A047318 -id:A047318 - cp's OEIS frontend

A308065 Nonnegative integers that are not the sum of two refactorable numbers whose difference is refactorable.

0, 1, 2, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74

Offset: 1

Wesley Ivan Hurt, Jul 27 2019

nonn

Differs from A047318 (not congruent to 3 (mod 7)) starting at a(n=22) = 24 = 3*7 + 3 which is in this sequence but not in A047318. - M. F. Hasler, Jun 30 2025

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Refactorable Number

Cf. A033950, A309347.

Different from A047318 (not congruent to 3 modulo 7) and A057904.

notref:= proc(n) option remember; n mod numtheory:-tau(n) <> 0 end proc:
filter:= proc(n)
  andmap(t -> notref(t) or notref(n-t) or notref(n-2*t), [$1 .. (n-1)/2])
end proc:
select(filter, [$0..100]); # Robert Israel, Jul 29 2025

Flatten[Table[If[Sum[(1 - Ceiling[(n - 2 i)/DivisorSigma[0, n - 2 i]] + Floor[(n - 2 i)/DivisorSigma[0, n - 2 i]]) (1 - Ceiling[i/DivisorSigma[0, i]] + Floor[i/DivisorSigma[0, i]]) (1 - Ceiling[(n - i)/DivisorSigma[0, n - i]] + Floor[(n - i)/DivisorSigma[0, n - i]]), {i, Floor[(n - 1)/2]}] == 0, n, {}], {n, 0, 100}]]

A057904 Positive integers that are not the sum of exactly three positive cubes.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 63, 64, 65, 67, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 88

Offset: 1

Eric W. Weisstein

nonn

Differs from A047318 = numbers not congruent to 3 modulo 7: for example, A047318(26) = 29 is not in this sequence. - M. F. Hasler, Jun 30 2025

			3 = 1^3 + 1^3 + 1^3, therefore 3 is not in this sequence. Similarly,
10 = 1^3 + 1^3 + 2^3, therefore 10 is not in this sequence.

T. D. Noe, Table of n, a(n) for n=1..1000
Eric Weisstein's World of Mathematics, Cubic Number.

Cf. A003072 (complement).

Cf. A047318 (not congruent to 3 mod 7), A308065 (not the same).

Select[Range[100], Count[ PowersRepresentations[#, 3, 3], pr_List /; FreeQ[pr, 0]] == 0 &] (* Jean-François Alcover, Oct 31 2012 *)

select( {is_A057904(n)=n<3 || !for(c=sqrtnint(n\/3,3),sqrtnint(n-2,3), isA003325(n-c^3)&&return)}, [1..99]) \\ M. F. Hasler, Jun 30 2025

A025456(a(n)) = 0. - Reinhard Zumkeller, Apr 23 2009

A298161 Nonnegative numbers n such that for any k > 0, n + k is not a multiple of prime(k) (where prime(k) denotes the k-th prime).

Original entry on oeis.org

0, 8, 18, 26, 36, 54, 56, 74, 84, 86, 134, 140, 156, 168, 170, 174, 194, 200, 216, 224, 236, 240, 246, 260, 300, 308, 324, 326, 366, 368, 386, 390, 414, 420, 440, 456, 464, 476, 494, 498, 518, 536, 560, 564, 576, 590, 594, 624, 630, 650, 660, 678, 698, 708

Offset: 1

Rémy Sigrist, Jan 14 2018

nonn

Equivalently, these are the numbers n >= 0 such that A298155(n) = 1.
Equivalently, these are the numbers n >= 0 such that the diagonal of A060175 starting at A060175(n+1, 1) contains only zeros.
All terms are even.
This sequence is a subsequence of A005843, A007494, A047207 and A047318.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A005843, A007494, A007814, A007949, A047207, A047318, A060175, A112765, A214411, A298155.

filter:= proc(n) local p, k;
  p:= 1:
  for k from 1 do
    p:= nextprime(p);
    if p > n+k then return true
    elif n+k mod p = 0 then return false
    fi
  od
end proc:
select(filter, [seq(i,i=0..1000,2)]); # Robert Israel, Jan 16 2018

A007814(a(n) + 1) = 0.
A007949(a(n) + 2) = 0.
A112765(a(n) + 3) = 0.
A214411(a(n) + 4) = 0.

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A308065 Nonnegative integers that are not the sum of two refactorable numbers whose difference is refactorable.

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A057904 Positive integers that are not the sum of exactly three positive cubes.

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A298161 Nonnegative numbers n such that for any k > 0, n + k is not a multiple of prime(k) (where prime(k) denotes the k-th prime).

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