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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A336851 a(n) = sigma(A003961(n)) - A003961(n), where A003961 is a prime shift towards larger primes, sigma is the sum of divisors.

Original entry on oeis.org

0, 1, 1, 4, 1, 9, 1, 13, 6, 11, 1, 33, 1, 15, 13, 40, 1, 49, 1, 41, 17, 17, 1, 105, 8, 21, 31, 57, 1, 87, 1, 121, 19, 23, 19, 178, 1, 27, 23, 131, 1, 123, 1, 65, 73, 33, 1, 321, 12, 81, 25, 81, 1, 249, 21, 183, 29, 35, 1, 309, 1, 41, 97, 364, 25, 141, 1, 89, 35, 153, 1, 565, 1, 45, 97, 105, 25, 177, 1, 401, 156, 47
Offset: 1

Views

Author

Antti Karttunen, Aug 05 2020

Keywords

Comments

Even terms occur on square n, odd terms on nonsquare n.
Numbers k such that a(k) = 2^e for e >= 1, are: 4, 25, 841, 12769, 66896041, etc., i.e., terms of A073715 squared.

Links

Crossrefs

Cf. A000203, A001065, A003961, A003973, A073715, A252748, A286385, A326042, A336852.
Cf. A336850 [= gcd(a(n),A003961(n))].

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336851(n) = (sigma(A003961(n)) - A003961(n));

Formula

a(n) = A003973(n) - A003961(n) = A000203(A003961(n)) - A003961(n).
a(n) = A001065(A003961(n)).
a(n) = A336852(n) - A286385(n).

Extensions

Comments edited by Antti Karttunen, Jul 03 2023

A285705 a(n) = 2*n - A285703(n) = 2*n - A000203(A064216(n)).

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 2, 4, 4, 2, 4, 4, 13, 13, 6, 2, 10, 12, 6, 4, 4, 2, 18, 4, 19, 10, 6, 24, 4, 6, 2, 22, 18, 6, 10, 4, 2, 37, 30, 6, 51, 4, 30, 16, 6, 20, 4, 24, 8, 44, 4, 2, 34, 4, 2, 16, 4, 36, 34, 36, 65, 10, 86, 14, 4, 4, 26, 76, 6, 2, 10, 48, 50, 55, 10, 2, 56, 36, 6, 16, 42, 6, 70, 4, 37, 46, 6, 98, 16, 6, 2, 4, 58, 76, 100, 10, 2, 52, 4, 2, 16, 60, 54
Offset: 1

Views

Author

Antti Karttunen, Apr 26 2017

Keywords

Comments

Question: Are all terms positive? - Yes, they are, see A286385. (Note added Jul 24 2022).
For listening: fast tempo and percussive instrument, default "modulo 88" pitch mapping, all 10000 terms.

Links

Crossrefs

Cf. A000203, A033879, A064989, A064216, A151799, A285703, A285704, A286385.

Programs

  • Mathematica
    Table[2 n - DivisorSigma[1, #] &@ If[n == 1, 1, Apply[Times, FactorInteger[2 n - 1] /. {p_, e_} /; p > 2 :> NextPrime[p, -1]^e]], {n, 103}] (* Michael De Vlieger, Apr 26 2017 *)
  • PARI
    A064989(n) = { my(f = factor(n)); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A285705(n) = (n+n - sigma(A064989(n+n-1))); \\ Antti Karttunen, Jul 24 2022
  • Scheme
    (define (A285705 n) (- (* 2 n) (A285703 n)))

Formula

a(n) = 2*n - A285703(n) = 2*n - A000203(A064216(n)).
a(n) = 1 + A286385(A064216(n)). - Antti Karttunen, Jul 24 2022
Sum_{k=1..n} a(k) ~ c * n^2, where c = 1 - Product_{p prime} (p^3/((p+1)*(p^2-q(p)))) = 0.1831523243..., where q(p) = prevprime(p) (A151799) if p > 2 and q(2) = 1. - Amiram Eldar, Dec 21 2023

A326182 Numbers for which A003961(n) <= A064989(sigma(A003961(n))).

