cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A341620 Number of nondeficient divisors of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 3, 0, 2, 0, 0, 0, 5, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 6, 0, 0, 0, 0, 0, 2, 0, 3, 0, 0, 0, 5, 0, 0, 0, 1, 0, 4, 0, 0, 0, 0, 0, 5, 0, 0, 0, 2, 0, 2, 0, 1, 0, 0, 0, 6, 0, 0, 0, 3, 0, 2, 0, 0, 0, 0, 0, 8
Offset: 1

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Author

Antti Karttunen, Feb 21 2021

Keywords

Comments

Number of nondeficient numbers (A023196) dividing n.

Links

Crossrefs

Cf. A000005, A006039 (positions of ones), A023196, A080224, A080225, A080226, A294927, A294936, A337690, A341619.
Differs from a derived sequence A341624 for the first time at n=120, where a(120)=8, while A341624(120)=1.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, 1 &, DivisorSigma[1, #] >= 2*# &]; Array[a, 120] (* Amiram Eldar, Feb 22 2021 *)
  • PARI
    A294936(n) = (sigma(n, -1)>=2); \\ From A294936.
A341620(n) = sumdiv(n,d,A294936(d));
  • PARI
    A341620(n) = sumdiv(n,d,(sigma(d)>=(2*d)));

Formula

a(n) = Sum_{d|n} A294936(d).
a(n) = A294927(n) + A294936(n).
a(n) = A080224(n) + A080225(n) = A000005(n) - A080226(n).
a(n) >= A337690(n) for all n.
a(n) = 1 iff A341619(n) = 1.

A341618 a(n) = 0 if n is not a primitive nondeficient number, otherwise a(n) is the number of primitive nondeficient divisors of the last number in the iteration x -> A003961(x) (starting from x=n) for which that count (A337690) is nonzero.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1
Offset: 1

Views

Author

Antti Karttunen, Feb 21 2021

Keywords

Links

Crossrefs

Differs from A337690 for the first time at n=120, where a(120)=1, while A337690(120)=2.
Cf. A006039, A341508, A341619, A341624.

Programs

  • PARI
    A341619(n) = if(sigma(n) < (2*n), 0, fordiv(n, d, if((d= 2*d), return(0))); (1)); \\ After code in A071395
A337690(n) = sumdiv(n,d,A341619(d));
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A341618(n) = { my(t, u=0); while((t=A337690(n))>0, u=t; n = A003961(n)); (u); };

A342669 Even numbers which are either primitively nondeficient (A006039), or become such after applying prime shift A003961 some number of times to them.

Original entry on oeis.org

6, 20, 28, 70, 88, 104, 120, 180, 272, 300, 304, 368, 420, 464, 496, 504, 550, 572, 630, 650, 660, 748, 780, 836, 924, 990, 1020, 1050, 1092, 1140, 1170, 1184, 1312, 1376, 1380, 1430, 1470, 1504, 1650, 1696, 1740, 1860, 1870, 1888, 1952, 2002, 2090, 2210, 2220, 2310, 2460, 2470, 2530, 2580, 2584, 2730, 2820, 2856, 2990
Offset: 1

Views

Author

Antti Karttunen, Mar 20 2021

Keywords

Comments

Even numbers k for which A341624(k) = 1.
Even numbers whose closure under map x -> A003961(x) contains a primitive non-deficient number (one of the terms of A006039). Shifting each term k exactly A336835(k)-1 times with A003961 towards larger primes gives those numbers, but not in monotonic order, producing instead a permutation of A006039.
Sequence 2*A246277(A006039(.)), sorted into ascending order.
If there are any two terms, x and y, such that the other is a multiple of the other, then A336835(x) != A336835(y), and furthermore, for any term k present here, for all its proper divisors (d|k, dA336835(d) < A336835(k), in other words, they reach the deficiency earlier (by prime shifting) than k itself.

Examples

			For n = 120 = 2^3 * 3 * 5, A341620(120) = 8, so it is not primitive nondeficient. However, prime-shifting it once gives A003961(120) = 945 = 3^3 * 5 * 7, which is one of the terms of A006039 as A341620(945) = 1. Therefore 120 is included in the sequence.

Links

Crossrefs

Cf. A000396, A006039 (even terms of these form a subsequence).
Cf. A003961, A246277, A336835, A341605/A341606, A341620, A341624.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A341620(n) = sumdiv(n,d,(sigma(d)>=(2*d)));
A341624(n) = { my(t, u=0); while((t=A341620(n))>0, u=t; n = A003961(n)); (u); };
isA342669(n) = (!(n%2)&&(1==A341624(n)));
Showing 1-3 of 3 results.

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