A356232 Numbers whose prime indices are all odd and cover an initial interval of odd positive integers.

1, 2, 4, 8, 10, 16, 20, 32, 40, 50, 64, 80, 100, 110, 128, 160, 200, 220, 250, 256, 320, 400, 440, 500, 512, 550, 640, 800, 880, 1000, 1024, 1100, 1210, 1250, 1280, 1600, 1760, 1870, 2000, 2048, 2200, 2420, 2500, 2560, 2750, 3200, 3520, 3740, 4000, 4096, 4400

Offset

1,2

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.

Also positions of first appearances of rows in A356226.

Links

Table of n, a(n) for n=1..51.

Example

The terms together with their prime indices begin:
      1: {}
      2: {1}
      4: {1,1}
      8: {1,1,1}
     10: {1,3}
     16: {1,1,1,1}
     20: {1,1,3}
     32: {1,1,1,1,1}
     40: {1,1,1,3}
     50: {1,3,3}
     64: {1,1,1,1,1,1}
     80: {1,1,1,1,3}
    100: {1,1,3,3}
    110: {1,3,5}
    128: {1,1,1,1,1,1,1}
    160: {1,1,1,1,1,3}
    200: {1,1,1,3,3}
    220: {1,1,3,5}
    250: {1,3,3,3}
    256: {1,1,1,1,1,1,1,1}
    320: {1,1,1,1,1,1,3}
    400: {1,1,1,1,3,3}

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
normQ[m_]:=Or[m=={}, Union[m]==Range[Max[m]]];
Select[Range[1000], normQ[(primeMS[#]+1)/2]&]

Prog

(Python)
from itertools import count
from functools import lru_cache
from sympy import prime, integer_log
from oeis_sequences.OEISsequences import bisection
def A356232(n):
    @lru_cache(maxsize=None)
    def g(x, m): return sum(g(x//(prime((m<<1)+1)**i), m-1) for i in range(1, integer_log(x, prime((m<<1)+1))[0]+1)) if m else x.bit_length()-1
    def f(x):
        c, p = n-1+x, 1
        for k in count(0):
            p *= prime((k<<1)+1)
            if p>x:
                break
            c -= g(x, k)
        return c
    return bisection(f, n, n) # Chai Wah Wu, Mar 21 2026
(Python)
from itertools import islice
from heapq import heappop, heappush
from sympy import primefactors, nextprime
def A356232_gen(): # generator of terms if first n terms of sequence is desired
    h, hset = [2], {2}
    yield 1
    while True:
        yield (m:=heappop(h))
        ps = primefactors(m)
        ps.append(nextprime(max(ps), 2))
        for p in ps:
            mp = m*p
            if mp not in hset:
                heappush(h, mp)
                hset.add(mp)
A356232_list = list(islice(A356232_gen(), 40)) # Chai Wah Wu, Mar 21 2026

Crossrefs

The partitions with these Heinz numbers are counted by A053251.

This is the odd restriction of A055932.

A subset of A066208 (numbers with all odd prime indices).

This is the sorted version of A356603.

These are the positions of first appearances of rows in A356226. Other statistics are:

- length: A287170, firsts A066205

- minimum: A356227

- maximum: A356228

- bisected length: A356229

- standard composition: A356230

- Heinz number: A356231

- positions of first appearances: A356232 (this sequence)

A001221 counts distinct prime factors, with sum A001414.

A001223 lists the prime gaps, reduced A028334.

A003963 multiplies together the prime indices.

A056239 adds up the prime indices, row sums of A112798.

A073491 lists numbers with gapless prime indices, complement A073492.

Cf. A000005, A001222, A061395, A073493, A132747, A137921, A193829, A286470, A356224, A356237.

Sequence in context: A287178 A093547 A068382 * A025612 A335239 A102248

Adjacent sequences: A356229 A356230 A356231 * A356233 A356234 A356235

Keyword

nonn

Author

Gus Wiseman, Aug 20 2022

Status

approved