A354362 -id:A354362 - cp's OEIS frontend

A228058 Odd 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. (Euler's criteria for odd perfect numbers).

45, 117, 153, 245, 261, 325, 333, 369, 405, 425, 477, 549, 605, 637, 657, 725, 801, 833, 845, 873, 909, 925, 981, 1017, 1025, 1053, 1233, 1325, 1341, 1377, 1413, 1421, 1445, 1525, 1557, 1573, 1629, 1737, 1773, 1805, 1813, 1825, 2009, 2057, 2061, 2097, 2169

Offset: 1

T. D. Noe, Aug 13 2013

nonn

It has been proved that if an odd perfect number exists, it belongs to this sequence. The first term of the form p^5 * n^2 is 28125 = 5^5 * 3^2, occurring in position 520.
Sequence A228059 lists the subsequence of these numbers that are closer to being perfect than smaller numbers. - T. D. Noe, Aug 15 2013
Sequence A326137 lists terms with at least five distinct prime factors. See further comments there. - Antti Karttunen, Jun 13 2019

T. D. Noe, Table of n, a(n) for n = 1..10000
Charles Greathouse and Eric W. Weisstein, MathWorld: Odd perfect number
Oliver Knill, The oldest open problem in mathematics, Handout for NEU Math Circle, December 2, 2007
P. P. Nielsen, Odd Perfect Numbers Have At Least Nine Distinct Prime Factors, arXiv:math/0602485 [math.NT], 2006.
Wikipedia, Perfect number: Odd perfect numbers
Index entries for sequences where any odd perfect numbers must occur

Subsequence of A191218, and also of A228056 and A228057 (simpler versions of this sequence).
For various subsequences with additional conditions, see A228059, A325376, A325380, A325822, A326137 (with omega(n)>=5), A324898 (conjectured, subsequence if it does not contain any prime powers), A354362, A386425 (conjectured), A386427 (nondeficient terms), A386428 (powerful terms), A386429 U A351574.
Cf. A027748, A124010, A005408, A324647, A325319, A325320, A325375, A325377, A325378, A325379, A325819, A325823, A325824.

import Data.List (partition)
a228058 n = a228058_list !! (n-1)
a228058_list = filter f [1, 3 ..] where
   f x = length us == 1 && not (null vs) &&
         fst (head us) `mod` 4 == 1 && snd (head us) `mod` 4 == 1
         where (us,vs) = partition (odd . snd) $
                         zip (a027748_row x) (a124010_row x)
-- Reinhard Zumkeller, Aug 14 2013

nn = 100; n = 1; t = {}; While[Length[t] < nn, n = n + 2; {p, e} = Transpose[FactorInteger[n]]; od = Select[e, OddQ]; If[Length[e] > 1 && Length[od] == 1 && Mod[od[[1]], 4] == 1 && Mod[p[[Position[e, od[[1]]][[1,1]]]], 4] == 1, AppendTo[t, n]]]; t (* T. D. Noe, Aug 15 2013 *)

up_to = 1000;
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));
A228058list(up_to) = { my(v=vector(up_to), k=0, n=0); while(kA228058(n), k++; v[k] = n)); (v); };
v228058 = A228058list(up_to);
A228058(n) = v228058[n]; \\ Antti Karttunen, Apr 22 2019

From Antti Karttunen, Apr 22 2019 & Jun 03 2019: (Start)
A325313(a(n)) = -A325319(n).
A325314(a(n)) = -A325320(n).
A001065(a(n)) = A325377(n).
A033879(a(n)) = A325379(n).
A034460(a(n)) = A325823(n).
A325814(a(n)) = A325824(n).
A324213(a(n)) = A325819(n).
(End)

Note in parentheses added to the definition by Antti Karttunen, Jun 03 2019

A354106 Numbers k for which A354102(k) = A354102(sigma(k)).

Original entry on oeis.org

1, 24, 2475, 2520, 2728, 5347, 6683, 8184, 8307, 8568, 9108, 9306, 10106, 11484, 12974, 16041, 17892, 20049, 23265, 25265, 26199, 30318, 32256, 32435, 38922, 39618, 40918, 44010, 44576, 44872, 50976, 55224, 55720, 56516, 58817, 63720, 63952, 64890, 65689, 66528, 67356, 69860, 72072, 73409, 74448, 75795, 79101, 83160

Offset: 1

Antti Karttunen, May 18 2022

nonn

No common terms with A006872 in range a(2) .. a(1001).

