The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A351979 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A351979

Numbers whose prime factorization has all odd prime indices and all even prime exponents.
(history; published version)

#12 by Joerg Arndt at Mon Sep 19 07:23:31 EDT 2022

STATUS

proposed

approved

#11 by Amiram Eldar at Mon Sep 19 04:49:51 EDT 2022

STATUS

editing

proposed

#10 by Amiram Eldar at Mon Sep 19 04:26:54 EDT 2022

LINKS

Amiram Eldar, <a href="/A351979/b351979.txt">Table of n, a(n) for n = 1..10000</a>

FORMULA

Sum_{n>=1} 1/a(n) = 1/Product_{k>=1} (1 - 1/prime(2*k-1)^2) = 1.4135142... . - Amiram Eldar, Sep 19 2022

STATUS

approved

editing

#9 by N. J. A. Sloane at Sun Mar 13 19:01:08 EDT 2022

STATUS

proposed

approved

#8 by Michael S. Branicky at Sat Mar 12 09:35:05 EST 2022

STATUS

editing

proposed

#7 by Michael S. Branicky at Sat Mar 12 09:35:01 EST 2022

PROG

(Python)

from sympy import factorint, primepi

def ok(n):

return all(primepi(p)%2==1 and e%2==0 for p, e in factorint(n).items())

print([k for k in range(15500) if ok(k)]) # Michael S. Branicky, Mar 12 2022

STATUS

proposed

editing

#6 by Gus Wiseman at Sat Mar 12 05:09:43 EST 2022

STATUS

editing

proposed

#5 by Gus Wiseman at Sat Mar 12 04:04:27 EST 2022

FORMULA

A257991(a(n)) = A001222(a(n)).

A162641(a(n)) = A001221(a(n)).

A162642(a(n)) = A257992(a(n)) = 0.

#4 by Gus Wiseman at Sat Mar 12 03:57:53 EST 2022

CROSSREFS

Cf. A000720, A028260, A045931, A055396, A061395, A106529, A181819, A195017, A241638, A276078, A324588, A325698, A325700.

#3 by Gus Wiseman at Sat Mar 12 03:54:42 EST 2022

COMMENTS

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, reverse A296150, sum A056239, length A001222.

A number's prime signature is the sequence of positive exponents in its prime factorization, which is row n of A124010, sorted A118914, length A001221, sum A001222.

Also Heinz numbers of integer partitions with all odd parts and all even multiplicities, counted by A055922 aeratedA035457 (see Emeric Deutsch's comment there).

EXAMPLE

100: prime(1)^2 * prime(3)^2

400: prime(1)^4 * prime(3)^2

484: prime(1)^2 * prime(5)^2

1156: prime(1)^2 * prime(7)^2

1600: prime(1)^6 * prime(3)^2

1936: prime(1)^4 * prime(5)^2

CROSSREFS

A162641 counts even prime exponents, odd A162642.

A257991 counts odd prime indices, even A257992.

Cf. A000720, A028260, `A045931, A055396, A061395, `A106529, `~A130780, `~A171966, A181819, A195017, ~`A239241, A241638, `A276078, `A324517, `~A324524, ~`A324525, `~A324571, ~A324572, `A324588, `A325127, `A325128, `~A325130, A325698, A325700.