A372299 -id:A372299 - cp's OEIS frontend

A372298 Primitive infinitary abundant numbers (definition 1): infinitary abundant numbers (A129656) whose all proper infinitary divisors are infinitary deficient numbers.

40, 56, 70, 72, 88, 104, 756, 924, 945, 1092, 1188, 1344, 1386, 1428, 1430, 1596, 1638, 1760, 1870, 2002, 2016, 2080, 2090, 2142, 2176, 2210, 2394, 2432, 2470, 2530, 2584, 2720, 2750, 2944, 2990, 3040, 3128, 3190, 3200, 3230, 3250, 3400, 3410, 3496, 3712, 3770

Offset: 1

Amiram Eldar, Apr 25 2024

			40 is a term since it is an infinitary abundant number and all its proper infinitary divisors, {1, 2, 4, 5, 8, 10, 20}, are infinitary deficient numbers.
24 and 30, which are infinitary abundant numbers, are not primitive, because they are divisible by 6 which is an infinitary perfect number.

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

Subsequence of A129656 and A372299.
A372300 is a subsequence.
Cf. A007357, A077609, A129657.
Similar sequences: A071395, A298973, A302573, A307112, A307114, A307115.

f[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]];
isigma[1] = 1; isigma[n_] := Times @@ f @@@ FactorInteger[n]; idefQ[n_] := isigma[n] < 2*n; idivs[1] = {1};
idivs[n_] := Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, e_Integer} :> p^Select[Range[0, e], BitOr[e, #] == e &])];
q[n_] := Module[{d = idivs[n]}, Total[d] > 2*n && AllTrue[Most[d], idefQ]]; Select[Range[4000], q]

isidiv(d, f) = {if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k, 2]); bde = binary(valuation(d, f[k, 1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)); ); ); return (1); }
idivs(n) = {my(f = factor(n), d = divisors(f), idiv = []); for (k=1, #d, if (isidiv(d[k], f), idiv = concat(idiv, d[k])); ); idiv; } \\ Michel Marcus at A077609
isigma(n) = {my(f = factor(n), b); prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], 1+f[i, 1]^(2^(#b-k)), 1)))} ;
is(n) = isigma(n) > 2*n && select(x -> x < n && isigma(x) >= 2*x, idivs(n)) == [];

A372300 Numbers k such that k and k+1 are both primitive infinitary abundant numbers (definition 1, A372298).

Original entry on oeis.org

812889, 3181815, 20787584, 181480695, 183872535, 307510664, 337206344, 350158808, 523403264, 744074624, 868421504, 1063361144, 1955365125, 2076191864, 2578966215, 3672231255, 4185590408, 5032685384, 7158001304, 8348108535, 10784978295, 16264812135, 20917209495, 24514454055

Offset: 1

Amiram Eldar, Apr 25 2024

nonn

The corresponding sequence with definition 2 (A372299) coincides with this sequence for the first 24 terms.

Subsequence of A129656, A327635 and A372298.
Cf. A372299.
Similar sequences: A283418, A330872, A361935.

isidiv(d, f) = {my(bne,bde); if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k, 2]); bde = binary(valuation(d, f[k, 1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)); ); ); return (1); }
idivs(n) = {my(f = factor(n), d = divisors(f), idiv = []); for (k=1, #d, if (isidiv(d[k], f), idiv = concat(idiv, d[k])); ); idiv; } \\ Michel Marcus at A077609
isigma(n) = {my(f = factor(n), b); prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], 1+f[i, 1]^(2^(#b-k)), 1)))} ;
isab(n) = isigma(n) > 2*n;
isprim(n) = select(x -> x= 2*x, idivs(n)) == [];
lista(kmax) = {my(is1 = 0, is2); for(k = 2, kmax, is2 = isab(k); if(is1 && is2, if(isprim(k-1) && isprim(k), print1(k-1, ", "))); is1 = is2);}

cp's OEIS Frontend

A372298 Primitive infinitary abundant numbers (definition 1): infinitary abundant numbers (A129656) whose all proper infinitary divisors are infinitary deficient numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI