A349759 -id:A349759 - cp's OEIS frontend

A349869 Nobly deficient numbers (A349759) that are not deficient numbers (A005100).

36, 48, 100, 112, 144, 162, 192, 196, 208, 324, 400, 448, 576, 784, 832, 900, 1296, 1458, 1600, 1764, 1936, 1984, 2304, 2368, 2500, 2704, 2752, 2916, 3072, 3136, 3600, 3904, 4288, 4356, 4624, 4672, 4900, 5184, 5776, 6084, 6208, 6400, 6976, 7056, 7168, 7744, 8100

Offset: 1

Amiram Eldar, Dec 03 2021

nonn

			36 is a term since it is not deficient, A000203(36) = 91 > 2*36 = 72, and both A000005(36) = 9 and A000203(36) = 91 are deficient.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A023196 and A349759.

Cf. A000005, A000203, A005100, A349868.

defQ[n_] := DivisorSigma[1, n] < 2*n; nobDefQ[n_] := And @@ defQ /@ DivisorSigma[{0, 1}, n]; Select[Range[10^4], !defQ[#] && nobDefQ[#] &]

A349758 Nobly abundant numbers: numbers k such that both d(k) = A000005(k) and sigma(k) = A000203(k) are abundant numbers (A005101).

Original entry on oeis.org

60, 84, 90, 96, 108, 126, 132, 140, 150, 156, 160, 180, 198, 204, 220, 224, 228, 234, 240, 252, 260, 276, 294, 300, 306, 308, 315, 336, 340, 342, 348, 350, 352, 360, 364, 372, 380, 396, 414, 416, 420, 432, 444, 460, 476, 480, 486, 490, 492, 495, 500, 504, 516

Offset: 1

Amiram Eldar, Nov 29 2021

nonn

Analogous to sublime numbers (A081357), with abundant numbers instead of perfect numbers.

The least odd term is a(27) = 315 and the least term that is coprime to 6 is a(298) = 1925.

			60 is a term since both d(60) = 12 and sigma(60) = 168 are abundant numbers: sigma(12) = 28 > 2*12 = 24 and sigma(168) = 480 > 2*168 = 336.

József Sándor and E. Egri, Arithmetical functions in algebra, geometry and analysis, Advanced Studies in Contemporary Mathematics, Vol. 14, No. 2 (2007), pp. 163-213.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jason Earls, Some Smarandache-type sequences and problems concerning abundant and deficient numbers, Smarandache Notions Journal, Vol. 14, No. 1 (2004), pp. 243-250.
József Sándor, Selected Chapters of Geometry, Analysis and Number Theory, 2005, pp. 132-134.
Shikha Yadav and Surendra Yadav, Multiplicatively perfect and related numbers, Journal of Rajasthan Academy of Physical Sciences, Vol. 15, No. 4 (2016), pp. 345-350.

Cf. A000005, A000203, A005101, A081357, A349759.

A349760 is a subsequence.

abQ[n_] := DivisorSigma[1, n] > 2*n; nobAbQ[n_] := And @@ abQ /@ DivisorSigma[{0, 1}, n]; Select[Range[500], nobAbQ]

isab(k) = sigma(k) > 2*k; \\ A005101
isok(k) = my(f=factor(k)); isab(numdiv(f)) && isab(sigma(f)); \\ Michel Marcus, Dec 02 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

Amiram Eldar, Nov 29 2021

nonn

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).

			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.

Amiram Eldar, Table of n, a(n) for n = 1..10000
József Sándor, Selected Chapters of Geometry, Analysis and Number Theory, 2005, pp. 132-134.

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

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

Amiram Eldar, Nov 29 2021

nonn

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.

			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.

Amiram Eldar, Table of n, a(n) for n = 1..10000
József Sándor, Selected Chapters of Geometry, Analysis and Number Theory, 2005, pp. 132-134.

Cf. A000005, A000010, A005100, A005101, A349758, A349759, A349761.

A164318 is a subsequence.

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]

A349763 Numbers k such that d(k) = A000005(k), sigma(k) = A000203(k) and phi(k) = A000010(k) are all deficient numbers (A005100).

Original entry on oeis.org

1, 2, 3, 4, 8, 16, 48, 64, 121, 128, 192, 256, 512, 529, 1024, 2116, 2209, 2809, 3072, 3481, 4096, 6889, 8192, 8836, 11449, 12288, 13924, 14641, 16384, 17161, 18769, 22201, 27556, 27889, 29282, 29929, 32041, 32768, 36481, 45796, 51529, 54289, 57121, 63001, 65536

Offset: 1

Amiram Eldar, Nov 29 2021

nonn

Sándor (2005) proved that this sequence is infinite by showing that any number of the form 2^(p-1), where p is a sufficiently large prime, is a term. d(2^(p-1)) = p and phi(2^(p-1)) = 2^(p-2) are deficient for all primes, while sigma(2^(p-1)) = 2^p - 1 is deficient for a sufficiently large prime, a result of a theorem by Bojanić (1954): lim_{p prime -> oo} sigma(2^p - 1)/(2^p - 1) = 1.

			2 is a term since d(2) = 2, sigma(2) = 3 and phi(2) = 1 are all deficient numbers.

R. Bojanić, Asymptotic evaluations of the sum of divisors of certain numbers (in Serbo-Croatian), Bull. Soc. Math.-Phys. R. P. Macédoine, Vol. 5 (1954), pp. 5-15.

Amiram Eldar, Table of n, a(n) for n = 1..10000
József Sándor, Selected Chapters of Geometry, Analysis and Number Theory, 2005, pp. 132-134.

Cf. A000005, A000010, A000203, A005100, A061286.

Subsequence of A349759.

defQ[n_] := DivisorSigma[1, n] < 2*n; q[n_] := And @@ defQ /@ Join[DivisorSigma[{0, 1}, n], {EulerPhi[n]}]; Select[Range[10^5], q]

isdef(k) = sigma(k) < 2*k;
isok(k) = my(f=factor(k)); isdef(numdiv(f)) && isdef(sigma(f)) && isdef(eulerphi(k)); \\ Michel Marcus, Dec 01 2021

cp's OEIS Frontend

A349869 Nobly deficient numbers (A349759) that are not deficient numbers (A005100).

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A349758 Nobly abundant numbers: numbers k such that both d(k) = A000005(k) and sigma(k) = A000203(k) are abundant numbers (A005101).

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

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

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A349763 Numbers k such that d(k) = A000005(k), sigma(k) = A000203(k) and phi(k) = A000010(k) are all deficient numbers (A005100).

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