A336315 -id:A336315 - cp's OEIS frontend

A336317 Numbers k such that A122111(k) [conjugated prime factorization of k] is one of Ore's Harmonic numbers (in A001599).

1, 6, 40, 126, 154, 204, 1716, 1914, 2772, 8580, 11264, 12090, 12540, 50960, 62790, 64350, 77748, 83200, 104720, 152320, 186116, 193440, 331890, 382720, 432768, 518364, 648788, 684684, 753480, 817344, 895356, 1083852, 1113840, 1619352, 1675044, 1743588, 1759680, 1991340, 2060322, 2360484, 2492028, 2621080, 2932800

Offset: 1

Antti Karttunen, Jul 19 2020

nonn

Numbers k for which A336314(k) = A323173(k).
Sequence A122111(A001599(n)), n >= 1, sorted into ascending order. Positions of zeros in A323174 (corresponding to perfect numbers similarly mapped) is a subsequence.
Note that all terms after 1 seem to be present in A102750. This observation is equal to Ore's conjecture that there are no odd Harmonic numbers larger than one.
Also, all terms after 1 seem to be even, which would imply that apart from its initial 1, A001599 were a subsequence of A102750. However, this is false, as there are terms of A001599 not in A102750, for example 8011798098793361832960 found by David A. Corneth. Note that A122111(8011798098793361832960) = 96922193555635754403846044921625, which is thus an odd term of this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..61
Index entries for sequences computed from indices in prime factorization

Cf. A001599, A088902, A102750, A122111, A323173, A323174, A336314, A336315, A336353, A336397 (the intersection with A001599).

isA001599(n) = !((sigma(n,0)*n)%sigma(n,1));
isA336317(n) = isA001599(A122111(n)); \\ Program for A122111 given under that entry.

\\ Standalone program:
isA336317(n) = if(1==n,1,my(f=factor(n),es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,d=1,s=1,x=1,p,e); for(i=1, #es, pri += es[i]; p = prime(pri); e = 1+is[i]-is[1+i]; d *= e; s *= ((p^e)-1)/(p-1); x *= (p^(e-1))); !((x*d)%s));

A336314 a(n) = A324121(A122111(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 12, 1, 2, 1, 4, 3, 2, 1, 12, 3, 2, 1, 12, 1, 6, 1, 4, 1, 8, 4, 36, 1, 10, 1, 24, 3, 2, 3, 4, 24, 4, 1, 12, 1, 56, 1, 24, 1, 2, 3, 4, 1, 4, 1, 6, 9, 6, 1, 4, 8, 8, 1, 12, 9, 48, 1, 4, 1, 2, 24, 120, 5, 2, 3, 18, 7, 12, 1, 36, 2, 10, 3, 24, 1, 12, 3, 4, 3, 112

Offset: 1

Antti Karttunen, Jul 19 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..12000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A122111, A324121, A336315.
Cf. A336317 (gives the positions where this coincides with A323173).
Cf. also A335914.

A122111(n) = if(1==n,n,my(f=factor(n), es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,m=1); for(i=1, #es, pri += es[i]; m *= prime(pri)^(is[i]-is[1+i])); (m));
A324121(n) = gcd(sigma(n),n*numdiv(n));
A336314(n) = A324121(A122111(n));

\\ Or as a standalone program:
A336314(n) = if(1==n,1,my(f=factor(n),es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,d=1,s=1,x=1,p,e); for(i=1, #es, pri += es[i]; p = prime(pri); e = 1+is[i]-is[1+i]; d *= e; s *= ((p^e)-1)/(p-1); x *= (p^(e-1))); gcd(s,x*d));

a(n) = A324121(A122111(n)) = gcd(A323173(n), A122111(n)*A336315(n)).

A336316 The number of non-unitary divisors in the conjugated prime factorization of n: a(n) = A048105(A122111(n)).

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 3, 0, 1, 2, 4, 0, 5, 4, 2, 0, 6, 0, 7, 2, 5, 6, 8, 0, 2, 8, 1, 4, 9, 0, 10, 0, 8, 10, 4, 0, 11, 12, 11, 2, 12, 4, 13, 6, 2, 14, 14, 0, 3, 2, 14, 8, 15, 0, 8, 4, 17, 16, 16, 0, 17, 18, 5, 0, 12, 8, 18, 10, 20, 4, 19, 0, 20, 20, 2, 12, 6, 12, 21, 2, 1, 22, 22, 4, 16, 24, 23, 6, 23, 0, 11, 14, 26, 26, 20, 0, 24

Offset: 1

Antti Karttunen, Jul 18 2020

nonn

Equally, the number of divisors in the conjugated prime factorization of n minus the number of its unitary divisors.

Note that A001221(A122111(n)) = A001221(n) for all n.

Antti Karttunen, Table of n, a(n) for n = 1..12000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A000005, A001221, A034444, A048105, A122111, A286621, A336315.

Cf. A055932 (the positions of zeros).

A336315(n) = if(1==n,n,my(p=apply(primepi,factor(n)[,1]~),m=1+p[1]); for(i=2, #p, m *= (1+p[i]-p[i-1])); (m));
A336316(n) = (A336315(n)-(2^omega(n)));

a(n) = A336315(n) - A034444(n) = A000005(A122111(n)) - 2^A001221(n).

a(n) = A048105(A122111(n)).

cp's OEIS Frontend

A336317 Numbers k such that A122111(k) [conjugated prime factorization of k] is one of Ore's Harmonic numbers (in A001599).

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A336314 a(n) = A324121(A122111(n)).

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A336316 The number of non-unitary divisors in the conjugated prime factorization of n: a(n) = A048105(A122111(n)).

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