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.

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A246281 Numbers k for which A003961(k) < 2*k; Numbers n such that if n = product_{k >= 1} (p_k)^(c_k), then product_{k >= 1} (p_{k+1})^(c_k) < 2*n, where p_k indicates the k-th prime, A000040(k).

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 19, 22, 23, 25, 26, 29, 31, 33, 34, 37, 38, 41, 43, 46, 47, 51, 53, 55, 58, 59, 61, 62, 65, 67, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 93, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115, 118, 119, 121, 122, 123, 127, 129, 131, 133, 134, 137, 139
Offset: 1

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Author

Antti Karttunen, Aug 24 2014

Keywords

Comments

Numbers n such that A003961(n) < 2*n.
Numbers n such that A048673(n) <= n.
All primes (A000040) are members. (Cf. Bertrand's postulate).
All terms are deficient (in A005100). See A286385. - Antti Karttunen, Aug 27 2020

Examples

			1 is present, as 1 = empty product and 1 < 2.
2 = p_1 is in the sequence, as p_2 = 3 and 3/2 < 2.
4 = p_1 * p_1 is not a member, as p_2 * p_2 = 3*3 = 9, and 9/4 > 2.
22 = 2*11 = p_1 * p_5 is a member, as p_2 * p_6 = 39, and 39/22 < 2.

Links

Crossrefs

Complement: A246282.
Union of A246351 and A048674.
Subsequence: A000040.
Subsequence of A005100.
Positions of zeros in A252742, in A336836, and in A337345.
Positions of negative terms in A252748.
Cf. A003961, A048673, A246361, A286385.

Programs

  • PARI
    A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i,1] = nextprime(f[i,1]+1)); factorback(f);
isA246281(n) = (A003961(n) < (n+n));
n = 0; i = 0; while(i < 10000, n++; if(isA246281(n), i++; write("b246281.txt", i, " ", n)));
  • Scheme
    ;; With Antti Karttunen's IntSeq-library.
(define A246281 (MATCHING-POS 1 1 (lambda (n) (<= (A048673 n) n))))

Extensions

A new shorter version of name prepended by Antti Karttunen, Aug 27 2020

A336835 Number of iterations of x -> A003961(x) needed before the result is deficient (sigma(x) < 2x), when starting from x=n.

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

Views

Author

Antti Karttunen, Aug 07 2020

Keywords

Comments

It holds that a(n) <= A336836(n) for all n, because sigma(n) <= A003961(n) for all n (see A286385 for a proof).
The first 3 occurs at n = 19399380, the first 4 at n = 195534950863140268380. See A336389.
If x and y are relatively prime (i.e., gcd(x,y) = 1), then a(x*y) >= max(a(x),a(y)). Compare to a similar comment in A336915.

Examples

			For n = 120, sigma(120) = 360 >= 2*120, thus 120 is not deficient, and we get the next number by applying the prime shift, A003961(120) = 945, and sigma(945) = 1920 >= 945*2, so neither 945 is deficient, so we prime shift once again, and A003961(945) = 9625, which is deficient, as sigma(9625) = 14976 < 2*9625. Thus after two iteration steps we encounter a deficient number, and therefore a(120) = 2.

Links

Crossrefs

Cf. A003961, A005100, A005940, A023196, A046523, A047802, A071364, A108951, A286385, A294934, A336834, A337474, A337475.
Cf. A336389 (position of the first occurrence of a term >= n).
Differs from A294936 for the first time at n=120.
Cf. also A246271, A252459, A336836 and A336915 for similar iterations.

Programs

  • Mathematica
    Array[-1 + Length@ NestWhileList[If[# == 1, 1, Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}]] &, #, DivisorSigma[1, #] >= 2 # &] &, 120] (* Michael De Vlieger, Aug 27 2020 *)
  • PARI
    A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A336835(n) = { my(i=0); while(sigma(n) >= (n+n), i++; n = A003961(n)); (i); };

Formula

If A294934(n) = 1, a(n) = 0, otherwise a(n) = 1 + a(A003961(n)).
From Antti Karttunen, Aug 21-Sep 01 2020: (Start)
For all n >= 1,
a(A046523(n)) >= a(n).
a(A071364(n)) >= a(n).
a(A108951(n)) = A337474(n).
a(A025487(n)) = A337475(n).
(End)

A372562 Square array A(n, k) = A246278(1+n, k) - 2*A246278(n, k), read by falling antidiagonals, where A246278 is the prime shift array.

Original entry on oeis.org

-1, 1, -1, 3, 7, -3, 11, 5, -1, -3, 1, 71, 7, 23, -9, 21, 13, 93, -11, -73, -9, 5, 85, -19, 645, -65, -49, -15, 49, -1, 189, 5, -465, -119, -217, -15, 39, 463, -11, 495, -127, 519, -209, -193, -17, 23, 95, 1151, -29, -273, -103, -2967, -207, -217, -27, -5, 149, 357, 9839, -119, -255, -231, -1551, -435, -721, -25
Offset: 1

Views

Author

Antti Karttunen, May 21 2024

Keywords

Comments

For all k >= 1, A(1+A336836(2*k), k) < 0, and it is the topmost negative number of the column k.
In those columns k where 2k is in A104210, 6, 12, 18, 24, ..., there is present a "prime thread" of successive primes (see the example).

Examples

			The top left corner of the array:
k=    1     2     3      4     5      6     7       8      9     10    11      12
2k=   2     4     6      8    10     12    14      16     18     20    22      24
--+-------------------------------------------------------------------------------
1 |  -1,    1,    3,    11,    1,    21,    5,     49,    39,    23,   -5,     87,
2 |  -1,    7,    5,    71,   13,    85,   -1,    463,    95,   149,    7,    605,
3 |  -3,   -1,    7,    93,  -19,   189,  -11,   1151,   357,    87,  -37,   2023,
4 |  -3,   23,  -11,   645,    5,   495,  -29,   9839,   165,   783,  -13,   9757,
5 |  -9,  -73,  -65,  -465, -127,  -273, -119,   -721,    39,  -903, -129,   2743,
6 |  -9,  -49, -119,   519, -103,  -255, -105,  26399, -1377,   225, -227,  18649,
7 | -15, -217, -209, -2967, -231, -2679, -397, -36721, -2223, -2825, -351, -28937,
...
Terms of column 9: 39 (3*13), 95 (5*19), 357 (3*7*17), 165 (3*5*11), 39 (3*13), -1377 (- 3^4 * 17), -2223 (- 3^2 * 13 * 19), ..., show an ascending "prime thread" (3, 5, 7, 11, 13, 17, 19, ...) that is mentioned in comments.

Links

Crossrefs

Cf. A003961, A104210, A246278, A252748, A336836.
Cf. A062234 (column 1 when values are negated).
Cf. also A252750 (same terms in irregular triangle), A372563.
See also conjecture 1 in A349753.

Programs

  • PARI
    up_to = 66;
A246278sq(row,col) = if(1==row,2*col, my(f = factor(2*col)); for(i=1, #f~, f[i,1] = prime(primepi(f[i,1])+(row-1))); factorback(f));
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A252748(n) = (A003961(n) - (2*n));
A372562sq(row,col) = A252748(A246278sq(row,col));
A372562list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A372562sq(col,(a-(col-1))))); (v); };
v372562 = A372562list(up_to);
A372562(n) = v372562[n];

Formula

A(n,k) = A252748(A246278(n,k)).
Showing 1-3 of 3 results.

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