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: Elmo R. Oliveira

Elmo R. Oliveira's wiki page.

Elmo R. Oliveira has authored 19 sequences. Here are the ten most recent ones:

A364689 Prime numbers that are the exact average of ten consecutive odd semiprimes.

Original entry on oeis.org

43, 53, 73, 83, 113, 373, 449, 577, 971, 1259, 1327, 1381, 1499, 1543, 1847, 2239, 2311, 2339, 2351, 2383, 2953, 3109, 3257, 3389, 4021, 4297, 4919, 5101, 5227, 5591, 5701, 5737, 5927, 6733, 6907, 7109, 7253, 7823, 8011, 9137, 9403, 9613, 10177, 11471, 11621, 11677, 12251, 12479, 12671, 12781
Offset: 1

Views

Author

Elmo R. Oliveira, Sep 25 2023

Keywords

Examples

			43 is a term because (21 + 25 + 33 + 35 + 39 + 49 + 51 + 55 + 57 + 65)/10 = 43 is prime.
449 is a term because (417 + 427 + 437 + 445 + 447 + 451 + 453 + 469 + 471 + 473)/10 = 449 is prime.

Crossrefs

Cf. A000040, A046315.
Cf. A363074, A363187, A363188, A364147, A364148, A364149, A364320, A364321.

Programs

  • Mathematica
    Select[Mean /@ Partition[Select[Range[1, 13000, 2], PrimeOmega[#] == 2 &], 10, 1], PrimeQ] (* Amiram Eldar, Sep 25 2023 *)

A364321 Prime numbers that are the exact average of nine consecutive odd semiprimes.

Original entry on oeis.org

97, 191, 293, 347, 401, 409, 479, 727, 1823, 1931, 2063, 2089, 2897, 2903, 2999, 3061, 3083, 3119, 3571, 3617, 3673, 3727, 3967, 4339, 4373, 4583, 4639, 4703, 4813, 5297, 5347, 5437, 5639, 5821, 6047, 6053, 6311, 6421, 6491, 6529, 6761, 6883, 7283, 7417, 7451, 7949, 8059, 8123, 8237
Offset: 1

Views

Author

Elmo R. Oliveira, Sep 25 2023

Keywords

Examples

			97 is a term because (77 + 85 + 87 + 91 + 93 + 95 + 111 + 115 + 119)/9 = 97 is prime.
401 is a term because (381 + 391 + 393 + 395 + 403 + 407 + 411 + 413 + 415)/9 = 401 is prime.

Crossrefs

Cf. A000040, A046315.
Cf. A363074, A363187, A363188, A364147, A364148, A364149, A364320, A364689.

Programs

  • Mathematica
    Select[Mean /@ Partition[Select[Range[1, 9000, 2], PrimeOmega[#] == 2 &], 9, 1], PrimeQ] (* Amiram Eldar, Sep 25 2023 *)

A364320 Prime numbers that are the exact average of eight consecutive odd semiprimes.

Original entry on oeis.org

43, 317, 607, 719, 853, 887, 919, 1231, 1237, 1283, 1303, 1951, 2179, 2609, 3001, 3271, 3389, 3491, 3547, 3643, 3889, 3931, 4241, 4297, 4447, 4517, 4567, 4621, 4817, 4831, 4871, 4909, 5479, 5623, 5647, 5653, 5953, 6211, 6301, 6869, 7019, 7559, 8011, 8191, 8297, 8311, 8317, 8369, 8447
Offset: 1

Views

Author

Elmo R. Oliveira, Sep 25 2023

Keywords

Examples

			43 is a term because (25 + 33 + 35 + 39 + 49 + 51 + 55 + 57)/8 = 43 is prime.
719 is a term because (703 + 707 + 713 + 717 + 721 + 723 + 731 + 737)/8 = 719 is prime.

Crossrefs

Cf. A000040, A046315.
Cf. A363074, A363187, A363188, A364147, A364148, A364149, A364321, A364689.

Programs

  • Mathematica
    Select[Mean /@ Partition[Select[Range[1, 9000, 2], PrimeOmega[#] == 2 &], 8, 1], PrimeQ] (* Amiram Eldar, Sep 25 2023 *)

A365202 Even semiprimes that are the exact average of six consecutive odd semiprimes.

Original entry on oeis.org

146, 194, 302, 478, 482, 614, 706, 1006, 1438, 1966, 1994, 2186, 2206, 2426, 2462, 2594, 2614, 3098, 3274, 3518, 3742, 3986, 4282, 4406, 4594, 4702, 5354, 5606, 6038, 6178, 6218, 6238, 6442, 6626, 6782, 7262, 7642, 7646, 7886, 8254, 9098, 9194, 9298, 9346, 9442, 9574, 9938
Offset: 1

Views

Author

Elmo R. Oliveira, Aug 25 2023

Keywords

Examples

			146 is a term because (133 + 141 + 143 + 145 + 155 + 159)/6 = 146 = 2*73 is an even semiprime.
302 is a term because (295 + 299 + 301 + 303 + 305 + 309)/6 = 302 = 2*151 is an even semiprime.

Crossrefs

Cf. A046315, A365200, A365201.
Subset of A100484.

