A242414 -id:A242414 - cp's OEIS frontend

A322136 Numbers whose number of prime factors counted with multiplicity exceeds half their sum of prime indices by at least 1.

4, 8, 12, 16, 24, 32, 36, 40, 48, 64, 72, 80, 96, 108, 112, 120, 128, 144, 160, 192, 216, 224, 240, 256, 288, 320, 324, 336, 352, 360, 384, 400, 432, 448, 480, 512, 576, 640, 648, 672, 704, 720, 768, 800, 832, 864, 896, 960, 972

Offset: 1

Gus Wiseman, Nov 27 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k). The sequence lists all Heinz numbers of integer partitions where the number of parts is at least 1 plus half the sum of parts.

Also Heinz numbers of integer partitions that are the vertex-degrees of some hypertree. We allow no singletons in a hypertree, so 2 is not included.

			The sequence of partitions with Heinz numbers in the sequence begins: (11), (111), (211), (1111), (2111), (11111), (2211), (3111), (21111), (111111), (22111), (31111), (211111), (22211), (41111), (32111), (1111111).

Cf. A000569, A025065, A030019, A056156, A056239, A056503, A112798, A181821, A242414, A304382, A320922, A320923, A320924, A320925.

Select[Range[1000],PrimeOmega[#]>=(Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]+2)/2&]

A319437 Number of series-reduced palindromic plane trees with n nodes.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 3, 4, 7, 10, 15, 23, 35, 52, 81, 121, 185, 280, 427, 645, 985, 1490, 2269, 3440, 5233, 7936, 12071, 18313, 27839, 42256, 64217, 97490, 148137, 224924, 341725, 518923, 788321, 1197178, 1818597, 2761926, 4195381, 6371808, 9678537, 14699771

Offset: 1

Gus Wiseman, Sep 18 2018

nonn

A rooted plane tree is series-reduced if every non-leaf node has at least two branches, and palindromic if the sequence of branches directly under any given node is a palindrome.

Andrew Howroyd, Table of n, a(n) for n = 1..500
Gus Wiseman, The a(15) = 81 series-reduced palindromic plane trees.

Cf. A000108, A000670, A001003, A005043, A008965, A025065, A118376, A242414, A317085, A317086, A317087, A319122, A319436.

srpanplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[srpanplane/@c],#==Reverse[#]&],{c,Join@@Permutations/@Select[IntegerPartitions[n-1],Length[#]>1&]}]];
Table[Length[srpanplane[n]],{n,15}]

PAL(p)={(1+p)/subst(1-p, x, x^2)}
seq(n)={my(p=O(1));for(i=1, n, p=PAL(x*p)-x*p); Vec(p)} \\ Andrew Howroyd, Sep 19 2018

Terms a(27) and beyond from Andrew Howroyd, Sep 19 2018

A335402 Numbers m such that the only normal integer partition of m whose run-lengths are a palindrome is (1)^m.

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269

Offset: 1

Gus Wiseman, Jun 06 2020

An integer partition is normal if it covers an initial interval of positive integers.
Conjecture: The sequence consists of 0, 1, 4, and all primes except 3.
From Chai Wah Wu, Jun 22 2020: (Start)
The above conjecture is true.
Proof: The cases of 0, 1, 4 can be checked by inspection. Next we show that if n is prime and not equal to 3, then n is a term. Let n be prime and consider a palindromic normal partition of n covering the integers 1,...,k with k > 1. Then the multiplicity of 1 and k are the same and the multiplicities of 2 and k-1 are the same, etc.
If k is even, then n is of the form (k+1)r. Since n is prime, this implies that n = k+1. Since n >= k(k+1)/2. this means that k = 2 and n = 3.
If k is odd, then n is of the form (k+1)r + w(k+1)/2. Let m = (k+1)/2, then n = m(2r+w). Since n is prime and r,w > 0, this means that m = 1, k = 1, a contradiction.
Next we show that if n is composite and not equal to 4, then n is not a term.
Suppose n = pq for 1 < p <= q. If p is odd, let k = p-1 > 1.
Consider the partition covering 1,..,k where the multiplicity is 1 except for 1 and k where the multiplicity is q-k/2 + 1 > 0. This is a normal palindromic partition summing up to pq = n.
If p is even, without loss of generality we can choose p = 2. Since n != 4, q >= 3. In this case, choosing k = 3 with 1 and 3 having multiplicity 1 and 2 having multiplicity q-2 > 0 results in a normal palindromic partition of 2q = n. QED
It is clear that if n is not a term, then any multiple of n is also not a term.
(End)

			There are 4 normal integer partitions of 10 whose sequence of multiplicities is a palindrome, namely (4321), (33211), (32221), (1111111111), so 10 does not belong to the sequence. The normal integer partitions of 7 are (3211), (2221), (22111), (211111), (1111111), none of which has palindromic multiplicities except the last, so 7 belongs to the sequence.

Positions of 1's in A317086.
Palindromic-multiplicity partitions are counted by A317085.
Normal integer partitions are counted by A000009.
Heinz numbers of normal palindromic-multiplicity partitions are A317087.
Cf. A000041, A000837, A025065, A046022, A124010, A242414.