Original entry on oeis.org

1, 4, 9, 36, 64, 169, 256, 576, 625, 676, 729, 1024, 1369, 1521, 1600, 2304, 2500, 2809, 2916, 3136, 4096, 4489, 5184, 5476, 5625, 6084, 6561, 6889, 7396, 7744, 9216, 9409, 10816, 11236, 12321, 14400, 14641, 16129, 17956, 19044, 22500, 23104, 24964, 25281, 26244, 26569, 27556, 27889, 28224, 28561, 33856, 36864, 37636, 40000, 40401
Offset: 1

Views

Author

Antti Karttunen, Jun 16 2019

Keywords

Comments

Not all terms are squares: the first nonsquare occurs as a(1427) = 97560000 = 2^6 * 3^2 * 5^4 * 271, and among the first 4011 terms (those <= 2^30), there are 24 of them, see A326183.

Links

Crossrefs

Cf. A000203, A000290, A003961, A064989, A286385, A326042, A326183 (nonsquares present).

Programs

  • PARI
    A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
isA326182(n) = { my(u=A003961(n)); (u <= A064989(sigma(u))); };

A337549 a(n) = A003972(n) - n.

Original entry on oeis.org

0, 0, 1, 2, 1, 2, 3, 10, 11, 2, 1, 12, 3, 6, 9, 38, 1, 22, 3, 16, 19, 2, 5, 48, 17, 6, 73, 32, 1, 18, 5, 130, 15, 2, 25, 84, 3, 6, 25, 68, 1, 38, 3, 28, 75, 10, 5, 168, 61, 34, 21, 44, 5, 146, 17, 124, 31, 2, 1, 84, 5, 10, 137, 422, 31, 30, 3, 40, 43, 50, 1, 288, 5, 6, 93, 56, 43, 50, 3, 244, 419, 2, 5, 156, 23, 6, 33
Offset: 1

Views

Author

Antti Karttunen, Sep 21 2020

Keywords

Comments

Möbius transform of A286385.

Links

Crossrefs

Cf. A003961, A003972, A286385, A336853.

Programs

  • PARI
    A003972(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); eulerphi(factorback(f)); };
A337549(n) = (A003972(n) - n);

Formula

a(n) = A003972(n) - n.

A349756 Numbers k such that the odd part of sigma(k) is equal to gcd(sigma(k), A003961(k)), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

Original entry on oeis.org

1, 2, 3, 6, 7, 14, 20, 21, 24, 27, 31, 42, 54, 57, 60, 62, 93, 114, 120, 127, 140, 160, 168, 186, 189, 216, 217, 220, 237, 254, 264, 301, 378, 381, 399, 408, 420, 434, 460, 474, 480, 513, 540, 552, 602, 620, 651, 660, 744, 762, 792, 798, 837, 840, 889, 903, 940, 1026, 1080, 1120, 1128, 1140, 1302, 1320, 1380, 1392, 1512
Offset: 1

Views

Author

Antti Karttunen, Dec 03 2021

Keywords

Comments

Numbers k for which A161942(k) = A342671(k).
From Antti Karttunen, Jul 23 2022: (Start)
Numbers k such that k is a multiple of A350073(k).
For any square s in this sequence, A349162(s) = 1, i.e. sigma(s) divides A003961(s), and also A286385(s). Question: Is 1 the only square in this sequence? (see the conjecture in A350072).
If both x and y are terms and gcd(x, y) = 1, then x*y is also present.
After 2, the only primes present are Mersenne primes, A000668.
(End)

Links

Crossrefs

Positions of 1's in A348992.
Positions where the powers of 2 (A000079) occur in A349162.
Cf. A000203, A003961, A161942, A286385, A342671, A350072, A350073, A355946 (characteristic function).
Cf. A000668, A046528 (subsequences).
Cf. also A348943.

