A351541 -id:A351541 - cp's OEIS frontend

A351543 Even numbers k such that there is an odd prime p that divides sigma(k), but valuation(k, p) differs from valuation(sigma(k), p), and p does not divide A003961(k), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

4, 8, 12, 16, 18, 26, 32, 36, 38, 44, 48, 50, 52, 56, 58, 64, 68, 72, 74, 76, 78, 80, 82, 86, 88, 90, 92, 96, 98, 100, 104, 108, 112, 116, 118, 122, 124, 126, 128, 132, 134, 136, 144, 146, 148, 150, 152, 156, 158, 162, 164, 166, 172, 176, 178, 180, 184, 188, 192, 194, 196, 200, 202, 204, 206, 208, 212, 218, 222, 226

Offset: 1

Antti Karttunen, Feb 16 2022

nonn

Even numbers k such that sigma(k) has an odd prime factor prime(i), but prime(i-1) is not a factor of k, and A286561(k, prime(i)) <> A286561(sigma(k), prime(i)). This differs from the definition of A351542 in that prime(i) is not here required to be a factor of k itself. The condition implies also that if there is any such odd prime factor prime(i) of sigma(k), it must be >= 5.
Even numbers k for which A351555(k) > 0.
Question: Is A351538 subsequence of this sequence?

			12 = 2^2 * 3 is present as sigma(12) = 28 = 2^2 * 7, whose prime factorization contains an odd prime 7 such that neither it nor the immediately previous prime, which is 5, divide 12 itself.
196 = 2^2 * 7^2 is present as sigma(196) = 399 = 3^1 * 7^1 * 19^1, which thus has a shared prime factor 7 with 196, but occurring with smaller exponent, and with no prime 5 (which is the previous prime before 7) present in the prime factorization of 196.
364 = 2^2 * 7^1 * 13^1 is present as sigma(364) = 784 = 2^4 * 7^2, which thus has a shared prime factor 7 with 364, but occurring with larger exponent, and with no prime 5 (which is the previous prime before 7) present in the prime factorization of 364.

Cf. A000203, A003961, A286561, A351555.
Subsequences: A351541, A351542, and also conjecturally A351538.
Cf. A351553 (complement among even numbers).
No common terms with A349745.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A351555(n) = { my(s=sigma(n),f=factor(s),u=A003961(n)); sum(k=1,#f~,if((f[k,1]%2) && 0!=(u%f[k,1]), (valuation(n,f[k,1])!=f[k,2]), 0)); };
isA351543(n) = (!(n%2) && A351555(n)>0);

A351540 Numbers k that have an odd prime factor p such that p^(1+valuation(k,p)) divides sigma(k).

Original entry on oeis.org

30, 51, 66, 96, 102, 120, 138, 159, 165, 174, 204, 210, 213, 246, 255, 264, 267, 282, 294, 306, 318, 321, 330, 345, 354, 357, 364, 390, 408, 426, 435, 462, 477, 480, 498, 510, 534, 537, 552, 561, 570, 591, 606, 615, 636, 642, 660, 663, 672, 678, 679, 690, 696, 699, 705, 714, 735, 745, 750, 753, 759, 760, 765, 786

Offset: 1

Antti Karttunen, Feb 16 2022

nonn

			30 = 2 * 3 * 5 is present as sigma(30) = 72 = 2^3 * 3^2, and thus there is at least one odd prime factor (in this case 3) such that a higher power of the same prime divides the sum of divisors of the same number.

Index entries for sequences related to sigma(n)

Positions of nonzero terms in A351539.
Cf. A000203, A351541 (subsequence).
Probably subsequence: A007691 \ (A323653 U A336702).
Cf. also A336353.

Select[Range[2, 800], Function[{k, s}, AnyTrue[DeleteCases[FactorInteger[k][[All, 1]], 2], Mod[s, #^(1 + IntegerExponent[k, #])] == 0 &]] @@ {#, DivisorSigma[1, #]} &] (* Michael De Vlieger, Feb 16 2022 *)

A351539(n) = { my(f=factor(n),s=sigma(n)); sum(k=1,#f~,(f[k,1]%2)&&(0==(s%(f[k,1]^(1+f[k,2]))))); };
isA351540(n) = (A351539(n)>0);

A351542 Even numbers k that have an odd prime factor p such that p also divides sigma(k), but valuation(k,p) differs from valuation(sigma(k), p), and p does not divide A003961(k), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

Original entry on oeis.org

196, 200, 364, 588, 760, 950, 1000, 1092, 1148, 1160, 1274, 1358, 1372, 1400, 1450, 1490, 1568, 1764, 1782, 1900, 1990, 2156, 2200, 2324, 2360, 2600, 2716, 2900, 2912, 2950, 2980, 3042, 3160, 3200, 3276, 3332, 3388, 3400, 3430, 3444, 3490, 3560, 3564, 3724, 3822, 3892, 3950, 3980, 4004, 4018, 4074, 4102, 4116, 4360

Offset: 1

Antti Karttunen, Feb 16 2022

nonn

Even numbers k that have an odd prime factor prime(i) such that it is also a prime factor of sigma(k), but prime(i-1) is not a factor of k, and A286561(k, prime(i)) <> A286561(sigma(k), prime(i)). This differs from the definition of A351543 in that A286561(k, prime(i)) must be > 0.

			196 = 2^2 * 7^2 is present as sigma(196) = 399 = 3^1 * 7^1 * 19^1, which thus has a shared prime factor 7 with 196, but occurring with smaller exponent, and with no prime 5 (which is the previous prime before 7) present in the prime factorization of 196.
364 = 2^2 * 7^1 * 13^1 is present as sigma(364) = 784 = 2^4 * 7^2, which thus has a shared prime factor 7 with 364, but occurring with larger exponent, and with no prime 5 (which is the previous prime before 7) present in the prime factorization of 364.

Cf. A000203, A003961, A286561.
Cf. A351541 (subsequence).
Subsequence of A351543.

Select[Range[2, 4400, 2], Function[{k, s, facs, t}, AnyTrue[DeleteCases[facs[[All, 1]], 2], And[Mod[s, #] == 0, IntegerExponent[s, #] != IntegerExponent[k, #], Mod[t, #] != 0] &]] @@ {#1, #2, #3, Apply[Times, (NextPrime[#1])^#2 & @@@ #3]} & @@ {#, DivisorSigma[1, #], FactorInteger[#]} &] (* Michael De Vlieger, Feb 16 2022 *)

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
Aux351542(n) = { my(f=factor(n),s=sigma(n),u=A003961(n),v); sum(k=1,#f~,if((f[k,1]%2) && 0!=(u%f[k,1]), v=valuation(s,f[k,1]); (v>0) && (v!=f[k,2]), 0)); };
isA351542(n) = (!(n%2) && Aux351542(n)>0);

cp's OEIS Frontend

A351543 Even numbers k such that there is an odd prime p that divides sigma(k), but valuation(k, p) differs from valuation(sigma(k), p), and p does not divide A003961(k), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

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A351540 Numbers k that have an odd prime factor p such that p^(1+valuation(k,p)) divides sigma(k).

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A351542 Even numbers k that have an odd prime factor p such that p also divides sigma(k), but valuation(k,p) differs from valuation(sigma(k), p), and p does not divide A003961(k), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

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