Programs

  • Mathematica
    sp=Select[Range[5,12000,2], PrimeOmega[#]==2&]; a={}; For[i=1, iStefano Spezia, Aug 25 2023 *)
  • PARI
    upto(n) = {my(res = List(), l = List([0,9,15,21,25,33]), s = sum(i = 1, #l, l[i]), i = l[#l]+2, ntimes6 = 6*n); while(1, if(bigomega(i) == 2, s += (i-l[1]); if(s > ntimes6, return(res)); if(s % 12 == 0 && isprime(s/12), listput(res, s/6)); listpop(l, 1); listput(l, i)); i+=2)} \\ David A. Corneth, Aug 26 2023

A365201 Even semiprimes that are the exact average of four consecutive odd semiprimes.

Original entry on oeis.org

74, 146, 194, 218, 302, 482, 586, 734, 746, 842, 914, 1042, 1138, 1262, 1346, 1438, 1574, 1646, 1654, 1838, 1874, 1894, 1906, 1942, 2026, 2186, 2206, 2458, 2462, 2762, 2906, 2962, 2974, 3098, 3106, 3202, 3218, 3826, 4198, 4274, 4286, 4322, 4414, 4502, 4534, 4622, 4666, 4754, 4934, 4946
Offset: 1

Views

Author

Elmo R. Oliveira, Aug 25 2023

Keywords

Examples

			74 is a term because (65 + 69 + 77 + 85)/4 = 74 = 2*37 is an even semiprime.
218 is a term because (215 + 217 + 219 + 221)/4 = 218 = 2*109 is an even semiprime.

Crossrefs

Cf. A046315, A365200, A365202.
Subset of A100484.

Programs

  • Mathematica
    sp=Select[Range[5,5200,2], PrimeOmega[#]==2&]; a={}; For[i=1, iStefano Spezia, Aug 25 2023 *)
  • PARI
    upto(n) = {my(res = List(), l = List([0, 9, 15, 21]), s = sum(i = 1, #l, l[i]), i = l[#l]+2, ntimes4 = 4*n); while(1, if(bigomega(i) == 2, s += (i-l[1]); if(s > ntimes4, return(res)); if(s % 8 == 0 && isprime(s/8), listput(res, s/4)); listpop(l, 1); listput(l, i)); i+=2)} \\ David A. Corneth, Aug 26 2023

A365200 Even semiprimes that are the exact average of two consecutive odd semiprimes.

Original entry on oeis.org

34, 86, 94, 122, 142, 194, 202, 214, 218, 262, 302, 314, 358, 386, 394, 422, 446, 562, 586, 626, 634, 698, 734, 838, 842, 922, 982, 1042, 1138, 1234, 1262, 1306, 1346, 1366, 1402, 1522, 1642, 1646, 1658, 1754, 1762, 1774, 1838, 1874, 1894, 1906, 1942, 1982, 2026, 2098, 2102, 2182, 2186, 2218
Offset: 1

Views

Author

Elmo R. Oliveira, Aug 25 2023

Keywords

Examples

			34 is a term because (33 + 35)/2 = 34 = 2*17 is an even semiprime.
86 is a term because (85 + 87)/2 = 86 = 2*43 is an even semiprime.

Crossrefs

Cf. A046315, A365201, A365202.
Subset of A100484.

Programs

  • Mathematica
    sp=Select[Range[5,2400,2], PrimeOmega[#]==2&]; a={}; For[i=1, iStefano Spezia, Aug 25 2023 *)
  • PARI
    upto(n) = {my(res = List(), l = List([0, 9]), s = sum(i = 1, #l, l[i]), i = l[#l]+2, ntimes2 = 2*n); while(1, if(bigomega(i) == 2, s += (i-l[1]); if(s > ntimes2, return(res)); if(s % 4 == 0 && isprime(s/4), listput(res, s/2)); listpop(l, 1); listput(l, i)); i+=2)} \\ David A. Corneth, Aug 26 2023

A364149 Prime numbers that are the exact average of seven consecutive odd semiprimes.

Original entry on oeis.org

31, 41, 617, 677, 937, 947, 1637, 1931, 1979, 2221, 2341, 2447, 2647, 2857, 3373, 3583, 3673, 3823, 3967, 4027, 4049, 4229, 4259, 4339, 4421, 4649, 4861, 4931, 5051, 5179, 5399, 5407, 5507, 5521, 5573, 5987, 6047, 6131, 6143, 6311, 6337, 6703, 6737, 7417, 7717, 7723, 7901, 8059, 8069, 8231, 8647
Offset: 1

Views

Author

Elmo R. Oliveira, Jul 10 2023

Keywords

Examples

			31 is a term because (15 + 21 + 25 + 33 + 35 + 39 + 49)/7 = 31 is prime.
617 is a term because (591 + 597 + 611 + 623 + 629 + 633 + 635)/7 = 617 is prime.

Crossrefs

Cf. A000040, A046315.
Cf. A363074, A363187, A363188, A364147, A364148.