Select[Range[0,30],Length[Select[IntegerPartitions[#],And[Or[#=={},Union[#]==Range[First[#]]],Length/@Split[#]==Reverse[Length/@Split[#]]]&]]==1&]

# from definition
from sympy.utilities.iterables import partitions
from sympy import integer_nthroot
A335402_list = []
for m in range(0,101):
    for d in partitions(m,k=integer_nthroot(2*m,2)[0]):
        l = len(d)
        if l > 0 and not(l == 1 and 1 in d):
            k = max(d)
            if l == k:
                for i in range(k//2):
                    if d[i+1] != d[k-i]:
                        break
                else:
                    break
    else:
        A335402_list.append(m) # Chai Wah Wu, Jun 22 2020

# from formula
from sympy import prime
A335402_list = [0,1,2,4] + [prime(i) for i in range(3,100)] # Chai Wah Wu, Jun 22 2020

n is a term if and only if n = 0, 1, 2, 4 or a prime > 3. - Chai Wah Wu, Jun 22 2020

a(22)-a(59) from Chai Wah Wu, Jun 22 2020

A344417 Number of palindromic factorizations of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 7, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1

Offset: 1

Gus Wiseman, May 22 2021

nonn

A palindrome is a sequence that is the same whether it is read forward or in reverse. A palindromic factorization of n is a finite multiset of positive integers > 1 with product n that can be permuted into a palindrome.

			The palindromic factorizations for n = 2, 4, 16, 36, 64, 144:
  (2)  (4)    (16)       (36)       (64)           (144)
       (2*2)  (4*4)      (6*6)      (8*8)          (12*12)
              (2*2*4)    (2*2*9)    (4*4*4)        (4*4*9)
              (2*2*2*2)  (3*3*4)    (2*2*16)       (4*6*6)
                         (2*2*3*3)  (2*2*4*4)      (2*2*36)
                                    (2*2*2*2*4)    (3*3*16)
                                    (2*2*2*2*2*2)  (2*2*6*6)
                                                   (3*3*4*4)
                                                   (2*2*2*2*9)
                                                   (2*2*3*3*4)
                                                   (2*2*2*2*3*3)

Positions of 1's are A005117.
The case of palindromic compositions is A016116.
The additive version (palindromic partitions) is A025065.
The case of palindromic prime signature is A242414.
The case of palindromic plane trees is A319436.
A001055 counts factorizations.
A229153 ranks non-palindromic partitions.
A265640 ranks palindromic partitions.
Cf. A000041, A000070, A000188, A004526, A030229, A056503, A057567, A082293, A344414.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
palQ[y_]:=Select[Permutations[y],#==Reverse[#]&]!={};
Table[Length[Select[facs[n],palQ]],{n,50}]

a(2^n) = A025065(n).

a(n) = A057567(A000188(n)). - Andrew Howroyd, May 22 2021

A383106 Numbers k such that A382883(k) != 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 100

Offset: 1

Peter Luschny, Apr 16 2025

nonn

See the comments in A382883.

Cf. A382883, A382943 (complement), A383016, A383017.

Cf. A005117 and similars: A062770, A072774, A085924, A242414, A304449.

V[n_, e_] := If[e == 1, 1, IntegerExponent[n, e]]; f[n_] := f[n] = -DivisorSum[n, V[n, #] * f[#] &, # < n &]; f[1] = 1; Select[Range[100], f[#] != 0 &] (* Amiram Eldar, Apr 29 2025 *)

def A383106List(upto): return [n for n in srange(1, upto) if A382883(n) != 0]

A322525 Numbers such that the list of exponents of their factorization is a palindromic list of primes.

Original entry on oeis.org

2700, 5292, 9000, 13068, 18252, 24300, 24500, 24696, 31212, 38988, 47628, 55125, 57132, 60500, 68600, 84500, 90828, 95832, 103788, 117612, 136125, 144500, 147852, 158184, 164268, 166012, 180500, 181548, 190125, 199692, 218700, 231525, 231868, 238572, 243000, 264500, 266200, 280908, 303372, 325125

Offset: 1

Pierandrea Formusa, Dec 13 2018

nonn

I mean nontrivial palindrome: more than one number and not all equal numbers.

Factorization is meant to produce p1^e1*...*pk^ek, with pi in increasing order.

			9000 is a term as 9000=2^3*3^2*5^3 and the correspondent exponents list [3,2,3] is a palindromic list of primes.

Subsequence of A242414.

aQ[s_] := Length[Union[s]]>1 && AllTrue[s, PrimeQ] && PalindromeQ[s]; Select[Range[1000], aQ[FactorInteger[#][[;;,2]]] &] (* Amiram Eldar, Dec 14 2018 *)

isok(n) = (ve=factor(n)[,2]~) && (Vecrev(ve)==ve) && (#ve>1) && (#Set(ve)>1) && (#select(x->(!isprime(x)), ve) == 0); \\ Michel Marcus, Dec 14 2018

from sympy.ntheory import factorint,isprime
def all_prime(l):
    for i in l:
        if not(isprime(i)): return(False)
    return(True)
def all_equal(l):
    ll=len(l)
    set_l=set(l)
    lsl=list(set_l)
    llsl=len(lsl)
    return(llsl==1)
def pal(l):
    return(l == l[::-1])
n=350000
r=""
lp=[]
lexp=[]
def calc(n):
    global lp,lexp
    a=factorint(n)
    lp=[]
    for p in a.keys():
        lp.append(p)
    lexp=[]
    for exp in a.values():
        lexp.append(exp)
    return
for i in range(4,n):
    calc(i)
    if len(lexp)>1:
        if all_prime(lexp):
            if not(all_equal(lexp)):
                if pal(lexp):
                    r += ","+str(i)
print(r[1:])

cp's OEIS Frontend

A322136 Numbers whose number of prime factors counted with multiplicity exceeds half their sum of prime indices by at least 1.

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A319437 Number of series-reduced palindromic plane trees with n nodes.

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A335402 Numbers m such that the only normal integer partition of m whose run-lengths are a palindrome is (1)^m.

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Formula

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A344417 Number of palindromic factorizations of n.

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Formula

A383106 Numbers k such that A382883(k) != 0.

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SageMath

A322525 Numbers such that the list of exponents of their factorization is a palindromic list of primes.

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Python