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 1-4 of 4 results.

A337345 Number of divisors d of n for which A003961(d) > 2*d, where A003961 is fully multiplicative with a(p) = nextprime(p).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 3, 0, 1, 1, 3, 0, 3, 0, 3, 1, 0, 0, 5, 0, 0, 2, 3, 0, 4, 0, 4, 0, 0, 1, 6, 0, 0, 1, 5, 0, 4, 0, 2, 3, 0, 0, 7, 1, 2, 0, 2, 0, 5, 0, 5, 1, 0, 0, 8, 0, 0, 3, 5, 0, 2, 0, 2, 1, 4, 0, 9, 0, 0, 2, 2, 0, 3, 0, 7, 3, 0, 0, 8, 0, 0, 0, 4, 0, 8, 1, 2, 0, 0, 0, 9, 0, 3, 2, 5, 0, 2, 0, 4, 4
Offset: 1

Views

Author

Antti Karttunen, Aug 27 2020

Keywords

Comments

Number of terms of A246282 that divide n.
Number of divisors d of n for which A048673(d) > d.

Links

Crossrefs

Inverse Möbius transform of A252742.
Cf. A003961, A048673, A246282, A337346, A337372 (positions of ones), A341609 (their characteristic function), A341610 (positions of terms > 1), A378658 [= a(A091191(n))], A378662, A378663.
Cf. also A080224, A337541, A341620.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A337345(n) = sumdiv(n,d,A003961(d)>(d+d));

Formula

a(n) = Sum_{d|n} A252742(d).
a(n) = A337346(n) + A252742(n).
From Antti Karttunen, Dec 10 2024: (Start)
a(n) = 1 <=> A341609(n) = 1.
a(n) = A378662(n) + A080224(n) = A378663(n) + A341620(n).
(End)

A337381 Numbers k for which A003973(k) >= 2*sigma(k).

Original entry on oeis.org

6, 8, 9, 12, 14, 15, 16, 18, 20, 21, 24, 27, 28, 30, 32, 35, 36, 40, 42, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 117, 120, 124, 125, 126, 128, 130, 132, 135, 136, 138, 140, 144, 147, 148, 150, 152, 153, 154
Offset: 1

Views

Author

Antti Karttunen, Aug 27 2020

Keywords

Comments

Note that A003973(n) >= sigma(n) for all n. See A336852.
Like the abundancy index (ratio A000203(n)/n), and ratio A003961(n)/n, the ratio A003973(n)/sigma(n) is also multiplicative and > 1 for all n > 1. Thus if the number has a proper divisor that is in this sequence, then the number itself is also. See A337543 for those terms included here, but which have no proper divisor in this sequence. - Antti Karttunen, Aug 31 2020
All terms are in A246282 because A341528(n) < A341529(n) for all n > 1. - Antti Karttunen, Feb 22 2021

Links

Crossrefs

Cf. A000203, A003961, A003973, A336852, A337541, A337542, A341528, A341529.
Cf. A337382 (complement), A337383 (characteristic function).
Subsequences: A337378, A337384, A337386, A337543 (primitive terms).
Subsequence of A246282.

Programs

  • PARI
    A003973(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); sigma(factorback(f)); };
isA337381(n) = (A003973(n)>=2*sigma(n));

A337543 Numbers that are in A337381, but that have no proper divisor in A337381.

Original entry on oeis.org

6, 8, 9, 14, 15, 20, 21, 35, 44, 49, 50, 52, 68, 76, 92, 110, 124, 125, 130, 148, 172, 188, 190, 212, 230, 244, 275, 286, 292, 310, 325, 338, 356, 363, 425, 429, 452, 470, 475, 494, 507, 575, 598, 627, 663, 715, 741, 759, 775, 806, 845, 847, 874, 897, 925, 969, 1001, 1023, 1058, 1075, 1083, 1131, 1173, 1175, 1183, 1209
Offset: 1

Views

Author

Antti Karttunen, Aug 31 2020

Keywords

Comments

Numbers k that are in A337381 [i.e., for which A003973(k) >= 2*A000203(k)], but none of whose proper divisors are in A337381.

Links

Crossrefs

Cf. A000203, A003961, A003973, A337381, A337383, A337541, A337542.
Cf. also A091191, A337372.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A337541(n) = sumdiv(n,d,sigma(A003961(d))>=2*sigma(d));
isA337543(n) = (1==A337541(n));

Formula

{k: 1==A337541(k)}.

A337542 Number of proper divisors d of n for which sigma(A003961(d)) >= 2*sigma(d), where sigma is the sum of divisors, and A003961(x) shifts the prime factorization of x one step towards larger primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 3, 0, 0, 1, 1, 0, 2, 0, 2, 0, 0, 0, 4, 0, 0, 0, 2, 0, 3, 0, 0, 2, 0, 0, 5, 0, 0, 0, 0, 0, 4, 0, 3, 0, 0, 0, 5, 0, 0, 2, 3, 0, 1, 0, 0, 0, 2, 0, 7, 0, 0, 1, 0, 0, 1, 0, 4, 2, 0, 0, 6, 0, 0, 0, 2, 0, 6, 0, 0, 0, 0, 0, 7, 0, 2, 1, 2, 0, 1, 0, 2, 3
Offset: 1

Views

Author

Antti Karttunen, Aug 31 2020

Keywords

Comments

Number of terms of A337381 less than n that divide n.

Links

Crossrefs

Cf. A000203, A003961, A337381, A337383, A337541, A337543.
Cf. also A337346.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A337542(n) = sumdiv(n,d,(dA003961(d))>=2*sigma(d));

Formula

a(n) = Sum_{d|n, dA337383(d).
a(n) = A337541(n) - A337383(n).
Showing 1-4 of 4 results.

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