A333950 -id:A333950 - cp's OEIS frontend

A333951 Numbers k such that both k and k+1 are recursive abundant numbers (A333928).

56924, 82004, 84524, 109395, 158235, 241604, 261260, 266475, 285075, 361844, 442035, 445004, 469755, 611324, 666315, 694484, 712844, 922635, 968715, 971684, 1102724, 1172115, 1190475, 1199835, 1239524, 1304324, 1338435, 1430715, 1442924, 1486275, 1523115, 1550835

Offset: 1

Amiram Eldar, Apr 11 2020

nonn

			56924 is a term since A333926(56924) = 120960 > 2 * 56924, and A333926(56925) = 116064 > 2 * 56925.

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

Subsequence of A333928.
Cf. A333926, A333950.
Analogous sequences: A096399, A283418 (primitive), A318167 (bi-unitary), A327635 (infinitary), A327942 (nonunitary), A331412 (unitary).

recDivQ[n_, 1] = True; recDivQ[n_, d_] := recDivQ[n, d] = Divisible[n, d] && AllTrue[FactorInteger[d], recDivQ[IntegerExponent[n, First[#]], Last[#]] &]; recDivs[n_] := Select[Divisors[n], recDivQ[n, #] &]; f[p_, e_] := 1 + Total[p^recDivs[e]]; recDivSum[1] = 1; recDivSum[n_] := Times @@ (f @@@ FactorInteger[n]); recAbQ[n_] := recDivSum[n] > 2*n; Select[Range[2*10^5], recAbQ[#] && recAbQ[# + 1] &]

A339936 Odd coreful abundant numbers: the odd terms of A308053.

Original entry on oeis.org

99225, 165375, 231525, 297675, 496125, 694575, 826875, 893025, 1091475, 1157625, 1225125, 1289925, 1488375, 1620675, 1686825, 1819125, 1885275, 2083725, 2149875, 2282175, 2480625, 2546775, 2679075, 2811375, 2877525, 3009825, 3075975, 3142125, 3274425, 3472875

Offset: 1

Amiram Eldar, Dec 23 2020

nonn

The asymptotic density of this sequence is Sum_{n>=1} f(A356871(n)) = 9.1348...*10^(-6), where f(n) = (4/(Pi^2*n))*Product_{prime p|n}(p/(p+1)) if n is odd and 0 otherwise. - Amiram Eldar, Sep 02 2022

			99225 is a term since it is odd and the sum of its coreful divisors is A057723(99225) = 201600 > 2 * 99225.

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

Intersection of A005408 and A308053.
Subsequence of A321147.
Cf. A057723, A356871.
Similar sequences: A005231, A094889, A112643, A127666, A129485, A293186, A321147, A333950.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 1; s[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[1, 10^6, 2], s[#] > 2*# &]

A339938 Odd non-coreful abundant numbers: the odd terms of A308127.

Original entry on oeis.org

15015, 19635, 21945, 23205, 25935, 26565, 31395, 33495, 33915, 35805, 39585, 41055, 42315, 42735, 45885, 47355, 49665, 50505, 51765, 54285, 55965, 58695, 61215, 64155, 68145, 70455, 72345, 75075, 77385, 80535, 82005, 83265, 84315, 91245, 95865, 102795, 105105

Offset: 1

Amiram Eldar, Dec 23 2020

nonn

First differs from A112643, A129485 and A249263 at n = 28.

			15015 is a term since it is odd and the sum of its non-coreful divisors is A308135(15015) = 17241 > 15015.

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

Intersection of A005408 and A308127.
Cf. A308135.
Similar sequences: A005231, A094889, A112643, A127666, A129485, A249263, A293186, A321147, A333950.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); fc[p_, e_] := f[p, e] - 1; s[1] = 0; s[n_] := Times @@ (f @@@ FactorInteger[n]) - Times @@ (fc @@@ FactorInteger[n]); Select[Range[1, 10^5, 2], s[#] > # &]

A357697 Odd cubefree abundant numbers.

Original entry on oeis.org

1575, 2205, 3465, 4095, 5355, 5775, 5985, 6435, 6825, 7245, 8085, 8415, 8925, 9135, 9555, 9765, 11025, 11655, 12705, 12915, 13545, 14805, 15015, 16695, 17325, 18585, 19215, 19635, 20475, 21105, 21945, 22365, 22995, 23205, 24255, 24885, 25935, 26145, 26565, 26775

Offset: 1

Amiram Eldar, Oct 10 2022

nonn

First differs from A333950 at n = 1258. Terms that are not in A333950 include 8564325, 8565795, 8567325, ... and terms of A333950 that are not here include 1126125, 2096325, 2207205, ... .

The numbers of terms not exceeding 10^k, for k = 4, 5, ..., are 16, 125, 1127, 11734, 116911, 1162781, 11638566, 116342286, ... . Apparently, the asymptotic density of this sequence exists and equals 0.00116... .

			1575 = 3^2 * 5^2 * 7 is a term since it is odd and cubefree and sigma(1575) = 3224 > 2*1575.

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

Intersection of A004709 and A005231.
Intersection of A005408 and A357695.
A112643 is a subsequence.
Cf. A000203 (sigma), A333950.

f[p_, e_] := (p^(e+1)-1)/(p-1); q[1] = 0; q[n_] := AllTrue[(fct = FactorInteger[n])[[;;, 2]], # < 3 &] && Times @@ f @@@ fct > 2*n; Select[Range[1, 30000, 2], q]

is(n) = {my(f); if(n%2 == 0, return(0)); f = factor(n); (n==1 || vecmax(f[,2]) < 3) && sigma(f, -1) > 2};

cp's OEIS Frontend

A333951 Numbers k such that both k and k+1 are recursive abundant numbers (A333928).

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A339936 Odd coreful abundant numbers: the odd terms of A308053.

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A339938 Odd non-coreful abundant numbers: the odd terms of A308127.

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A357697 Odd cubefree abundant numbers.

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