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: Patrick De Geest

Patrick De Geest's wiki page.

Patrick De Geest has authored 2275 sequences. Here are the ten most recent ones:

A379400 Numbers that can be written in exactly five different ways as a sum of at most nine positive third powers.

Original entry on oeis.org

133, 145, 152, 153, 159, 160, 161, 163, 182, 190, 192, 196, 197, 198, 200, 201, 209, 218, 220, 223, 225, 226, 233, 234, 235, 236, 237, 238, 246, 248, 249, 255, 258, 265, 268, 269, 274, 277, 279, 292, 300, 302, 304, 305, 310, 311, 312, 329, 331, 340, 356, 364, 365, 429
Offset: 1

Views

Author

Patrick De Geest, Dec 23 2024

Keywords

Comments

The 'nine' is not arbitrary. Waring stated that every natural number can be expressed as a sum of at most nine cubes (A002804).

Examples

			133 is in the sequence since 2^3+5^3 = 1^3+1^3+1^3+1^3+1^3+4^3+4^3 = 1^3+2^3+2^3+2^3+3^3+3^3+3^3+3^3 = 1^3+1^3+1^3+1^3+1^3+1^3+1^3+1^3+5^3 = 1^3+1^3+2^3+2^3+2^3+2^3+2^3+3^3+4^3.

Links

Crossrefs

Cf. A002804, A025453, A379396, A379397, A379398, A379399.

Programs

  • PARI
    upto(n) = my(v=vector(n), maxb=sqrtnint(n, 3)); forvec(x=vector(9, i, [0, maxb]), s=sum(i=1, 9, x[i]^3); if(0x==5, v, 1) \\ David A. Corneth, Dec 23 2024

A379399 Numbers that can be written in exactly four different ways as a sum of at most nine positive third powers.

Original entry on oeis.org

72, 91, 126, 128, 129, 131, 132, 134, 135, 136, 137, 138, 139, 140, 144, 146, 147, 154, 155, 156, 157, 158, 162, 164, 165, 166, 168, 170, 171, 172, 173, 179, 180, 181, 184, 185, 187, 191, 193, 194, 195, 199, 203, 205, 206, 207, 210, 211, 215, 221, 228, 229, 230, 231, 232, 241, 242, 266, 267, 293, 295, 319, 330, 338, 366, 455
Offset: 1

Views

Author

Patrick De Geest, Dec 23 2024

Keywords

Comments

The 'nine' is not arbitrary. Waring stated that every natural number can be expressed as a sum of at most nine cubes (cf. A002804).
Conjecture: this sequence is finite and a(66) = 455 is the last term. Verified up to 10^8. - Charles R Greathouse IV, Dec 28 2024

Examples

			215 is in the sequence since 1^3+2^3+3^3+3^3+3^3+5^3 = 1^3+1^3+2^3+2^3+2^3+4^3+5^3 = 2^3+2^3+3^3+3^3+3^3+3^3+3^3+4^3 = 1^3+2^3+2^3+2^3+2^3+3^3+3^3+4^3+4^3.

Links

Crossrefs

Cf. A002804, A025453, A379396, A379397, A379398, A379400.

Programs

  • PARI
    upto(n) = my(v=vector(n), maxb=sqrtnint(n, 3)); forvec(x=vector(9, i, [0, maxb]), s=sum(i=1, 9, x[i]^3); if(0x==4, v, 1) \\ David A. Corneth, Dec 23 2024

A379396 Numbers that can be written in exactly one way as a sum of at most nine positive third powers.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 31, 38, 39, 45, 46, 47, 50, 52, 53
Offset: 1

Views

Author

Patrick De Geest, Dec 22 2024

Keywords

Comments

The 'nine' is not arbitrary. Waring stated that every natural number can be expressed as a sum of at most nine cubes. (Cf. A002804)

Examples

			7 is in the sequence since there is only 1^3+1^3+1^3+1^3+1^3+1^3+1^3.
53 is in the sequence since there is only 1^3+1^3+2^3+2^3+2^3+3^3;

Links

Crossrefs

Cf. A002804, A025453, A379397, A379398, A379399, A379400

Programs

  • PARI
    upto(n) = my(v=vector(n), maxb=sqrtnint(n,3)); forvec(x=vector(9,i,[0,maxb]), s=sum(i=1,9,x[i]^3); if(0x==1,v,1) \\ David A. Corneth, Dec 23 2024

A379398 Numbers that can be written in exactly three different ways as a sum of at most nine positive third powers.

