786432, 448, 1920, 126, 1572864, 39

Offset: 1

Antti Karttunen, Jan 04 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..1024

Even bisection of A122111.
Topmost row of A253561.
Permutation of A102750.
Cf. A243285, A244982.

Scheme
```
(define (A253568 n) (A122111 (* 2 n)))
```

a(n) = A122111(2*n).
Other identities. For all n >= 1:
A244982(n) = A243285(a(n)).

A255363 Numbers with the property that A006255(k) = A070229(k).

Original entry on oeis.org

5, 7, 11, 13, 14, 17, 19, 21, 22, 23, 26, 29, 31, 33, 34, 37, 38, 39, 41, 43, 44, 46, 47, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, 67, 68, 69, 71, 73, 74, 76, 78, 79, 82, 83, 85, 86, 87, 88, 89, 92, 93, 94, 95, 97, 101, 102, 103, 106, 107, 109, 111, 113, 114

Offset: 1

Peter Kagey, Feb 21 2015

nonn

A070229(n) is a lower bound of A006255(n) for all n in A102750.
This list contains all primes greater than 3 and no perfect squares.
Let k be a fixed integer, then k*p is found in this list for all sufficiently large primes p.

Peter Kagey, Table of n, a(n) for n = 1..5000

A364193 Number of integer partitions of n where the least part is the unique mode.

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 7, 9, 13, 17, 24, 32, 43, 58, 75, 97, 130, 167, 212, 274, 346, 438, 556, 695, 865, 1082, 1342, 1655, 2041, 2511, 3067, 3756, 4568, 5548, 6728, 8130, 9799, 11810, 14170, 16980, 20305, 24251, 28876, 34366, 40781, 48342, 57206, 67597, 79703

Offset: 0

Gus Wiseman, Jul 16 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 in {a,a,b,b,b,c,d,d,d} are {b,d}.

			The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (322)      (44)
                    (211)   (2111)   (222)     (511)      (422)
                    (1111)  (11111)  (411)     (3211)     (611)
                                     (3111)    (4111)     (2222)
                                     (21111)   (22111)    (4211)
                                     (111111)  (31111)    (5111)
                                               (211111)   (32111)
                                               (1111111)  (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

For greatest part and multiple modes we have A171979.
Allowing multiple modes gives A240303.
For greatest instead of least part we have A362612, ranks A362616.
For mean instead of least part we have A363723.
These partitions have ranks A364160.
A000041 counts integer partitions.
A362611 counts modes in prime factorization, A362613 co-modes.
A362614 counts partitions by number of modes, co-modes A362615.
A363486 gives least mode in prime indices, A363487 greatest.
A363952 counts partitions by low mode, A363953 high.
Ranking and counting partitions:
- A356862 = unique mode, counted by A362608
- A359178 = unique co-mode, counted by A362610
- A362605 = multiple modes, counted by A362607
- A362606 = multiple co-modes, counted by A362609
Cf. A002865, A008284, A070003, A098859, A102750, A237984, A327472, A360015.

Table[If[n==0,0,Length[Select[IntegerPartitions[n], Last[Length/@Split[#]]>Max@@Most[Length/@Split[#]]&]]],{n,0,30}]

cp's OEIS Frontend

A243058 Fixed points of A243057 and A243059.

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A243059 If n = p_a * p_b * ... * p_h * p_i * p_j * p_k, with p_a <= p_b <= ... <= p_k, then a(n) = p_{k-j} * p_{k-i} * p_{k-h} * ... * p_{k-a} * p_k, where p_{0} = 0 and for k>=1, p_{k} = A000040(k). a(1)=1 by convention.

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A253561 Square array read by antidiagonals: A(row,col) = A122111(A246278(row,col)).

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A336353 Numbers k such that sigma(k) does not have any prime factor larger than the largest prime factor of k.

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A336317 Numbers k such that A122111(k) [conjugated prime factorization of k] is one of Ore's Harmonic numbers (in A001599).

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PARI

A348044 The nearest common ancestor of n and n^2 in the Doudna tree (A005940).

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A243074 a(1) = 1, a(n) = n/p^(k-1), where p = largest prime dividing n and p^k = highest power of p dividing n.

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A253568 Even bisection of A122111: a(n) = A122111(2*n).

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A255363 Numbers with the property that A006255(k) = A070229(k).

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