A348938 -id:A348938 - cp's OEIS frontend

A348748 Odd numbers k for which A064989(sigma(k)) < A064989(k), where A064989 shifts the prime factorization one step towards lower primes, and sigma is the sum of divisors function.

3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 27, 29, 31, 33, 35, 37, 39, 41, 43, 47, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 119, 123, 125, 127, 129, 131, 133, 135, 137, 139, 141, 143, 145, 147, 149, 151, 153, 155, 157, 159, 161, 163

Offset: 1

Antti Karttunen, Nov 02 2021

nonn

Sequence obtained when A003961 is applied to A348738 and the terms are sorted into ascending order.

The first squares in this sequence are: 169, 361, 961, 1369, 1849, 2209, 2809, 3721, 4489, 5329, 6241, 6889, ...

Cf. A000203, A003961, A064989, A326042, A348738, A348749, A348938 (terms of A228058 that occur here).

Cf. also A348741, A348753.

f[2, e_] := 1; f[p_, e_] := NextPrime[p, -1]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[1, 200, 2], s[DivisorSigma[1, #]] < s[#] &] (* Amiram Eldar, Nov 04 2021 *)

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
isA348748(n) = ((n%2)&&(A064989(sigma(n)) < A064989(n)));

A348939 Odd numbers k for which A064989(sigma(k)) > A064989(k), and which are 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

45, 117, 325, 333, 405, 549, 605, 657, 925, 1053, 1413, 1445, 1525, 1737, 1825, 2205, 2493, 2817, 2825, 2925, 2997, 3033, 3573, 3645, 3789, 3825, 3925, 4113, 4825, 4869, 4941, 5445, 5517, 5733, 5913, 5949, 6057, 6425, 6525, 6597, 6813, 6925, 7025, 7497, 7605, 7825, 7893, 8125, 8325, 8425, 8973, 9225, 9477, 9837, 9925

Offset: 1

Antti Karttunen, Nov 04 2021

nonn

Obviously, any hypothetical odd perfect number would be neither in this sequence nor in A348938.

Intersection of A228058 and A348749.

Cf. A000203, A064989, A326042, A348938.

q[n_] := Module[{f = FactorInteger[n]}, p = f[[;; , 1]]; e = f[[;; , 2]]; odde = Select[e, OddQ]; Length[e] > 1 && Length[odde] == 1 && Divisible[odde[[1]] - 1, 4] && Divisible[p[[Position[e, odde[[1]]][[1, 1]]]] - 1, 4]]; f[2, e_] := 1; f[p_, e_] := NextPrime[p, -1]^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[1, 10000, 2], q[#] && s[DivisorSigma[1, #]] > s[#] &] (* Amiram Eldar, Nov 04 2021 *)

A064989(n) = { my(f = factor(n)); if((n>1 && f[1, 1]==2), f[1, 2] = 0); for (i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f) };
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));
isA348749(n) = ((n%2)&&(A064989(sigma(n)) > A064989(n)));
isA348939(n) = (isA228058(n)&&isA348749(n));

A351535 Odd numbers k that are not multiples of 3 and for which sigma(k) is congruent to 2 modulo 4.

Original entry on oeis.org

5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 245, 257, 269, 277, 281, 293, 313, 317, 325, 337, 349, 353, 373, 389, 397, 401, 409, 421, 425, 433, 449, 457, 461, 509, 521, 541, 557, 569, 577, 593, 601, 605, 613, 617, 637, 641, 653, 661, 673, 677, 701, 709

Offset: 1

Antti Karttunen, Feb 13 2022

a(25) = 245 = 5* 7^2 is the first term that is not prime.

The sequence shows a steady high percentage of primes. The percentages of the number of prime terms in the first 10^3, ..., 10^8 terms are 86.9, 86.6, 87.1, 87.8, 88.4, 88.8. Additionally, approx 99% of the composite terms indivisible by 5 belong to A348938. - Bill McEachen, Aug 21 2025

Bill McEachen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n)

Intersection of A001651 and A191218. Complement of A351533 in A191218.

Cf. A002144 (subsequence).

Select[Range[1, 700, 2], !Divisible[#, 3] && Mod[DivisorSigma[1, #], 4] == 2 &] (* Amiram Eldar, Feb 13 2022 *)

isA351535(n) = ((n%2) && (0!=(n%3)) && (2 == (sigma(n)%4)));

cp's OEIS Frontend

A348748 Odd numbers k for which A064989(sigma(k)) < A064989(k), where A064989 shifts the prime factorization one step towards lower primes, and sigma is the sum of divisors function.

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A348939 Odd numbers k for which A064989(sigma(k)) > A064989(k), and which are 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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A351535 Odd numbers k that are not multiples of 3 and for which sigma(k) is congruent to 2 modulo 4.

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