Original entry on oeis.org

35, 56, 64, 65, 67, 68, 70, 75, 81, 82, 83, 84, 86, 89, 92, 93, 94, 96, 97, 98, 99, 100, 105, 107, 108, 110, 112, 113, 118, 119, 120, 121, 124, 125, 127, 130, 141, 142, 143, 148, 149, 150, 151, 167, 169, 174, 175, 176, 177, 178, 183, 186, 188, 202, 204, 212, 213, 214, 240, 247, 303
Offset: 1

Views

Author

Patrick De Geest, Dec 22 2024

Keywords

Comments

The 'nine' is not arbitrary. Waring stated that every natural number can be expressed as a sum of at most nine cubes. (Cf. A002804)

Examples

			67 is in the sequence since 1^3+1^3+1^3+4^3 = 2^3+2^3+2^3+2^3+2^3+3^3 = 1^3+1^3+1^3+1^3+1^3+2^3+3^3+3^3.

Links

Crossrefs

Cf. A002804, A025453, A379396, A379397, A379399, A379400

Programs

  • PARI
    upto(n) = my(v=vector(n), maxb=sqrtnint(n,3)); forvec(x=vector(9,i,[0,maxb]), s=sum(i=1,9,x[i]^3); if(0x==3,v,1)) \\ David A. Corneth, Dec 23 2024

A379397 Numbers that can be written in exactly two different ways as a sum of at most nine positive third powers.

Original entry on oeis.org

8, 9, 16, 27, 28, 29, 30, 32, 33, 34, 36, 37, 40, 41, 42, 43, 44, 48, 49, 51, 54, 55, 57, 58, 59, 60, 61, 62, 63, 66, 69, 71, 73, 74, 76, 77, 78, 79, 80, 85, 87, 88, 90, 95, 101, 102, 103, 104, 106, 109, 111, 114, 115, 116, 117, 122, 123, 239
Offset: 1

Views

Author

Patrick De Geest, Dec 22 2024

Keywords

Comments

The 'nine' is not arbitrary. Waring stated that every natural number can be expressed as a sum of at most nine cubes (cf. A002804).

Examples

			29 is in the sequence since 1^3 + 1^3 + 3^3 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 2^3 + 2^3 + 2^3.
123 is in the sequence since 2^3 + 2^3 + 2^3 + 2^3 + 3^3 + 4^3 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 3^3 + 3^3 + 4^3.

Links

Crossrefs

Cf. A002804, A025453, A379396, A379398, A379399, A379400.

Programs

  • PARI
    upto(n) = my(v=vector(n), maxb=sqrtnint(n,3)); forvec(x=vector(9,i,[0,maxb]), s=sum(i=1,9,x[i]^3); if(0x==2,v,1) \\ David A. Corneth, Dec 23 2024

A226487 First available increasing palindromes (A002113) found in the decimal expansion of the number e-2 (A001113).

Original entry on oeis.org

7, 8, 818, 2662, 9669, 39193, 94349, 99699, 985589, 988890, 5065605, 6609066, 7193917, 7390937, 8316138, 43488434, 563303365, 799929997, 1149559411, 68088588086, 85367376358, 208532235802, 991964469199
Offset: 1

Views

Author

Patrick De Geest and Robert G. Wilson v, Jun 09 2013

Keywords

Comments

The entry 988890 is actually 0988890.

Crossrefs

Cf. A001113, A226486.

Programs

  • Mathematica
    e = RealDigits[E-2, 10, 2500000][[1]]; palQ[n_] := n == Reverse[n]; mx = 0; k = 1;   While[k < 1000000, j = 1; While[j <= k, If[ palQ[ Take[ e, {j, k}]], p = FromDigits[ Take[e, {j, k}]]; If[p > mx, mx = p; Print[p]; e = Drop[e, k]; k = 0; Break[]]]; j++]; k++]

A226486 First available increasing palindromes (A002113) found in the decimal expansion of Pi-3 (A000796).

Original entry on oeis.org

1, 4, 5, 9, 535, 979, 46264, 59195, 73637, 77477, 99999, 467764, 8683868, 23911932, 398989893, 559555955, 769646967, 972464279, 992868299, 21348884312, 49612121694, 450197791054, 9475082805749
Offset: 1

Views

Author

Patrick De Geest and Robert G. Wilson v, Jun 09 2013

Keywords

Crossrefs

Cf. A000796, A068046, A226487.

