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.

User: Antonio Graciá Llorente

Antonio Graciá Llorente's wiki page.

Antonio Graciá Llorente has authored 4 sequences.

A368497 Decimal expansion of the fixed point c = S(c) of S(x) = Sum_{k>=1} (prime(k) - x) / Product_{i=1..k-1} prime(i).

Original entry on oeis.org

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

Views

Author

Antonio Graciá Llorente, Dec 27 2023

Keywords

Comments

S(x) = (1-x)*(1+A064648) + A249270 is linear so the fixed point is unique.
With this constant as h(1) = c, sequence h(n+1) = ceiling(h(n)) * (h(n) - ceiling(h(n)) + c) is real numbers with the property that ceiling(h(n)) = prime(n).

Examples

			1.709755124475931301268259070090809...

Crossrefs

Cf. A064648, A249270, A000040.
Cf. A341930 (S(3/2)), A340469 (S(2)).

Programs

  • PARI
    solve(x=1,2,suminf(k=1,(prime(k)-x)/prod(i=1,k-1,prime(i)))-x) \\ Michal Paulovic, Dec 28 2023

Formula

Equals (A249270 + A064648 + 1)/(A064648 + 2).

A235399 Numbers which are the digital sum of the cube of some prime.

Original entry on oeis.org

8, 9, 10, 17, 19, 26, 28, 35, 37, 44, 46, 53, 55, 62, 64, 71, 73, 80, 82, 89, 91, 98, 100, 107, 109, 116, 118, 125, 127, 134, 136, 143, 145, 152, 154, 161, 163, 170, 172, 179, 181, 188, 190, 197, 199, 206, 208, 215, 217, 224, 226, 233, 235, 242, 244, 251, 253
Offset: 1

Views

Author

Antonio Graciá Llorente, Jan 09 2014

Keywords

Comments

A235398 sorted and duplicates removed.

Crossrefs

Cf. A235398, A004164, A007953, A123157.

Programs

  • Mathematica
    Total[IntegerDigits[#]] & /@ (Prime[Range[5000000]]^3) // Union (* The program generates the first 39 terms of the sequence. To generate more, increase the Range constant. *) (* Harvey P. Dale, Sep 19 2021 *)
  • PARI
    list(maxx)={v=List();n=2; while(n<=maxx,q=n^3;summ=sumdigits(q);
if(setsearch(v,summ)<1,listput(v,summ));n=nextprime(n+1));vecsort(v,,8) ;} \\ Bill McEachen, Jan 29 2014

Formula

Conjecture: for n > 4, a(n) = a(n-2) + 9 = A056020(n).
Conjecture: a(n) = (1/4)*(-1)^n*(9*(-1)^n*(2*n-1) + 5), n >= 3. - Bill McEachen, Feb 13 2021

Extensions

a(37)-a(57) from Lars Blomberg, Feb 10 2016

A235398 Sum of digits of the cubes of prime numbers.

Original entry on oeis.org

8, 9, 8, 10, 8, 19, 17, 28, 17, 26, 28, 19, 26, 28, 17, 35, 26, 28, 19, 26, 28, 28, 35, 35, 28, 8, 28, 17, 28, 35, 28, 26, 26, 37, 35, 28, 46, 28, 35, 35, 35, 37, 44, 37, 35, 46, 37, 37, 35, 37, 35, 35, 37, 26, 44, 35, 35, 28, 28, 26, 37, 35, 37, 17, 37, 26
Offset: 1

Views

Author

Antonio Graciá Llorente, Jan 09 2014

Keywords

Crossrefs

Cf. A004164, A007953, A123157.

Programs

  • Maple
    read("transforms") :
A235398 := proc(n)
    digsum(ithprime(n)^3) ;
end proc:
seq(A235398(n),n=1..40) ; # R. J. Mathar, Jul 19 2024
  • PARI
    a(n) = sumdigits(prime(n)^3); \\ Michel Marcus, Jan 09 2014

Formula

a(n) = A007953(A030078(n)). - R. J. Mathar, Jul 19 2024

A234429 Numbers which are the digital sum of the square of some prime.

Original entry on oeis.org

4, 7, 9, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 61, 64, 67, 70, 73, 76, 79, 82, 85, 88, 91, 94, 97, 100, 103, 106, 109, 112, 115, 118, 121, 124, 127, 130, 133, 136, 139, 142, 145, 148, 151, 154, 157, 160, 163, 166, 169, 172, 175, 178
Offset: 1

Views

Author

Antonio Graciá Llorente, Dec 26 2013

Keywords

Comments

A123157 sorted and duplicates removed.

Crossrefs

Cf. A067180, A067523, A123157, A056991.
Cf. A062397, A226803, A226802, A165492, A165459 A165493, A229058, A165502, A165503, A165504.

Programs

  • PARI
    terms(nn) = {v = []; forprime (p = 1, nn, v = concat(v, sumdigits(p^2));); vecsort(v,,8);} \\ Michel Marcus, Jan 08 2014

Formula

From Robert G. Wilson v, Sep 28 2014: (Start)
Except for 3, all primes are congruent to +-1 (mod 3). Therefore, (3n +- 1)^2 = 9n^2 +- 6n + 1 which is congruent to 1 (mod 3).
4: 2, 11, 101, ... (A062397);
7: 5, 149, 1049, ... (A226803);
9: only 3;
10: 19, 71, 179, 251, 449, 20249, 24499, 100549, ... (A226802);
13: 7, 29, 47, 61, 79, 151, 349, 389, 461, 601, 1051, 1249, 1429, ... (A165492);
16: 13, 23, 31, 41, 59, 103, 131, 139, 211, 229, 239, 347, 401, ... (A165459);
19: 17, 37, 53, 73, 89, 107, 109, 127, 181, 199, 269, 271, 379, ... (A165493);
22: 43, 97, 191, 227, 241, 317, 331, 353, 421, 439, 479, 569, 619, 641, ...;
25: 67, 113, 157, 193, 257, 283, 311, 337, 373, 409, 419, 463, ... (A229058);
28: 163, 197, 233, 307, 359, 397, 431, 467, 487, 523, 541, 577, 593, 631, ...;
31: 83, 137, 173, 223, 263, 277, 281, 367, 443, 457, 547, 587, ... (A165502);
34: 167, 293, 383, 563, 607, 617, 733, 787, 823, 859, 877, 941, 967, 977, ...;
37: 433, 613, 683, 773, 827, 863, 1063, 1117, 1187, 1223, 1567, ... (A165503);
40: 313, 947, 983, 1303, 1483, 1609, 1663, 1933, 1973, 1987, 2063, 2113, ...;
43: 887, 1697, 1723, 1867, 1913, 2083, 2137, 2417, 2543, 2633, ... (A165504);
46: 883, 937, 1367, 1637, 2213, 2447, 2683, 2791, 2917, 3313, 3583, 3833, ...;
49: 1667, 2383, 2437, 2617, 2963, 4219, 4457, 5087, 5281, 6113, 6163, ...;
... Also see A229058. (End)
Conjectures from Chai Wah Wu, Apr 16 2025: (Start)
a(n) = 2*a(n-1) - a(n-2) for n > 5.
G.f.: x*(2*x^4 - x^3 - x^2 - x + 4)/(x - 1)^2. (End)

Extensions

a(36) from Michel Marcus, Jan 08 2014
a(37)-a(54) from Robert G. Wilson v, Sep 28 2014
More terms from Giovanni Resta, Aug 15 2019

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.