A386426 -id:A386426 - cp's OEIS frontend

A386424 Numbers k such that sigma(k) has the same powerful part as k, where sigma is the sum of divisors function.

1, 2, 5, 12, 13, 26, 29, 37, 41, 44, 56, 61, 73, 74, 76, 90, 101, 109, 113, 122, 137, 146, 153, 157, 172, 173, 181, 193, 218, 229, 236, 257, 268, 277, 281, 312, 313, 314, 317, 353, 362, 373, 386, 389, 397, 401, 409, 421, 433, 457, 458, 461, 509, 522, 524, 528, 541, 554, 560, 569, 601, 613, 617, 626, 641, 652, 653

Offset: 1

Antti Karttunen, Aug 17 2025

Conjecture 1: the initial 1 is the only square in this sequence, and a(2) = 2 is the only term that is twice a square.

Conjecture 2: A323653 is a subsequence (which would follow from conjecture 1 (c) given there).

Cf. A000203, A003557, A057521, A387156.
Subsequences: A323653 (conjectured), A351549, A386425 (odd composites), A386426 (nondeficient terms).
Cf. also A006872, A351446, A387158.

rad[n_] := Times @@ First /@ FactorInteger[n];a057521[n_] := n/Denominator[n/rad[n]^2];Select[Range[653],a057521[DivisorSigma[1,#]]==a057521[#]&] (* James C. McMahon, Aug 18 2025 *)

A057521(n)=my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1))
isA386424(n) = (A057521(sigma(n))==A057521(n));

{k | A057521(A000203(k)) = A057521(k)}, or equally, {k | A387156(k) = A003557(k)}.

A386425 Odd composites k such that sigma(k) has the same powerful part as k, where sigma is the sum of divisors function.

Original entry on oeis.org

153, 801, 1773, 3725, 4689, 4753, 5013, 6957, 8577, 8725, 9549, 9873, 11493, 13437, 14409, 15381, 18621, 19269, 21213, 21537, 23481, 25101, 26073, 26225, 28989, 29161, 29313, 29961, 32229, 33849, 34173, 36117, 38061, 39033, 40653, 42597, 43893, 47457, 47781, 48725, 48753, 51669, 52317, 54261, 56953, 57177, 57501

Offset: 1

Antti Karttunen, Aug 17 2025

nonn

By definition, the sequence contains all odd perfect numbers, and also includes any hypothetical odd triperfect number that is not a multiple of 3 (see A005820 and A347391), and similarly, any odd term of A046060 that is not a multiple of 5, etc. If there are no squares in this sequence (see conjecture in A386424), then the latter categories of numbers certainly do not exist, and this is then a subsequence of A228058.

The first nondeficient term is a(32315) = 81022725. See A386426.

Intersection of A071904 and A386424.
Nonsquare terms form a subsequence of A228058.
Cf. A000203, A003557, A057521, A386426 (nondeficient terms).
Cf. also A324647, A349749.

rad[n_] := Times @@ First /@ FactorInteger[n];a057521[n_] := n/Denominator[n/rad[n]^2];Select[Range[9,57501,2],!PrimeQ[#]&&a057521[DivisorSigma[1,#]]==a057521[#]&] (* James C. McMahon, Aug 18 2025 *)

A057521(n)=my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1))
isA386425(n) = ((n>1) && (n%2) && !isprime(n) && (A057521(sigma(n))==A057521(n)));

{k | k is odd composite and A003557(A000203(k)) = A003557(k)}.

A386427 Odd nondeficient numbers of the form p^(1+4k) * r^2, where p is prime of the form 1+4m, r > 1, and gcd(p,r) = 1.

Original entry on oeis.org

2205, 19845, 108045, 143325, 178605, 187425, 236925, 266805, 319725, 353925, 372645, 407925, 452025, 462825, 584325, 637245, 646425, 658125, 672525, 789525, 796005, 804825, 845325, 920205, 972405, 981225, 1007325, 1055925, 1069425, 1102725, 1113525, 1116225, 1166445, 1201725, 1245825, 1289925, 1378125, 1380825

Offset: 1

Antti Karttunen, Aug 18 2025

Nondeficient numbers (A023196) that satisfy Euler's condition for odd perfect numbers (A228058).

This is not equal to A348743, as that sequence contains also terms like 1279741205456530915782536871495922949062895982530933679752838870798129159675 and 15388519572341080054329140040512468358441210638435506649120749687401476705908239675, that are lacking from this sequence.

Intersection of A023196 and A228058.
Also the intersection of A083207 and A228058, and probably also of A005835 and A228058. - Antti Karttunen, Aug 21 2025
Subsequence of A348743, from which this eventually differs at some very large n.
Cf. A386426 (conjectured subsequence).

isA228058(n) = if(!(n%2)||(omega(n)<2), 0, my(f=factor(n), y=0); for(i=1, #f~, if(1==(f[i, 2]%4), if((1==y)||(1!=(f[i, 1]%4)), return(0), y=1), if(f[i, 2]%2, return(0)))); (y));
isA386427(n) = ((sigma(n)>=(2*n)) && isA228058(n));

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A386424 Numbers k such that sigma(k) has the same powerful part as k, where sigma is the sum of divisors function.

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A386425 Odd composites k such that sigma(k) has the same powerful part as k, where sigma is the sum of divisors function.

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A386427 Odd nondeficient numbers of the form p^(1+4k) * r^2, where p is prime of the form 1+4m, r > 1, and gcd(p,r) = 1.

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