Among the 1001 initial terms, only 3119744 and 13890816 occur also in A191217.

Cf. A000010, A000203, A354102.

Cf. also A006872, A191217, A354189, A354362.

\\ Needs also the program from A267099:
A354102(n) = eulerphi(A267099(n));
isA354106(k) = (A354102(k)==A354102(sigma(k)));

A260021 Solutions to phi(n) = phi(sigma(n)) that are not given by Theorem 3 of Golomb's manuscript.

Original entry on oeis.org

1, 3, 15, 45, 175, 357, 585, 608, 646, 962, 1071, 1292, 1443, 1508, 1586, 1664, 1665, 1898, 2275, 2295, 2379, 2745, 2847, 3285, 3848, 4082, 4329, 4514, 4641, 4736, 4845, 5018, 5402, 6123, 6232, 6344, 6475, 6771, 7052, 7065, 7137, 7202, 7215, 7527, 7592, 7803, 7808, 8103, 8138, 8398, 8541, 8685, 8906, 9344, 9526, 10322

Offset: 1

N. J. A. Sloane, Jul 14 2015

nonn

S. W. Golomb, Equality among number-theoretic functions, Manuscript, no date; Second update, Dec 29, 1992.

Antti Karttunen, Table of n, a(n) for n = 1..20000
S. W. Golomb, Equality among number-theoretic functions, Unpublished manuscript. (Annotated scanned copy)
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..103800
Index entries for sequences related to sigma(n)

Setwise difference A006872 \ A354345. Subset of positions of zeros in A353636.
Cf. A354362 (subsequence).
Cf. also A005383, A353637, A354344.

A354344(n) = { if(!(n%15),n/=15,if(!(n%9),n/=9,if(!(n%8),n/=8,if(!(n%3),n/=3,if(!(n%2),n/=2,return(0)))))); ((n>5) && isprime(n) && isprime((1+n)/2)); };
A353637(n) = (eulerphi(sigma(n))==eulerphi(n));
isA260021(n) = (A353637(n) && !A354344(n)); \\ Antti Karttunen, May 24 2022

{k | 1==A353637(k) and 0==A354344(k)}. - Antti Karttunen, May 25 2022

Term a(1) = 1 prepended and terms a(14) .. a(56) added by Antti Karttunen, May 24 2022

A386419 Odd numbers k that are closer to being perfect than previous terms, and also satisfy the condition that phi(k) = phi(sigma(k)).

Original entry on oeis.org

1, 3, 15, 45, 585, 2295, 11475, 29835, 72675, 424575, 7977165, 28851975, 29277885, 39317175

Offset: 1

Antti Karttunen, Jul 21 2025

Questions: Is 45 the only term also in A228058? (See also A354362). Are there only multiples of 5 after the two initial terms?

If it exists, a(15) > 2^30 (1073741824).

Index entries for sequences where (the least) odd perfect number must occur

Subsequence of A353679.
Cf. A000010, A000203, A000396, A228058, A354362.
Cf. also A171929, A228059, A386420, A386421, A386422.

A353680(n) = ((n%2) && (eulerphi(sigma(n))==eulerphi(n)));
isA353679(n) = A353680(n);
m=-1; n=0; k=0; while(m!=0, n++; if(!(n%(2^25)),print1("("n")")); if(isA353679(n), if((m<0) || abs((sigma(n)/n)-2)

cp's OEIS Frontend

A228058 Odd 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. (Euler's criteria for odd perfect numbers).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions

A354106 Numbers k for which A354102(k) = A354102(sigma(k)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A260021 Solutions to phi(n) = phi(sigma(n)) that are not given by Theorem 3 of Golomb's manuscript.

Views

Author

Keywords

References

Links

Crossrefs

Programs

PARI

Formula

Extensions

A386419 Odd numbers k that are closer to being perfect than previous terms, and also satisfy the condition that phi(k) = phi(sigma(k)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI