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.

Showing 1-7 of 7 results.

A349868 Nobly abundant numbers (A349758) that are not abundant numbers (A005101).

Original entry on oeis.org

315, 495, 525, 585, 675, 693, 735, 765, 819, 825, 850, 855, 884, 950, 975, 988, 1012, 1035, 1071, 1078, 1125, 1150, 1196, 1197, 1210, 1215, 1274, 1275, 1276, 1287, 1292, 1305, 1323, 1364, 1395, 1425, 1449, 1450, 1508, 1550, 1564, 1612, 1617, 1628, 1665, 1666
Offset: 1

Views

Author

Amiram Eldar, Dec 03 2021

Keywords

Comments

This sequence if infinite. For example, 45*p is a term if p is a prime > 5.

Examples

			315 is a term since it is not abundant, A000203(315) = 624 < 2*315 = 630, and both A000005(315) = 12 and A000203(315) = 624 are abundant numbers.

Links

Crossrefs

Intersection of A263837 and A349758.
Cf. A000005, A000203, A005101, A349869.

Programs

  • Mathematica
    abQ[n_] := DivisorSigma[1, n] > 2*n; nobAbQ[n_] := And @@ abQ /@ DivisorSigma[{0, 1}, n]; Select[Range[2000], !abQ[#] && nobAbQ[#] &]

A349870 Numbers k such that k and k+1 are both nobly abundant numbers (A349758).

Original entry on oeis.org

735, 819, 1035, 1196, 1274, 1275, 1449, 1665, 1700, 1924, 1925, 1952, 1988, 2204, 2324, 2331, 2540, 2655, 2960, 2975, 3068, 3195, 3267, 3324, 3339, 3380, 3549, 3555, 3626, 3717, 4004, 4059, 4164, 4220, 4235, 4256, 4556, 4563, 4598, 4599, 4635, 4655, 4675, 4719
Offset: 1

Views

Author

Amiram Eldar, Dec 03 2021

Keywords

Examples

			735 is a term since 735 and 736 are both nobly abundant numbers: A000005(735) = A000005(736) = 12, A000203(735) = 1368 and A000203(736) = 1512 are all abundant numbers.

Links

Crossrefs

Cf. A000005, A000203, A005101, A096399.
Subsequence of A349758.

Programs

  • Mathematica
    abQ[n_] := DivisorSigma[1, n] > 2*n; nobAbQ[n_] := And @@ abQ /@ DivisorSigma[{0, 1}, n]; Select[Range[50000], nobAbQ[#] && nobAbQ[# + 1] &]

A349871 a(n) is the start of the least run of exactly n consecutive nobly abundant numbers (A349758).

Original entry on oeis.org

60, 735, 1274, 28674, 19940, 1699947, 4630544, 24816974, 111500620, 262315467, 30477326444
Offset: 1

Views

Author

Amiram Eldar, Dec 03 2021

Keywords

Comments

a(12) > 1.6*10^11, if it exists.

Examples

			a(2) = 735 since 735 and 736 are the least pair of consecutive numbers that are nobly abundant numbers.

Crossrefs

Cf. A349758, A349870.

Programs

  • Mathematica
    abQ[n_] := DivisorSigma[1, n] > 2*n; nobAbQ[n_] := And @@ abQ /@ DivisorSigma[{0, 1}, n]; seq[len_, nmax_] := Module[{s = Table[0, {len}], count = 0, n = 1, n1, d}, While[n < nmax && count < len, n1 = n; If[nobAbQ[n], While[nobAbQ[++n1]]; d = n1 - n; If[d <= len && s[[d]] == 0, count++; s[[d]] = n]]; n = n1 + 1]; TakeWhile[s, # > 0 &]]; seq[6, 10^6]

A349759 Nobly deficient numbers: numbers k such that both d(k) = A000005(k) and sigma(k) = A000203(k) are deficient numbers (A005100).

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 9, 13, 16, 21, 25, 31, 36, 37, 43, 48, 49, 61, 64, 67, 73, 81, 93, 97, 100, 109, 111, 112, 121, 127, 128, 144, 151, 157, 162, 163, 169, 181, 183, 192, 193, 196, 208, 211, 217, 219, 225, 229, 241, 256, 277, 283, 289, 313, 324, 331, 337, 361, 373
Offset: 1

Views

Author

Amiram Eldar, Nov 29 2021

Keywords

Comments

Analogous to sublime numbers (A081357), with deficient numbers instead of perfect numbers.
If p != 5 is a prime such that (p+1)/2 is also a prime (i.e., p is in A005383 \ {5}), then p is a term of this sequence.

Examples

			2 is a term since both d(2) = 2 and sigma(2) = 3 are deficient numbers.

Links

Crossrefs

Cf. A000005, A000203, A005100, A005383, A081357, A349758.

Programs

  • Mathematica
    defQ[n_] := DivisorSigma[1, n] < 2*n; nobDefQ[n_] := And @@ defQ /@ DivisorSigma[{0, 1}, n]; Select[Range[400], nobDefQ]
  • PARI
    isdef(k) = sigma(k) < 2*k; \\ A005100
isok(k) = my(f=factor(k)); isdef(numdiv(f)) && isdef(sigma(f)); \\ Michel Marcus, Dec 03 2021

A349760 Numbers k such that d(k) = A000005(k), sigma(k) = A000203(k) and phi(k) = A000010(k) are all abundant numbers (A005101).

Original entry on oeis.org

84, 90, 108, 126, 132, 140, 150, 156, 180, 198, 220, 224, 228, 234, 252, 260, 276, 294, 300, 306, 308, 315, 336, 342, 348, 350, 352, 360, 364, 372, 380, 396, 414, 416, 420, 432, 444, 460, 476, 486, 490, 492, 495, 500, 504, 516, 522, 525, 528, 532, 540, 550, 558
Offset: 1

Views

Author

Amiram Eldar, Nov 29 2021

Keywords

Comments

Sándor (2005) proved that this sequence is infinite by showing that it includes all the numbers of the form 11 * p^11 * k where p != 11 is a prime and k is any number coprime to 11*p.

Examples

			84 is a term since d(84) = 12, sigma(84) = 224 and phi(84) = 24 are all abundant numbers: sigma(12) = 28 > 2*12 = 24, sigma(224) = 504 > 2*224 = 448 and sigma(24) = 60 > 2*24 = 48.

Links

Crossrefs

Cf. A000005, A000010, A000203, A005101.
Subsequence of A349758.

Programs

  • Mathematica
    abQ[n_] := DivisorSigma[1, n] > 2*n; q[n_] := And @@ abQ /@ Join[DivisorSigma[{0, 1}, n], {EulerPhi[n]}]; Select[Range[500], q]
  • PARI
    isab(k) = sigma(k) > 2*k; \\ A005101
isok(k) = my(f=factor(k)); isab(numdiv(f)) && isab(sigma(f)) && isab(eulerphi(f)); \\ Michel Marcus, Dec 03 2021

A349761 Numbers k such that sigma(k) = A000203(k) is an abundant number (A005101) and phi(k) = A000010(k) is a deficient number (A005100).

Original entry on oeis.org

6, 10, 11, 15, 17, 20, 22, 23, 24, 30, 34, 40, 46, 47, 51, 53, 59, 60, 68, 69, 80, 83, 85, 92, 94, 96, 102, 106, 107, 118, 120, 131, 136, 137, 138, 141, 149, 160, 166, 167, 170, 173, 177, 179, 188, 191, 204, 214, 227, 233, 235, 236, 239, 240, 249, 251, 255, 257
Offset: 1

Views

Author

Amiram Eldar, Nov 29 2021

Keywords

Comments

Sándor (2005) proved that this sequence is infinite by showing that it includes all the numbers of the form 3 * 2^k where k == 11 (mod 12). If gcd((k+1)/12, 6) = 1, then this number is also nobly abundant (A349758).

Examples

			6 is a term since sigma(6) = 12 is an abundant number, sigma(12) = 28 > 2*12 = 24, and phi(6) = 2 is a deficient number, sigma(2) = 3 < 2*2 = 4.

Links

Crossrefs

Cf. A000010, A000203, A005100, A005101, A349758, A349759, A349762.

Programs

  • Mathematica
    abQ[n_] := DivisorSigma[1, n] > 2*n; defQ[n_] := DivisorSigma[1, n] < 2*n; q[n_] := abQ[DivisorSigma[1, n]] && defQ[EulerPhi[n]]; Select[Range[250], q]

A349762 Numbers k such that phi(k) = A000010(k) is an abundant number (A005101) and d(k) = A000005(k) is a deficient number (A005100).

Original entry on oeis.org

13, 19, 21, 25, 26, 27, 31, 33, 35, 36, 37, 38, 39, 41, 42, 43, 49, 54, 55, 56, 57, 61, 62, 65, 66, 67, 70, 71, 73, 74, 77, 78, 79, 81, 82, 86, 87, 88, 89, 91, 93, 95, 97, 100, 101, 103, 104, 105, 109, 110, 111, 112, 113, 114, 115, 119, 122, 123, 125, 127, 129
Offset: 1

Views

Author

Amiram Eldar, Nov 29 2021

Keywords

Comments

Sándor (2005) proved that this sequence is infinite by showing that it includes all the numbers of the form 3^(p^2-1) where p is a prime.

Examples

			13 is a term since phi(13) = 12 is an abundant number, sigma(12) = 28 > 2*12 = 24, and d(13) = 2 is a deficient number, sigma(2) = 3 < 2*2 = 4.

Links

Crossrefs

Cf. A000005, A000010, A005100, A005101, A349758, A349759, A349761.
A164318 is a subsequence.

Programs

  • Mathematica
    abQ[n_] := DivisorSigma[1, n] > 2*n; defQ[n_] := DivisorSigma[1, n] < 2*n; q[n_] := abQ[EulerPhi[n]] && defQ[DivisorSigma[0, n]]; Select[Range[150], q]
Showing 1-7 of 7 results.

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