Programs

  • Mathematica
    f[p_, e_] := NextPrime[p]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; oddpart[n_] := n/2^IntegerExponent[n, 2]; q[n_] := oddpart[(sigma = DivisorSigma[1, n])] == GCD[sigma, s[n]]; Select[Range[1500], q] (* Amiram Eldar, Dec 04 2021 *)
  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A355946(n) = { my(s=sigma(n)); !(A003961(n)%((s>>=valuation(s,2)))); };
isA349756(n) = A355946(n);

A378981 a(n) = (A003961(n)-sigma(n)) mod (A003961(n)-2*n), where A003961 is fully multiplicative with a(prime(i)) = prime(i+1).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 5, 0, 1, 17, 3, 4, 1, 1, 1, 36, 3, 21, 10, 3, 5, 75, 0, 0, 14, 0, 1, 33, 5, 1, 0, 3, 1, 134, 3, 2, 1, 99, 1, 69, 3, 4, 12, 0, 5, 281, 18, 7, 2, 6, 5, 255, 0, 177, 0, 3, 1, 147, 5, 2, 22, 1, 2, 51, 3, 10, 0, 87, 1, 480, 5, 9, 26, 12, 3, 87, 3, 381, 41, 3, 5, 271, 25, 9, 16, 171, 7, 291, 0, 16, 0, 15, 12
Offset: 1

Views

Author

Antti Karttunen, Dec 12 2024

Keywords

Links

Crossrefs

Cf. A000203, A003961, A252748, A286385.
Cf. A378980 (positions of 0's), A349753 (positions of 0's at odd n).

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A378981(n) = ((A003961(n)-sigma(n))%((2*n)-A003961(n)));

Formula

a(n) = A286385(n) mod -A252748(n) = (A003961(n)-A000203(n)) mod ((2*n)-A003961(n)).

A336836 Number of iterations of x -> A003961(x) needed before A003961(x) < 2x, when starting from x=n, or -1 if such a number is never reached.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Aug 07 2020

Keywords

Comments

Starting from n, the number of prime shifts needed before a term of A246281 is reached.
It holds that a(n) >= A336835(n) for all n, because sigma(n) <= A003961(n) for all n (see A286385 for a proof).
Note that in contrast to abundancy used in A336835, the condition [A003961(x) > 2x] (= A252742) is not monotonic when iterating with A003961. For example, we have A003961(9) = 25 > 2*9, A003961(25) = 49 < 2*25, and then again A003961(49) = 121 > 2*49.
Question: Is the escape clause necessary in the definition?

Links

Crossrefs

Cf. A246281 (positions of zeros, numbers k for which A003961(k) < 2*k).
Cf. A005940, A252742, A252743, A286385.
Cf. also A246271, A252459, A336835 for similar iterations.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336836(n) = for(i=0,oo,my(n2 = n+n); n = A003961(n); if(n < n2, return(i)));

A341615 Numbers k for which sigma(k) < 2k < A003961(k).

Original entry on oeis.org

4, 8, 9, 10, 14, 15, 16, 21, 27, 32, 35, 39, 44, 45, 49, 50, 52, 57, 63, 64, 68, 69, 75, 76, 81, 91, 92, 98, 99, 105, 110, 116, 117, 124, 125, 128, 130, 135, 136, 147, 148, 152, 153, 154, 164, 165, 170, 171, 172, 175, 182, 184, 188, 189, 190, 195, 207, 212, 225, 230, 231, 232, 236, 238, 242, 243, 244, 245, 248, 250, 255, 256
Offset: 1

Views

Author

Antti Karttunen, Feb 22 2021

Keywords

Comments

Deficient numbers in A246282.
Not all odd terms are in A337372.

Crossrefs

Intersection of A005100 and A246282. Subsequence of A341614.
Cf. A341613 (characteristic function).
Cf. A000203, A003961, A252748, A286385, A326057, A326134, A337372.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA341615(n) = A341613(n);

A351547 a(n) = sigma(n) / A351545(n).