Programs

  • Mathematica
    Select[Mean /@ Partition[Select[Range[1, 9000, 2], PrimeOmega[#] == 2 &], 7, 1], PrimeQ] (* Amiram Eldar, Jul 11 2023 *)

A364148 Prime numbers that are the exact average of six consecutive odd semiprimes.

Original entry on oeis.org

23, 79, 109, 491, 599, 797, 809, 853, 953, 1021, 1171, 1289, 1361, 1531, 1543, 1559, 1811, 1951, 1987, 2143, 2179, 2239, 2273, 2309, 2381, 2399, 3169, 3271, 3343, 3371, 3433, 3613, 3701, 4051, 4157, 4297, 4327, 4357, 4457, 4603, 4789, 4871, 5227, 5233, 5443, 5479, 5623, 5711, 5737, 5927, 6073
Offset: 1

Views

Author

Elmo R. Oliveira, Jul 10 2023

Keywords

Examples

			23 is a term because (9 + 15 + 21 + 25 + 33 + 35)/6 = 23 is prime.
109 is a term because (93 + 95 + 111 + 115 + 119 + 121)/6 = 109 is prime.

Crossrefs

Cf. A000040, A046315.
Cf. A363074, A363187, A363188, A364147, A364149.

Programs

  • Mathematica
    Select[Mean /@ Partition[Select[Range[1, 6000, 2], PrimeOmega[#] == 2 &], 6, 1], PrimeQ] (* Amiram Eldar, Jul 11 2023 *)

A364147 Prime numbers that are the exact average of five consecutive odd semiprimes.

Original entry on oeis.org

101, 677, 743, 811, 907, 1039, 1109, 1129, 1301, 1373, 1381, 1567, 1789, 1931, 1949, 1979, 2029, 2447, 2621, 2663, 2731, 2879, 2909, 2971, 3119, 3187, 3221, 3319, 3529, 3631, 3677, 3803, 3823, 3943, 4201, 4253, 4549, 4597, 4637, 4643, 4649, 4801, 4951, 5119, 5189, 5431, 5987, 6053, 6151, 6311
Offset: 1

Views

Author

Elmo R. Oliveira, Jul 10 2023

Keywords

Examples

			101 is a term because (91 + 93 + 95 + 111 + 115)/5 = 101 is prime.
743 is a term because (737 + 737 + 745 + 749 + 753)/5 = 743 is prime.

Links

Crossrefs

Cf. A000040, A046315.
Cf. A363074, A363187, A363188, A364148, A364149.

Programs

  • Maple
    N:= 10^4: # for terms involving semiprimes up to N
OSP:= select(t -> numtheory:-bigomega(t) = 2, [seq(i,i=1..N,2)]):
select(t -> t::integer and isprime(t), add(OSP[i..-6+i],i=1..5)/5); # Robert Israel, Aug 11 2023
  • Mathematica
    Select[Mean /@ Partition[Select[Range[1, 6500, 2], PrimeOmega[#] == 2 &], 5, 1], PrimeQ] (* Amiram Eldar, Jul 11 2023 *)

A363187 Prime numbers that are the average of three consecutive odd semiprimes.

Original entry on oeis.org

31, 41, 59, 83, 107, 139, 163, 191, 197, 281, 311, 383, 397, 443, 521, 673, 677, 757, 821, 887, 997, 1061, 1109, 1151, 1171, 1229, 1237, 1373, 1423, 1453, 1619, 1823, 1889, 1931, 2053, 2141, 2203, 2221, 2309, 2339, 2437, 2473, 2477, 2749, 2801, 2837, 2953, 3019, 3119, 3163, 3209, 3257, 3347
Offset: 1

Views

Author

Elmo R. Oliveira, May 20 2023

Keywords

Examples

			31 is a term because (25 + 33 + 35)/3 = 31 is prime.
41 is a term because (35 + 39 + 49)/3 = 41 is prime.

Links

Crossrefs

Cf. A000040, A046315.
Cf. A363074, A363188.

Programs

  • Maple
    OP:= select(isprime, [seq(i, i=3..10000, 2)]):
OSP:= sort(select(`<=`, [seq(seq(OP[i]*OP[j], j=1..i), i=1..nops(OP))], 3*OP[-1])):
SA:= [seq(add(OSP[i+j], j=0..2)/3, i=1..nops(OSP)-2)]:
select(t -> t::integer and isprime(t), SA); # Robert Israel, May 22 2023
  • Mathematica
    Select[Plus @@@ Partition[Select[Range[1, 3400, 2], PrimeOmega[#] == 2 &], 3, 1] / 3, PrimeQ] (* Amiram Eldar, May 21 2023 *)
  • Python
    from itertools import count, islice
from sympy import factorint, isprime
def semiprime(n): return sum(e for e in factorint(n).values()) == 2
def nextoddsemiprime(n): return next(k for k in count(n+1+(n&1), 2) if semiprime(k))
def agen(): # generator of terms
    osp = [9, 15, 21]
    while True:
        q, r = divmod(sum(osp), len(osp))
        if r == 0 and isprime(q):
            yield q
        osp = osp[1:] + [nextoddsemiprime(osp[-1])]
print(list(islice(agen(), 53))) # Michael S. Branicky, May 21 2023

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

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