cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A351555 a(n) is the number of odd prime factors of sigma(n) that do not divide A003961(n) and the valuation(n, p) is different from valuation(sigma(n), p).

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 2, 1, 1, 1, 0, 2, 0, 1, 1, 2, 0, 1, 1, 2, 1, 1, 1, 1, 0, 1, 1, 0, 1, 2, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 0, 1, 2, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 2, 0, 1, 2, 0
Offset: 1

Views

Author

Antti Karttunen, Feb 16 2022

Keywords

Links

Crossrefs

Cf. A000203, A003961, A351551, A351554 (positions of zeros), A351553 (even terms there), A351543.

Programs

  • PARI
    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)); };

A351538 Numbers k such that both k and sigma(k) are congruent to 2 modulo 4 and the 3-adic valuation of sigma(k) is exactly 1.

Original entry on oeis.org

26, 74, 122, 146, 194, 218, 234, 314, 362, 386, 458, 482, 554, 626, 650, 666, 674, 698, 746, 794, 818, 842, 866, 914, 1082, 1098, 1154, 1202, 1226, 1314, 1322, 1346, 1418, 1466, 1514, 1538, 1658, 1706, 1746, 1754, 1850, 1874, 1962, 1994, 2018, 2042, 2066, 2106, 2138, 2186, 2234, 2258, 2306, 2402, 2426, 2474, 2498
Offset: 1

Views

Author

Antti Karttunen, Feb 14 2022

Keywords

Comments

All the terms of the form 4u+2 in A349745 (if they exist) are found in this sequence. As A351537 is the intersection of A191218 and A329963, and the latter has asymptotic density zero, so has this sequence also. It is conjectured that A351555(a(n)) is nonzero for all n, which would imply that the intersection with A349745 is empty. - Antti Karttunen, Feb 19 2022

Links

Crossrefs

Cf. A191218, A329963, A349745, A351537.
Probably a subsequence of A351543. (See also A351550, A351555).

Programs

  • PARI
    isA351538(n) = if(!(2==(n%4)),0, my(s=sigma(n)); (2 == (s%4)) && (1==valuation(s,3)));

Formula

a(n) = 2 * A351537(n).

A351553 Even numbers k such that there are no odd prime factors p of sigma(k) such that p does not divide A003961(k) and the valuation(k, p) is different from valuation(sigma(k), p), where A003961 is fully multiplicative with a(p) = nextprime(p), and sigma is the sum of divisors function.

Original entry on oeis.org

2, 6, 10, 14, 20, 22, 24, 28, 30, 34, 40, 42, 46, 54, 60, 62, 66, 70, 84, 94, 102, 106, 110, 114, 120, 130, 138, 140, 142, 154, 160, 168, 170, 174, 182, 186, 190, 198, 210, 214, 216, 220, 224, 230, 238, 254, 260, 264, 270, 280, 282, 290, 308, 310, 318, 322, 330, 340, 354, 374, 378, 380, 382, 390, 408, 410, 420, 426
Offset: 1

Views

Author

Antti Karttunen, Feb 16 2022

Keywords

Comments

Even numbers k for which A351555(k) = 0.

Links

Crossrefs

Cf. A000203, A005820, A003961, A046060, A351552.
Even terms in A351554, positions of zeros at even indices in A351555.
Cf. A351543 (complement among even numbers), A386430.

Programs

  • PARI
    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)); };
isA351553(n) = (!(n%2) && 0==A351555(n));

Extensions

Definition corrected by Antti Karttunen, Aug 27 2025

A351541 Even numbers k that have an odd prime factor p such that p^(1+valuation(k,p)) divides sigma(k), but 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

364, 760, 1092, 1148, 1160, 1358, 1490, 1782, 1990, 2324, 2360, 2716, 2912, 2980, 3160, 3276, 3388, 3430, 3444, 3490, 3560, 3564, 3892, 3980, 4004, 4074, 4102, 4360, 4490, 4676, 4990, 5068, 5302, 5320, 5432, 5510, 5560, 5960, 5990, 6188, 6244, 6804, 6860, 6916, 6972, 6980, 7028, 7128, 7160, 7462, 7960, 8120, 8148
Offset: 1

Views

Author

Antti Karttunen, Feb 16 2022

Keywords

Comments

Even numbers k that have an odd prime factor prime(i) such that prime(i-1) is not a factor of k, and prime(i)^(1+A286561(k,prime(i))) divides sigma(k).

Examples

			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.

Links

Crossrefs

Cf. A000203, A003961.
Subsequence of A351540, and of A351542 and of A351543.

Programs

  • Mathematica
    Select[Range[2, 8200, 2], Function[{k, s, facs, t}, AnyTrue[DeleteCases[facs[[All, 1]], 2], And[Mod[s, #^(1 + IntegerExponent[k, #])] == 0, Mod[t, #] != 0] &]] @@ {#1, #2, #3, Apply[Times, (NextPrime[#1])^#2 & @@@ #3]} & @@ {#, DivisorSigma[1, #], FactorInteger[#]} &] (* Michael De Vlieger, Feb 16 2022 *)
  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
Aux351541(n) = { my(f=factor(n),s=sigma(n),u=A003961(n)); sum(k=1,#f~,(f[k,1]%2) && 0!=(u%f[k,1]) && (0==(s%(f[k,1]^(1+f[k,2]))))); };
isA351541(n) = (!(n%2) && Aux351541(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

Views

Author

Antti Karttunen, Feb 16 2022

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    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);
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