Programs

  • Mathematica
    pi = RealDigits[Pi-3, 10, 2500000][[1]]; palQ[n_] := n == Reverse[n]; mx = 0; k = 1; While[k < 1000, j = 1; While[j <= k, If[ palQ[ Take[ pi, {j, k}]], p = FromDigits[ Take[ pi, {j, k}]]; If[p > mx, mx = p; Print[p]; pi = Drop[pi, k]; k = 0; Break[]]]; j++]; k++]

A111524 a(1) = 10; a(n) is smallest number >= a(n-1) such that the juxtaposition a(1)a(2)...a(n) is a prime.

Original entry on oeis.org

10, 13, 23, 49, 111, 113, 171, 211, 293, 309, 309, 469, 639, 759, 951, 1037, 1057, 1083, 1257, 1269, 1287, 1341, 1551, 1637, 1677, 1981, 1989, 2021, 2059, 2357, 2583, 2697, 2967, 3289, 6789, 7073, 7323, 7369, 7463, 7501, 7709, 7869, 8029, 8069, 8077, 8519
Offset: 1

Views

Author

Patrick De Geest, Zak Seidov and Robert G. Wilson v, Aug 05 2005

Keywords

Links

Crossrefs

Cf. A111525, A074346, A033680, A033679, A033681.
Cf. A046254, A046255, A046256, A046257, A046258, A046259.

Programs

  • Mathematica
    a[1] = 10; a[n_] := a[n] = Block[{k = a[n - 1] + 1 + Mod[a[n - 1], 2], c = IntegerDigits @ Table[ a[i], {i, n - 1}]}, While[ !PrimeQ[ FromDigits @ Flatten @ Append[c, IntegerDigits[k]]], k += 2]; k]; Table[ a[n], {n, 46}] (* Robert G. Wilson v, Aug 05 2005 *)
  • Python
    from sympy import isprime
def aupton(terms):
    alst, astr = [10], "10"
    while len(alst) < terms:
        k = alst[-1] + (1 - alst[-1]%2)
        while not isprime(int(astr+str(k))): k += 2
        alst.append(k)
        astr += str(k)
    return alst
print(aupton(46)) # Michael S. Branicky, Oct 13 2021

A084843 Numbers n such that no palindrome m > 0 exists with n + m also a palindrome.

Original entry on oeis.org

919, 1020, 1029, 1031, 1038, 1041, 1047, 1051, 1061, 1065, 1071, 1074, 1081, 1091, 1101, 1130, 1131, 1139, 1141, 1148, 1151, 1157, 1161, 1171, 1175, 1181, 1191, 1201, 1231, 1240, 1241, 1249, 1251, 1258, 1261, 1267, 1271, 1281, 1291, 1301, 1314, 1341
Offset: 1

Views

Author

Patrick De Geest, Jun 08 2003

Keywords

Examples

			There is no palindrome x > 0 such that x + 919 (itself coincidentally also palindromic) is a palindrome.

Crossrefs

Cf. A002113, A082272, A082273, A104444.

A084836 Palindromic primes with nondecreasing digits up to the middle and then nonincreasing.

Original entry on oeis.org

2, 3, 5, 7, 11, 131, 151, 181, 191, 353, 373, 383, 787, 797, 11311, 11411, 12421, 12721, 12821, 13331, 13831, 13931, 14741, 15551, 16661, 17971, 19991, 33533, 34543, 34843, 35753, 77977, 78887, 79997, 1114111, 1117111, 1123211, 1126211, 1129211, 1134311
Offset: 1

Views

Author

Patrick De Geest, Jun 07 2003

Keywords

Links

Crossrefs

Cf. A084837, A062351, A062352.

Programs

  • Python
    from sympy import isprime
from itertools import count, islice, combinations_with_replacement as mc
def agen():
    yield from (2, 3, 5, 7, 11)
    for d in count(2):
        ndni = (int("".join(m+m[:-1][::-1])) for m in mc("123456789", d))
        yield from filter(isprime, ndni)
print(list(islice(agen(), 40))) # Michael S. Branicky, Jun 26 2022

Extensions

a(39) and beyond from Michael S. Branicky, Jun 26 2022

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