Original entry on oeis.org

1, 1, 4, 7, 6, 4, 8, 15, 13, 2, 12, 28, 14, 8, 24, 31, 18, 13, 20, 6, 32, 4, 24, 12, 31, 14, 40, 56, 30, 8, 32, 63, 48, 2, 48, 91, 38, 20, 56, 90, 42, 32, 44, 84, 78, 8, 48, 124, 57, 31, 72, 98, 54, 40, 72, 120, 16, 10, 60, 24, 62, 32, 104, 127, 12, 16, 68, 14, 96, 16, 72, 195, 74, 38, 124, 140, 96, 56, 80, 186, 121
Offset: 1

Views

Author

Antti Karttunen, Feb 16 2022

Keywords

Links

Crossrefs

Cf. A000203, A003961, A351545, A351546, A354997.
Cf. A286385, A342671, A349161, A349162.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A351547(n) = { my(s=sigma(n),f=factor(s),u=A003961(n)); s/prod(k=1,#f~,if(!(u%f[k,1]) && (f[k,2]>=valuation(u,f[k,1])), f[k,1]^f[k,2], 1)); };

Formula

a(n) = A000203(n) / A351545(n).
a(n) = A351546(n) * A354997(n). - Antti Karttunen, Jul 09 2022

A372563 Square array A(n, k) = A246278(1+n, k) - sigma(A246278(n, k)), read by falling antidiagonals, where A246278 is the prime shift array.

Original entry on oeis.org

0, 2, 1, 3, 12, 1, 12, 11, 18, 3, 3, 85, 29, 64, 1, 17, 23, 187, 47, 36, 3, 9, 97, 19, 931, 53, 106, 1, 50, 17, 291, 75, 733, 71, 54, 3, 36, 504, 35, 889, 31, 2533, 77, 148, 5, 21, 121, 1620, 65, 1011, 111, 1639, 187, 288, 1, 3, 171, 505, 11840, 59, 2197, 119, 4927, 179, 90, 5
Offset: 1

Views

Author

Antti Karttunen, May 21 2024

Keywords

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  | 0,   2,   3,    12,   3,    17,   9,     50,    36,    21,   3,     75,
2  | 1,  12,  11,    85,  23,    97,  17,    504,   121,   171,  29,    635,
3  | 1,  18,  29,   187,  19,   291,  35,   1620,   505,   265,  25,   2525,
4  | 3,  64,  47,   931,  75,   889,  65,  11840,   795,  1259,  93,  12503,
5  | 1,  36,  53,   733,  31,  1011,  59,  12456,  1561,   817,  89,  16853,
6  | 3, 106,  71,  2533, 111,  2197, 157,  52580,  1839,  2987, 107,  50507,
7  | 1,  54,  77,  1639, 119,  2163,  49,  41580,  3193,  3101, 127,  53357,
8  | 3, 148, 187,  4927, 113,  6197, 211, 142280,  8283,  4969, 183, 179083,
9  | 5, 288, 179, 11669, 305,  9481, 277, 414720,  6965, 13421, 239, 374459,
10 | 1,  90, 187,  4531, 131,  7685,  73, 190980, 12649,  6303, 137, 293947,
11 | 5, 376, 301, 19869, 247, 18395, 331, 919856, 17173, 17161, 425, 906981,
12 | 3, 274, 167, 16861, 255, 13189, 349, 899540, 10335, 17099, 367, 777083,

Links

Crossrefs

Cf. A000203, A003961, A246278, A286385, A355927.
Cf. A046933 (column 1).
Cf. also A355924, A372562.

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); };
A286385(n) = (A003961(n)-sigma(n));
A372563sq(row,col) = A286385(A246278sq(row,col));
A372563list(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] = A372563sq(col,(a-(col-1))))); (v); };
v372563 = A372563list(up_to);
A372563(n) = v372563[n];

Formula

A(n, k) = A286385(A246278(n, k)) = A246278(1+n, k) - A355927(n, k).
Previous Showing 21-30 of 41 results. Next

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