A105441 -id:A105441 - cp's OEIS frontend

A110473 Integers not in "array with primes".

1, 9, 15, 18, 21, 25, 27, 30, 33, 35, 36, 39, 42, 45, 49, 50, 51, 54, 55, 57, 60, 63, 65, 66, 69, 70, 72, 75, 77, 78, 81, 84, 85, 87, 90, 91, 93, 95, 98, 99, 100, 102, 105, 108, 110, 111, 114, 115, 117, 119, 120, 121, 123, 125, 126, 129, 130, 132, 133, 135, 138, 140

Offset: 1

Eric Angelini and Alexandre Wajnberg, Sep 08 2005

This is 1 + A105441.

			"Array with primes":
consider this array where the first column is made of all prime numbers p and the n-th term of a row is two times the previous one on the same row [p(n)=p*2^n]:
p p*2 p*4 p*8 p*16 p*32 p*64 p*128 p*256 p*512 ...
2 4 8 16 32 64 128 256 512 1024 ...
3 6 12 24 48 96 192 384 768 1536 ...
5 10 20 40 80 160 320 640 1280 2560 ...
7 14 28 56 112 224 448 896 1792 3584 ...
11 22 44 88 176 352 704 1408 2816 5632 ...
13 26 52 104 208 416 832 1664 3328 6656 ...
17 34 68 136 272 544 1088 2176 4352 8704 ...
19 38 76 152 304 608 1216 2432 4864 9728 ...
23 46 92 184 368 736 1472 2944 5888 11776 ...
29 58 116 232 464 928 1856 3712 7424 14848 ...
31 62 124 248 496 992 1984 3968 7936 15872 ...
37 74 148 296 592 1184 2368 4736 9472 18944 ...
41 82 164 328 656 1312 2624 5248 10496 20992 ...
43 86 172 344 688 1376 2752 5504 11008 22016 ...
47 94 188 376 752 1504 3008 6016 12032 24064 ...
53 106 212 424 848 1696 3392 6784 13568 27136 ...
...

# The function tries to represent the idea; an
# implementation via A105441 would be more efficient.
A110473 := proc(n) local T;
T := proc(n) local A,i,l,h,k; A := [];
k := simplify(floor(log[2](n+1))-1);
for i from 0 to k do
    l := iquo(n,2^(i+1))+1; h := iquo(n,2^i);
    A := [op(A), select(isprime, [$l..h])] od;
convert(A,set) end;
`if`(T(n-1) minus T(n) = {}, n, NULL) end:
seq(A110473(i),i=1..140); # Peter Luschny, Mar 18 2013

With[{nn=150},Complement[Range[nn],Sort[Flatten[Table[Prime[n]*2^Range[ 0,nn],{n,nn}]]]]] (* Harvey P. Dale, Oct 18 2013 *)

A105442 Numbers with at least two odd prime factors (not necessarily distinct) such that in binary representation all divisors of n are contained in n.

Original entry on oeis.org

55, 215, 407, 1403, 1681, 3223, 3362, 3415, 6724, 13448, 13655, 15487, 25751, 80089, 146621, 160178, 218455, 237169, 320356, 445663, 464711, 474338, 873815, 948676, 1662743, 1897352, 1932377, 1975531, 2484187, 3223001, 3410639, 3872639

Offset: 1

Reinhard Zumkeller, Apr 09 2005

nonn

Intersection of A093641 and A105441;

A087436(a(n)) > 1.

Cf. A093642, A093641.

A344338 Smallest number that is the sum of two or more consecutive positive n-th powers in more than one way.

Original entry on oeis.org

9, 365, 33075

Offset: 1

Ilya Gutkovskiy, May 15 2021

a(4) > 10^24. - Bert Dobbelaere, May 16 2021

Conjecture: no terms exist for n >= 4. - Jon E. Schoenfield, May 16 2021

			9 = 2 + 3 + 4 = 4 + 5.
365 = 10^2 + 11^2 + 12^2 = 13^2 + 14^2.
33075 = 11^3 + 12^3 + 13^3 + 14^3 + 15^3 + 16^3 + 17^3 + 18^3 + 19^3 = 15^3 + 16^3 + 17^3 + 18^3 + 19^3 + 20^3.

Cf. A062681, A062682, A091414, A105441, A230718.

N=3 # <== Adapt here
import heapq
sigma=1+2**N
h=[(sigma,1,2)]
nextcount=3
oldv,olds,oldl=0,0,0
while True:
    (v,s,l)=heapq.heappop(h)
    if v==oldv:
        break
    if v>=sigma:
        sigma += nextcount**N
        heapq.heappush(h, (sigma,1,nextcount))
        nextcount+=1
    oldv,olds,oldl = v,s,l
    v-=s**N ; s+=1 ; l+=1 ;    v+=l**N
    heapq.heappush(h,(v,s,l))
print("a(%d) = %d = sum(i^%d, i=%d..%d) = sum(i^%d, i=%d..%d)"%
    (N,v,N,olds,oldl,N,s,l))
# Bert Dobbelaere, May 16 2021

A362983 Number of prime factors of n (with multiplicity) that are greater than the least.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 18 2023

nonn

			The prime factorization of 360 is 2*2*2*3*3*5, with factors greater than the least 3*3*5, so a(360) = 3.

Positions of 0's are A000961.
Positions of numbers > 0 are A024619.
Positions of first appearances appear to be A099856.
For "less than greatest" instead of "greater than least" we have A325226.
For multiplicities instead of parts we have A363131.
A027746 lists prime factors, A112798 indices, A124010 exponents.
A047966 counts uniform partitions, ranks A072774.
A363128 counts partitions with more than one non-mode, complement A363129.
Cf. A001221, A001222, A052126, A053585, A061395, A064989, A071178, A105441, A307517, A325230.

Table[PrimeOmega[n]-If[n==1,0,FactorInteger[n][[1,2]]],{n,30}]

a(n) = A001222(n) - A067029(n).

a(n) = A001222(A028234(n)).

cp's OEIS Frontend

A110473 Integers not in "array with primes".

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A105442 Numbers with at least two odd prime factors (not necessarily distinct) such that in binary representation all divisors of n are contained in n.

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A344338 Smallest number that is the sum of two or more consecutive positive n-th powers in more than one way.

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Python

A362983 Number of prime factors of n (with multiplicity) that are greater than the least.

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Mathematica

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